[{"data":1,"prerenderedAt":39404},["ShallowReactive",2],{"learn-topics":3},[4,29,46,62,79,96,113,131,147,163,179,195,211,227,244,260,276,292,309,370,386,402,419,435,451,467,483,499,516,532,548,564,580,597,613,629,645,661,677,11253,13230,22292,32045],{"id":5,"title":6,"body":7,"category":18,"description":19,"extension":20,"meta":21,"navigation":22,"order":23,"path":24,"seo":25,"sitemap":26,"stem":27,"__hash__":28},"learn\u002Flearn\u002Fcad-bim\u002Fautocad-fundamentals.md","AutoCAD Fundamentals for Surveyors",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"This content is being developed.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],"cad-bim","Essential AutoCAD skills for surveying professionals.","md",{},true,1,"\u002Flearn\u002Fcad-bim\u002Fautocad-fundamentals",{"title":6,"description":19},{"loc":24},"learn\u002Fcad-bim\u002Fautocad-fundamentals","o_OIBZFWXkQLrsZkmXtkrFL6QIT-c0XvQf4XNMuNXhM",{"id":30,"title":31,"body":32,"category":18,"description":38,"extension":20,"meta":39,"navigation":22,"order":40,"path":41,"seo":42,"sitemap":43,"stem":44,"__hash__":45},"learn\u002Flearn\u002Fcad-bim\u002Fbim-fundamentals.md","BIM Fundamentals for Survey Professionals",{"type":8,"value":33,"toc":36},[34],[11,35,13],{},{"title":15,"searchDepth":16,"depth":16,"links":37},[],"Introduction to Building Information Modeling for surveyors.",{},5,"\u002Flearn\u002Fcad-bim\u002Fbim-fundamentals",{"title":31,"description":38},{"loc":41},"learn\u002Fcad-bim\u002Fbim-fundamentals","RBzoTvsSysPcKcxBjqXK9FIlcwgQH57NdtQm9g-K9qg",{"id":47,"title":48,"body":49,"category":18,"description":55,"extension":20,"meta":56,"navigation":22,"order":16,"path":57,"seo":58,"sitemap":59,"stem":60,"__hash__":61},"learn\u002Flearn\u002Fcad-bim\u002Fcoordinate-import-export.md","Coordinate Import & Export Workflows",{"type":8,"value":50,"toc":53},[51],[11,52,13],{},{"title":15,"searchDepth":16,"depth":16,"links":54},[],"Managing coordinate data between surveying software and CAD.",{},"\u002Flearn\u002Fcad-bim\u002Fcoordinate-import-export",{"title":48,"description":55},{"loc":57},"learn\u002Fcad-bim\u002Fcoordinate-import-export","pL1mbgw6aeaELLVrptsdqTK1AtW8uus3HCTmawo_TZE",{"id":63,"title":64,"body":65,"category":18,"description":71,"extension":20,"meta":72,"navigation":22,"order":73,"path":74,"seo":75,"sitemap":76,"stem":77,"__hash__":78},"learn\u002Flearn\u002Fcad-bim\u002Fdrawing-standards-layers.md","Drawing Standards & Layer Management",{"type":8,"value":66,"toc":69},[67],[11,68,13],{},{"title":15,"searchDepth":16,"depth":16,"links":70},[],"Organizing CAD drawings using industry standards and layer conventions.",{},3,"\u002Flearn\u002Fcad-bim\u002Fdrawing-standards-layers",{"title":64,"description":71},{"loc":74},"learn\u002Fcad-bim\u002Fdrawing-standards-layers","UetLgdHRW02XEWiT6Z3v7KOkW3za3KdtvuHsdcHcBEY",{"id":80,"title":81,"body":82,"category":18,"description":88,"extension":20,"meta":89,"navigation":22,"order":90,"path":91,"seo":92,"sitemap":93,"stem":94,"__hash__":95},"learn\u002Flearn\u002Fcad-bim\u002Fdxf-dwg-format.md","DXF & DWG Format Deep Dive",{"type":8,"value":83,"toc":86},[84],[11,85,13],{},{"title":15,"searchDepth":16,"depth":16,"links":87},[],"Understanding DXF and DWG file formats for surveying data exchange.",{},4,"\u002Flearn\u002Fcad-bim\u002Fdxf-dwg-format",{"title":81,"description":88},{"loc":91},"learn\u002Fcad-bim\u002Fdxf-dwg-format","gab5wxkQQhI-lJ8LPa3wJgyST9Pn5LkzdgagxZR4Zxw",{"id":97,"title":98,"body":99,"category":18,"description":105,"extension":20,"meta":106,"navigation":22,"order":107,"path":108,"seo":109,"sitemap":110,"stem":111,"__hash__":112},"learn\u002Flearn\u002Fcad-bim\u002Fscan-to-bim.md","Scan-to-BIM: Point Cloud to Model",{"type":8,"value":100,"toc":103},[101],[11,102,13],{},{"title":15,"searchDepth":16,"depth":16,"links":104},[],"Converting point cloud data into BIM models.",{},6,"\u002Flearn\u002Fcad-bim\u002Fscan-to-bim",{"title":98,"description":105},{"loc":108},"learn\u002Fcad-bim\u002Fscan-to-bim","h2RSpZ89umDfrAtwPQPL9zf_ZplO7ooxw8k-M2fKCnM",{"id":114,"title":115,"body":116,"category":122,"description":123,"extension":20,"meta":124,"navigation":22,"order":125,"path":126,"seo":127,"sitemap":128,"stem":129,"__hash__":130},"learn\u002Flearn\u002Fdrone\u002Faccuracy-assessment.md","Accuracy Assessment & Reporting",{"type":8,"value":117,"toc":120},[118],[11,119,13],{},{"title":15,"searchDepth":16,"depth":16,"links":121},[],"drone","Assessing and reporting accuracy of drone survey deliverables.",{},7,"\u002Flearn\u002Fdrone\u002Faccuracy-assessment",{"title":115,"description":123},{"loc":126},"learn\u002Fdrone\u002Faccuracy-assessment","WuJop8hvprtByVDJiwCjJJdpckUBIDf-ukagP5u6sko",{"id":132,"title":133,"body":134,"category":122,"description":140,"extension":20,"meta":141,"navigation":22,"order":16,"path":142,"seo":143,"sitemap":144,"stem":145,"__hash__":146},"learn\u002Flearn\u002Fdrone\u002Fcamera-calibration.md","Camera Calibration & Overlap Planning",{"type":8,"value":135,"toc":138},[136],[11,137,13],{},{"title":15,"searchDepth":16,"depth":16,"links":139},[],"Calibrating cameras and planning image overlap for photogrammetry.",{},"\u002Flearn\u002Fdrone\u002Fcamera-calibration",{"title":133,"description":140},{"loc":142},"learn\u002Fdrone\u002Fcamera-calibration","zau0DRjgvLmiReNpONhlahlpzWtjPRDSQgcezumOQ7s",{"id":148,"title":149,"body":150,"category":122,"description":156,"extension":20,"meta":157,"navigation":22,"order":73,"path":158,"seo":159,"sitemap":160,"stem":161,"__hash__":162},"learn\u002Flearn\u002Fdrone\u002Fground-control-points.md","Ground Control Points: Design & Placement",{"type":8,"value":151,"toc":154},[152],[11,153,13],{},{"title":15,"searchDepth":16,"depth":16,"links":155},[],"Designing and placing ground control points for drone surveys.",{},"\u002Flearn\u002Fdrone\u002Fground-control-points",{"title":149,"description":156},{"loc":158},"learn\u002Fdrone\u002Fground-control-points","xFrhtU07GqI9uN4Xu6u2KEU9umRzZBO9jMZrXgeokdk",{"id":164,"title":165,"body":166,"category":122,"description":172,"extension":20,"meta":173,"navigation":22,"order":107,"path":174,"seo":175,"sitemap":176,"stem":177,"__hash__":178},"learn\u002Flearn\u002Fdrone\u002Forthomosaic-dtm-production.md","Orthomosaic & DTM Production",{"type":8,"value":167,"toc":170},[168],[11,169,13],{},{"title":15,"searchDepth":16,"depth":16,"links":171},[],"Creating orthomosaics and digital terrain models from drone data.",{},"\u002Flearn\u002Fdrone\u002Forthomosaic-dtm-production",{"title":165,"description":172},{"loc":174},"learn\u002Fdrone\u002Forthomosaic-dtm-production","LsCwUQsDxeCN-pXUH8RDUgIPBhYd_kHVCP6n9_1x7us",{"id":180,"title":181,"body":182,"category":122,"description":188,"extension":20,"meta":189,"navigation":22,"order":90,"path":190,"seo":191,"sitemap":192,"stem":193,"__hash__":194},"learn\u002Flearn\u002Fdrone\u002Fphotogrammetric-processing.md","Photogrammetric Processing Workflow",{"type":8,"value":183,"toc":186},[184],[11,185,13],{},{"title":15,"searchDepth":16,"depth":16,"links":187},[],"Processing drone imagery into 3D models and orthophotos.",{},"\u002Flearn\u002Fdrone\u002Fphotogrammetric-processing",{"title":181,"description":188},{"loc":190},"learn\u002Fdrone\u002Fphotogrammetric-processing","ObLvArC6uRKn54ibXni76V3x-moezGJP85O5IAt-CFk",{"id":196,"title":197,"body":198,"category":122,"description":204,"extension":20,"meta":205,"navigation":22,"order":40,"path":206,"seo":207,"sitemap":208,"stem":209,"__hash__":210},"learn\u002Flearn\u002Fdrone\u002Fpoint-cloud-generation.md","Point Cloud Generation & Cleaning",{"type":8,"value":199,"toc":202},[200],[11,201,13],{},{"title":15,"searchDepth":16,"depth":16,"links":203},[],"Generating and cleaning point clouds from drone surveys.",{},"\u002Flearn\u002Fdrone\u002Fpoint-cloud-generation",{"title":197,"description":204},{"loc":206},"learn\u002Fdrone\u002Fpoint-cloud-generation","dOQvDVh-rURCxDp21i3wWb1MDE4oAzUib5wRR36L2P0",{"id":212,"title":213,"body":214,"category":122,"description":220,"extension":20,"meta":221,"navigation":22,"order":23,"path":222,"seo":223,"sitemap":224,"stem":225,"__hash__":226},"learn\u002Flearn\u002Fdrone\u002Fuav-flight-planning.md","UAV Flight Planning Fundamentals",{"type":8,"value":215,"toc":218},[216],[11,217,13],{},{"title":15,"searchDepth":16,"depth":16,"links":219},[],"Planning and executing drone survey flights.",{},"\u002Flearn\u002Fdrone\u002Fuav-flight-planning",{"title":213,"description":220},{"loc":222},"learn\u002Fdrone\u002Fuav-flight-planning","_CLHOYSPPczKgug8liPSMm7qhyvMa7kb67BQcOdd9Gc",{"id":228,"title":229,"body":230,"category":236,"description":237,"extension":20,"meta":238,"navigation":22,"order":73,"path":239,"seo":240,"sitemap":241,"stem":242,"__hash__":243},"learn\u002Flearn\u002Ffoundations\u002Fangles-bearings-directions.md","Angles, Bearings & Directions",{"type":8,"value":231,"toc":234},[232],[11,233,13],{},{"title":15,"searchDepth":16,"depth":16,"links":235},[],"foundations","Measuring and calculating angles, bearings, and directions in surveying.",{},"\u002Flearn\u002Ffoundations\u002Fangles-bearings-directions",{"title":229,"description":237},{"loc":239},"learn\u002Ffoundations\u002Fangles-bearings-directions","tMw03wnfl1Hn9kdBkreXGAINxpbNNUKrwLRhfkd8gGc",{"id":245,"title":246,"body":247,"category":236,"description":253,"extension":20,"meta":254,"navigation":22,"order":40,"path":255,"seo":256,"sitemap":257,"stem":258,"__hash__":259},"learn\u002Flearn\u002Ffoundations\u002Fcoordinate-computation.md","Coordinate Computation (Polar\u002FRectangular)",{"type":8,"value":248,"toc":251},[249],[11,250,13],{},{"title":15,"searchDepth":16,"depth":16,"links":252},[],"Converting between polar and rectangular coordinates in surveying calculations.",{},"\u002Flearn\u002Ffoundations\u002Fcoordinate-computation",{"title":246,"description":253},{"loc":255},"learn\u002Ffoundations\u002Fcoordinate-computation","VvkhtYfly1-B-Anw-z4pQqCSXaPAPJQhfPkv-_gAcbA",{"id":261,"title":262,"body":263,"category":236,"description":269,"extension":20,"meta":270,"navigation":22,"order":16,"path":271,"seo":272,"sitemap":273,"stem":274,"__hash__":275},"learn\u002Flearn\u002Ffoundations\u002Fcoordinate-systems.md","Coordinate Systems & Grids",{"type":8,"value":264,"toc":267},[265],[11,266,13],{},{"title":15,"searchDepth":16,"depth":16,"links":268},[],"Understanding coordinate systems, map projections, and grid references in surveying.",{},"\u002Flearn\u002Ffoundations\u002Fcoordinate-systems",{"title":262,"description":269},{"loc":271},"learn\u002Ffoundations\u002Fcoordinate-systems","vAztjnMkXf19VSD9L72tY4ks06YByU487ka5bZhU6tc",{"id":277,"title":278,"body":279,"category":236,"description":285,"extension":20,"meta":286,"navigation":22,"order":90,"path":287,"seo":288,"sitemap":289,"stem":290,"__hash__":291},"learn\u002Flearn\u002Ffoundations\u002Fdistance-measurement.md","Distance Measurement",{"type":8,"value":280,"toc":283},[281],[11,282,13],{},{"title":15,"searchDepth":16,"depth":16,"links":284},[],"Methods and techniques for accurate distance measurement in surveying.",{},"\u002Flearn\u002Ffoundations\u002Fdistance-measurement",{"title":278,"description":285},{"loc":287},"learn\u002Ffoundations\u002Fdistance-measurement","HvRRYXpaGbaSWqkjpmxMwSSBYb_soGz_1n4cBlcMoxc",{"id":293,"title":294,"body":295,"category":236,"description":301,"extension":20,"meta":302,"navigation":22,"order":303,"path":304,"seo":305,"sitemap":306,"stem":307,"__hash__":308},"learn\u002Flearn\u002Ffoundations\u002Fintroduction-to-setting-out.md","Introduction to Setting Out",{"type":8,"value":296,"toc":299},[297],[11,298,13],{},{"title":15,"searchDepth":16,"depth":16,"links":300},[],"Fundamentals of setting out for construction and engineering projects.",{},8,"\u002Flearn\u002Ffoundations\u002Fintroduction-to-setting-out",{"title":294,"description":301},{"loc":304},"learn\u002Ffoundations\u002Fintroduction-to-setting-out","fmxbLZQVF-ZnMCWHdoejtLiweRYD8inkgru1c7Egn5I",{"id":310,"title":311,"body":312,"category":236,"description":363,"extension":20,"meta":364,"navigation":22,"order":23,"path":365,"seo":366,"sitemap":367,"stem":368,"__hash__":369},"learn\u002Flearn\u002Ffoundations\u002Fthe-geomatics-profession.md","The Geomatics Profession",{"type":8,"value":313,"toc":358},[314,322,327,330,334,347,351],[11,315,316,317,321],{},"Surveying Core ",[318,319,320],"strong",{},"Learn"," turns decades of field knowledge into clear, progressive lessons — for students, junior surveyors, and anyone building foundations in geomatics.",[323,324,326],"h2",{"id":325},"what-youll-find-here","What you'll find here",[11,328,329],{},"Theory in plain language, worked examples, and links to calculators and reference material when you're ready to go deeper. Nothing is a dead end: coordinate geometry connects to instruments, data formats, and CAD workflows.",[323,331,333],{"id":332},"who-this-is-for","Who this is for",[335,336,337,341,344],"ul",{},[338,339,340],"li",{},"Students starting in surveying or civil engineering",[338,342,343],{},"Early-career surveyors strengthening core skills",[338,345,346],{},"Professionals expanding into GNSS, traverse adjustment, or setting out",[323,348,350],{"id":349},"whats-next","What's next",[11,352,353,354,357],{},"Structured ",[318,355,356],{},"learning paths"," — starting with Engineering Surveying Foundations — are on the roadmap. Articles will appear here as we publish them.",{"title":15,"searchDepth":16,"depth":16,"links":359},[360,361,362],{"id":325,"depth":16,"text":326},{"id":332,"depth":16,"text":333},{"id":349,"depth":16,"text":350},"What geomatics is, who surveying professionals are, and how this learning path is structured.",{},"\u002Flearn\u002Ffoundations\u002Fthe-geomatics-profession",{"title":311,"description":363},{"loc":365},"learn\u002Ffoundations\u002Fthe-geomatics-profession","fe3kXF3LJnXe4C_vSt3KQ12KBe1jbHDKgMJJvFU0-Js",{"id":371,"title":372,"body":373,"category":236,"description":379,"extension":20,"meta":380,"navigation":22,"order":125,"path":381,"seo":382,"sitemap":383,"stem":384,"__hash__":385},"learn\u002Flearn\u002Ffoundations\u002Ftraverse-adjustment-bowditch.md","Traverse Adjustment (Bowditch)",{"type":8,"value":374,"toc":377},[375],[11,376,13],{},{"title":15,"searchDepth":16,"depth":16,"links":378},[],"Adjusting traverse measurements using the Bowditch method.",{},"\u002Flearn\u002Ffoundations\u002Ftraverse-adjustment-bowditch",{"title":372,"description":379},{"loc":381},"learn\u002Ffoundations\u002Ftraverse-adjustment-bowditch","iTDOsJ0q8D3FbjCG4xQTWC6dQGvdJ4496GKMopo5tt4",{"id":387,"title":388,"body":389,"category":236,"description":395,"extension":20,"meta":396,"navigation":22,"order":107,"path":397,"seo":398,"sitemap":399,"stem":400,"__hash__":401},"learn\u002Flearn\u002Ffoundations\u002Ftraversing-control-networks.md","Traversing & Control Networks",{"type":8,"value":390,"toc":393},[391],[11,392,13],{},{"title":15,"searchDepth":16,"depth":16,"links":394},[],"Establishing and working with traverse networks and control points.",{},"\u002Flearn\u002Ffoundations\u002Ftraversing-control-networks",{"title":388,"description":395},{"loc":397},"learn\u002Ffoundations\u002Ftraversing-control-networks","ZHVTM1w-_hu8Rvhftcw-zTHxgDE2HUtUYyJ9_yFl00s",{"id":403,"title":404,"body":405,"category":411,"description":412,"extension":20,"meta":413,"navigation":22,"order":16,"path":414,"seo":415,"sitemap":416,"stem":417,"__hash__":418},"learn\u002Flearn\u002Fgeodesy\u002Fcoordinate-systems-national-global.md","National & Global Coordinate Systems",{"type":8,"value":406,"toc":409},[407],[11,408,13],{},{"title":15,"searchDepth":16,"depth":16,"links":410},[],"geodesy","Understanding national and global coordinate reference systems.",{},"\u002Flearn\u002Fgeodesy\u002Fcoordinate-systems-national-global",{"title":404,"description":412},{"loc":414},"learn\u002Fgeodesy\u002Fcoordinate-systems-national-global","yG_esDWD8sRdOsCB2tJEA73zd6x47mQZ7g92hMhmkEc",{"id":420,"title":421,"body":422,"category":411,"description":428,"extension":20,"meta":429,"navigation":22,"order":90,"path":430,"seo":431,"sitemap":432,"stem":433,"__hash__":434},"learn\u002Flearn\u002Fgeodesy\u002Fdatum-transformations.md","Datum Transformations (Helmert & OSTN)",{"type":8,"value":423,"toc":426},[424],[11,425,13],{},{"title":15,"searchDepth":16,"depth":16,"links":427},[],"Transforming coordinates between different datums using Helmert and OSTN methods.",{},"\u002Flearn\u002Fgeodesy\u002Fdatum-transformations",{"title":421,"description":428},{"loc":430},"learn\u002Fgeodesy\u002Fdatum-transformations","DmsDF8ghA2GJxGuF80nisqMzoLryIG1OHbYvjSlIeW0",{"id":436,"title":437,"body":438,"category":411,"description":444,"extension":20,"meta":445,"navigation":22,"order":23,"path":446,"seo":447,"sitemap":448,"stem":449,"__hash__":450},"learn\u002Flearn\u002Fgeodesy\u002Fearth-ellipsoids-geoids.md","Earth Shape: Ellipsoids & Geoids",{"type":8,"value":439,"toc":442},[440],[11,441,13],{},{"title":15,"searchDepth":16,"depth":16,"links":443},[],"Understanding Earth's shape through ellipsoids and geoids.",{},"\u002Flearn\u002Fgeodesy\u002Fearth-ellipsoids-geoids",{"title":437,"description":444},{"loc":446},"learn\u002Fgeodesy\u002Fearth-ellipsoids-geoids","62RNLKblRALW7ebjTbxP8jsjQ_dOUCjmN0cf9zGJcDg",{"id":452,"title":453,"body":454,"category":411,"description":460,"extension":20,"meta":461,"navigation":22,"order":40,"path":462,"seo":463,"sitemap":464,"stem":465,"__hash__":466},"learn\u002Flearn\u002Fgeodesy\u002Fgnss-national-grids.md","Using GNSS with National Grids",{"type":8,"value":455,"toc":458},[456],[11,457,13],{},{"title":15,"searchDepth":16,"depth":16,"links":459},[],"Integrating GNSS measurements with national grid coordinate systems.",{},"\u002Flearn\u002Fgeodesy\u002Fgnss-national-grids",{"title":453,"description":460},{"loc":462},"learn\u002Fgeodesy\u002Fgnss-national-grids","LWp6OmehR_gtwPWr9ziGvsDXgP8bFdQ-4oHcvFinXXo",{"id":468,"title":469,"body":470,"category":411,"description":476,"extension":20,"meta":477,"navigation":22,"order":73,"path":478,"seo":479,"sitemap":480,"stem":481,"__hash__":482},"learn\u002Flearn\u002Fgeodesy\u002Fheight-systems-vertical-datums.md","Height Systems & Vertical Datums",{"type":8,"value":471,"toc":474},[472],[11,473,13],{},{"title":15,"searchDepth":16,"depth":16,"links":475},[],"Understanding height systems and vertical datums in geodesy.",{},"\u002Flearn\u002Fgeodesy\u002Fheight-systems-vertical-datums",{"title":469,"description":476},{"loc":478},"learn\u002Fgeodesy\u002Fheight-systems-vertical-datums","5B6hQbrF6dH8Gs0j_IGhYJgoPR3xvmcgZ7lcaP_rfxE",{"id":484,"title":485,"body":486,"category":411,"description":492,"extension":20,"meta":493,"navigation":22,"order":107,"path":494,"seo":495,"sitemap":496,"stem":497,"__hash__":498},"learn\u002Flearn\u002Fgeodesy\u002Fpractical-coordinate-conversions.md","Practical Coordinate System Conversions",{"type":8,"value":487,"toc":490},[488],[11,489,13],{},{"title":15,"searchDepth":16,"depth":16,"links":491},[],"Practical methods for converting between coordinate systems.",{},"\u002Flearn\u002Fgeodesy\u002Fpractical-coordinate-conversions",{"title":485,"description":492},{"loc":494},"learn\u002Fgeodesy\u002Fpractical-coordinate-conversions","WHCHVLMiRd48mhWNYVUn7PDooLt9-Hk2T13cz1m6FVQ",{"id":500,"title":501,"body":502,"category":508,"description":509,"extension":20,"meta":510,"navigation":22,"order":16,"path":511,"seo":512,"sitemap":513,"stem":514,"__hash__":515},"learn\u002Flearn\u002Fgnss\u002Ferror-sources-corrections.md","Error Sources & Corrections",{"type":8,"value":503,"toc":506},[504],[11,505,13],{},{"title":15,"searchDepth":16,"depth":16,"links":507},[],"gnss","Understanding and correcting GNSS measurement errors.",{},"\u002Flearn\u002Fgnss\u002Ferror-sources-corrections",{"title":501,"description":509},{"loc":511},"learn\u002Fgnss\u002Ferror-sources-corrections","XdD1OBPDMbtM1pZnt0l1BEd8T6gbuWChjNdAicCnoVk",{"id":517,"title":518,"body":519,"category":508,"description":525,"extension":20,"meta":526,"navigation":22,"order":40,"path":527,"seo":528,"sitemap":529,"stem":530,"__hash__":531},"learn\u002Flearn\u002Fgnss\u002Fgnss-data-processing.md","GNSS Data Processing",{"type":8,"value":520,"toc":523},[521],[11,522,13],{},{"title":15,"searchDepth":16,"depth":16,"links":524},[],"Processing and analyzing GNSS survey data.",{},"\u002Flearn\u002Fgnss\u002Fgnss-data-processing",{"title":518,"description":525},{"loc":527},"learn\u002Fgnss\u002Fgnss-data-processing","xu18Qup2Ui9bMP1gWOCzOSmdhTjLeV7J4ebf0KXoeCE",{"id":533,"title":534,"body":535,"category":508,"description":541,"extension":20,"meta":542,"navigation":22,"order":23,"path":543,"seo":544,"sitemap":545,"stem":546,"__hash__":547},"learn\u002Flearn\u002Fgnss\u002Fmulti-constellation-gnss.md","Multi-Constellation GNSS Fundamentals",{"type":8,"value":536,"toc":539},[537],[11,538,13],{},{"title":15,"searchDepth":16,"depth":16,"links":540},[],"Understanding GPS, GLONASS, Galileo, and BeiDou satellite systems.",{},"\u002Flearn\u002Fgnss\u002Fmulti-constellation-gnss",{"title":534,"description":541},{"loc":543},"learn\u002Fgnss\u002Fmulti-constellation-gnss","_SLR_7WiM4DXL8bvN11qr2DVNwU2-5ShD2zna_ZlkQg",{"id":549,"title":550,"body":551,"category":508,"description":557,"extension":20,"meta":558,"navigation":22,"order":90,"path":559,"seo":560,"sitemap":561,"stem":562,"__hash__":563},"learn\u002Flearn\u002Fgnss\u002Fntrip-cors-networks.md","NTRIP & CORS Networks",{"type":8,"value":552,"toc":555},[553],[11,554,13],{},{"title":15,"searchDepth":16,"depth":16,"links":556},[],"Using NTRIP streaming and Continuously Operating Reference Stations.",{},"\u002Flearn\u002Fgnss\u002Fntrip-cors-networks",{"title":550,"description":557},{"loc":559},"learn\u002Fgnss\u002Fntrip-cors-networks","rypnoKk7BV7-sxeIRjajDr_5ertxq6HqqaO2Ay1-V04",{"id":565,"title":566,"body":567,"category":508,"description":573,"extension":20,"meta":574,"navigation":22,"order":73,"path":575,"seo":576,"sitemap":577,"stem":578,"__hash__":579},"learn\u002Flearn\u002Fgnss\u002Frtk-surveying.md","RTK Surveying",{"type":8,"value":568,"toc":571},[569],[11,570,13],{},{"title":15,"searchDepth":16,"depth":16,"links":572},[],"Real-Time Kinematic surveying principles and practices.",{},"\u002Flearn\u002Fgnss\u002Frtk-surveying",{"title":566,"description":573},{"loc":575},"learn\u002Fgnss\u002Frtk-surveying","ZAvadiSnfR0x9JhidQdsHM_INzjGvZf-SVSrzzBpNJs",{"id":581,"title":582,"body":583,"category":589,"description":590,"extension":20,"meta":591,"navigation":22,"order":90,"path":592,"seo":593,"sitemap":594,"stem":595,"__hash__":596},"learn\u002Flearn\u002Fstructural\u002Fcolumn-beam-slab-verification.md","Column, Beam & Slab Verification",{"type":8,"value":584,"toc":587},[585],[11,586,13],{},{"title":15,"searchDepth":16,"depth":16,"links":588},[],"structural","Verification techniques for structural elements in construction.",{},"\u002Flearn\u002Fstructural\u002Fcolumn-beam-slab-verification",{"title":582,"description":590},{"loc":592},"learn\u002Fstructural\u002Fcolumn-beam-slab-verification","Mx-Kovx7sc9PBbwgYOzpG7IwZzTDWFeSlL3dthawq9c",{"id":598,"title":599,"body":600,"category":589,"description":606,"extension":20,"meta":607,"navigation":22,"order":23,"path":608,"seo":609,"sitemap":610,"stem":611,"__hash__":612},"learn\u002Flearn\u002Fstructural\u002Fconstruction-surveying-overview.md","Construction Surveying Overview",{"type":8,"value":601,"toc":604},[602],[11,603,13],{},{"title":15,"searchDepth":16,"depth":16,"links":605},[],"Introduction to construction surveying principles and practices.",{},"\u002Flearn\u002Fstructural\u002Fconstruction-surveying-overview",{"title":599,"description":606},{"loc":608},"learn\u002Fstructural\u002Fconstruction-surveying-overview","wsi3ZtyNOr9FyHbbaH-uwvu6unwAo7hvnH7_2oYvyMQ",{"id":614,"title":615,"body":616,"category":589,"description":622,"extension":20,"meta":623,"navigation":22,"order":107,"path":624,"seo":625,"sitemap":626,"stem":627,"__hash__":628},"learn\u002Flearn\u002Fstructural\u002Fcooling-tower-shape-surveys.md","Cooling Tower & Shell Structure Surveys",{"type":8,"value":617,"toc":620},[618],[11,619,13],{},{"title":15,"searchDepth":16,"depth":16,"links":621},[],"Specialized surveying techniques for cooling towers and shell structures.",{},"\u002Flearn\u002Fstructural\u002Fcooling-tower-shape-surveys",{"title":615,"description":622},{"loc":624},"learn\u002Fstructural\u002Fcooling-tower-shape-surveys","YPXedcPuwJq23gCoZViVUYctLRtLqSHsDOEWF6BRfNs",{"id":630,"title":631,"body":632,"category":589,"description":638,"extension":20,"meta":639,"navigation":22,"order":40,"path":640,"seo":641,"sitemap":642,"stem":643,"__hash__":644},"learn\u002Flearn\u002Fstructural\u002Fhigh-rise-verticality-monitoring.md","High-Rise & Verticality Monitoring",{"type":8,"value":633,"toc":636},[634],[11,635,13],{},{"title":15,"searchDepth":16,"depth":16,"links":637},[],"Monitoring verticality and deformation in high-rise structures.",{},"\u002Flearn\u002Fstructural\u002Fhigh-rise-verticality-monitoring",{"title":631,"description":638},{"loc":640},"learn\u002Fstructural\u002Fhigh-rise-verticality-monitoring","JlkKSm7DFJmuI2qE-JJYVYhpIEbGzD7yOnLvkeookoA",{"id":646,"title":647,"body":648,"category":589,"description":654,"extension":20,"meta":655,"navigation":22,"order":16,"path":656,"seo":657,"sitemap":658,"stem":659,"__hash__":660},"learn\u002Flearn\u002Fstructural\u002Fsetting-out-principles.md","Setting Out Principles",{"type":8,"value":649,"toc":652},[650],[11,651,13],{},{"title":15,"searchDepth":16,"depth":16,"links":653},[],"Fundamental principles of setting out for structural engineering.",{},"\u002Flearn\u002Fstructural\u002Fsetting-out-principles",{"title":647,"description":654},{"loc":656},"learn\u002Fstructural\u002Fsetting-out-principles","Q_hrtuYIxoEywjclS8Cw3XYa18cA612ZK8lk1Dr1BY0",{"id":662,"title":663,"body":664,"category":589,"description":670,"extension":20,"meta":671,"navigation":22,"order":73,"path":672,"seo":673,"sitemap":674,"stem":675,"__hash__":676},"learn\u002Flearn\u002Fstructural\u002Fstructural-tolerance-standards.md","Structural Tolerance Standards (NSSS\u002FNCSC)",{"type":8,"value":665,"toc":668},[666],[11,667,13],{},{"title":15,"searchDepth":16,"depth":16,"links":669},[],"Understanding structural tolerance standards for surveying and construction.",{},"\u002Flearn\u002Fstructural\u002Fstructural-tolerance-standards",{"title":663,"description":670},{"loc":672},"learn\u002Fstructural\u002Fstructural-tolerance-standards","1w1vq2nWEFmiPC43T0azlQukOAoprcoRkWJMzpaYncw",{"id":678,"title":679,"body":680,"category":11245,"description":11246,"extension":20,"meta":11247,"navigation":22,"order":23,"path":11248,"seo":11249,"sitemap":11250,"stem":11251,"__hash__":11252},"learn\u002Flearn\u002Ftools\u002Fbearing-distance.md","Bearing & Distance",{"type":8,"value":681,"toc":11215},[682,686,689,698,702,705,710,713,717,720,724,727,731,734,738,741,755,761,765,768,850,1594,1598,1601,1610,2010,2018,2389,2397,2722,2726,2729,2856,2865,2926,2929,3092,3096,3225,3380,3509,3803,4125,4128,4136,4211,4218,4293,4300,4461,4468,4629,4633,4971,4974,5296,5328,5537,5571,5770,5869,5907,6054,6352,6939,7016,7020,7024,7027,7175,7236,7388,7393,7664,7667,7738,7900,7903,8068,8071,8076,8202,8207,8437,8440,8685,8688,8694,8698,8701,8706,8934,8937,9170,9173,9206,9575,9794,9801,9834,9840,9999,10002,10271,10278,10283,10286,10289,10509,10512,10515,10519,10522,10526,10782,10786,11027,11031,11037,11158,11161,11165,11168,11172,11175,11189,11192,11196,11199,11202,11206,11209],[323,683,685],{"id":684},"introduction","Introduction",[11,687,688],{},"In plane surveying, translating physical field observations into structured digital plans requires a deep understanding of coordinate geometry (COGO). While physical surveying instruments, such as electronic total stations, record spatial relationships in polar formats using distances and angles, civil engineering designs, geographic information systems (GIS), and site layout plans are built on rectangular grid coordinate systems. The mathematical transition between these two spatial representations is a foundational task performed daily by field surveyors, civil engineering technicians, and office personnel.",[11,690,691,692,697],{},"The calculations required to convert raw polar vectors into localized grid coordinates must be executed with absolute mathematical precision. Even minor procedural mistakes can lead to costly errors during construction site layout and boundary definition. To eliminate manual calculation errors and instantly verify spatial transformations, engineering technicians and surveying professionals can utilize the ",[693,694,696],"a",{"href":695},"\u002Ftools\u002Fbearing-distance","Bearing & distance calculator",". This article provides a comprehensive guide to the mathematical and practical principles that govern these coordinate transformations.",[323,699,701],{"id":700},"the-practical-role-of-coordinate-geometry-in-surveying","The Practical Role of Coordinate Geometry in Surveying",[11,703,704],{},"Coordinate geometry serves as the mathematical bridge between physical observations on the Earth's surface and their representations on a flat engineering design plane. In professional practice, these conversions are utilized across three core operational phases:",[706,707,709],"h3",{"id":708},"polar-data-acquisition-with-modern-instruments","Polar Data Acquisition with Modern Instruments",[11,711,712],{},"Standard field measurements rely on polar coordinates. An electronic total station setup at a known control monument (the occupied station) establishes a baseline by backsighting another known point. When the instrument observes a target, it measures a horizontal angle from that reference baseline and a horizontal distance to the target. This polar dataset must be converted into rectangular coordinate differences to determine the absolute coordinate location of the observed point.",[706,714,716],{"id":715},"traverse-computations-and-quality-control","Traverse Computations and Quality Control",[11,718,719],{},"A traverse is a sequence of connected lines whose lengths and directions are measured to establish spatial control across a project site. For closed-loop or closed-line traverses, surveyors turn interior angles at each station. Before calculating final coordinate values, these angles are mathematically adjusted in the office to eliminate angular misclosure. Once the adjusted whole-circle bearings are established, the polar-to-rectangular equations are applied to each traverse leg to calculate grid differences. The algebraic sum of these differences reveals the linear error of closure, which must be balanced using standard adjustment computations, such as the Bowditch or transit rule, before final coordinate publication.",[706,721,723],{"id":722},"construction-layout-and-stakeout-offsets","Construction Layout and Stakeout Offsets",[11,725,726],{},"During the construction phase, the workflow reverses. Designing engineers establish structural features—such as column lines, utility pipelines, and roadway curb points—directly as rectangular coordinates on a grid system. To physically place these elements on the ground, a surveyor must calculate the inverse relationship. By executing a rectangular-to-polar calculation between a known on-site control monument and the design coordinate, the surveyor computes the exact azimuth and distance needed to stake out the point.",[323,728,730],{"id":729},"azimuth-and-bearing-conventions","Azimuth and Bearing Conventions",[11,732,733],{},"Defining spatial direction requires a reliable angular reference system. In geomatics and civil engineering, direction is expressed relative to a reference meridian, which is typically Grid North. Two primary angular conventions are utilized to define these directions: Azimuths (Whole Circle Bearings) and Quadrant Bearings (Reduced Bearings).",[706,735,737],{"id":736},"whole-circle-bearings-azimuths","Whole Circle Bearings (Azimuths)",[11,739,740],{},"An azimuth is an angular direction measured continuously clockwise from Grid North, ranging from 0° to 360°. Under this convention:",[335,742,743,746,749,752],{},[338,744,745],{},"North is represented by 0° (or 360°).",[338,747,748],{},"East is represented by 90°.",[338,750,751],{},"South is represented by 180°.",[338,753,754],{},"West is represented by 270°.",[11,756,757,758,760],{},"Azimuths are widely used in programmatic computations, digital data collectors, and the ",[693,759,696],{"href":695}," because they represent direction as a single, continuous numerical value, which simplifies algorithmic processing.",[706,762,764],{"id":763},"quadrant-bearings-reduced-bearings","Quadrant Bearings (Reduced Bearings)",[11,766,767],{},"A quadrant bearing defines a direction by referencing an acute angle (ranging strictly between 0° and 90°) measured from either the North or South end of the meridian toward the East or West. The notation consists of a prefix indicating the primary reference axis (N or S), the acute numerical angle, and a suffix indicating the direction of deviation (E or W).",[11,769,770,771,819,820,849],{},"Because historical deeds, boundary descriptions, and older construction plans often use quadrant bearings, surveyors must be adept at converting these values. The table below outlines the mathematical rules for converting between azimuths (",[772,773,776,799],"span",{"className":774},[775],"katex",[772,777,780],{"className":778},[779],"katex-mathml",[781,782,784],"math",{"xmlns":783},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[785,786,787,794],"semantics",{},[788,789,790],"mrow",{},[791,792,793],"mi",{},"θ",[795,796,798],"annotation",{"encoding":797},"application\u002Fx-tex","\\theta",[772,800,804],{"className":801,"ariaHidden":803},[802],"katex-html","true",[772,805,808,813],{"className":806},[807],"base",[772,809],{"className":810,"style":812},[811],"strut","height:0.6944em;",[772,814,793],{"className":815,"style":818},[816,817],"mord","mathnormal","margin-right:0.0278em;",") and quadrant bearings (",[772,821,823,836],{"className":822},[775],[772,824,826],{"className":825},[779],[781,827,828],{"xmlns":783},[785,829,830,834],{},[788,831,832],{},[791,833,693],{},[795,835,693],{"encoding":797},[772,837,839],{"className":838,"ariaHidden":803},[802],[772,840,842,846],{"className":841},[807],[772,843],{"className":844,"style":845},[811],"height:0.4306em;",[772,847,693],{"className":848},[816,817],") across the four geographic quadrants:",[851,852,853,875],"table",{},[854,855,856],"thead",{},[857,858,859,863,866,869,872],"tr",{},[860,861,862],"th",{},"Quadrant",[860,864,865],{},"Azimuth range",[860,867,868],{},"Quadrant notation",[860,870,871],{},"Azimuth → quadrant bearing",[860,873,874],{},"Quadrant bearing → azimuth",[876,877,878,1021,1220,1407],"tbody",{},[857,879,880,884,887,927,963],{},[881,882,883],"td",{},"I (NE)",[881,885,886],{},"0°–90°",[881,888,889,890,926],{},"N ",[772,891,893,911],{"className":892},[775],[772,894,896],{"className":895},[779],[781,897,898],{"xmlns":783},[785,899,900,908],{},[788,901,902,904],{},[791,903,693],{},[791,905,907],{"mathvariant":906},"normal","°",[795,909,910],{"encoding":797},"a°",[772,912,914],{"className":913,"ariaHidden":803},[802],[772,915,917,920,923],{"className":916},[807],[772,918],{"className":919,"style":812},[811],[772,921,693],{"className":922},[816,817],[772,924,907],{"className":925},[816]," E",[881,928,889,929,926],{},[772,930,932,948],{"className":931},[775],[772,933,935],{"className":934},[779],[781,936,937],{"xmlns":783},[785,938,939,945],{},[788,940,941,943],{},[791,942,793],{},[791,944,907],{"mathvariant":906},[795,946,947],{"encoding":797},"\\theta°",[772,949,951],{"className":950,"ariaHidden":803},[802],[772,952,954,957,960],{"className":953},[807],[772,955],{"className":956,"style":812},[811],[772,958,793],{"className":959,"style":818},[816,817],[772,961,907],{"className":962},[816],[881,964,965],{},[772,966,968,988],{"className":967},[775],[772,969,971],{"className":970},[779],[781,972,973],{"xmlns":783},[785,974,975,985],{},[788,976,977,979,983],{},[791,978,793],{},[980,981,982],"mo",{},"=",[791,984,693],{},[795,986,987],{"encoding":797},"\\theta = a",[772,989,991,1012],{"className":990,"ariaHidden":803},[802],[772,992,994,997,1000,1005,1009],{"className":993},[807],[772,995],{"className":996,"style":812},[811],[772,998,793],{"className":999,"style":818},[816,817],[772,1001],{"className":1002,"style":1004},[1003],"mspace","margin-right:0.2778em;",[772,1006,982],{"className":1007},[1008],"mrel",[772,1010],{"className":1011,"style":1004},[1003],[772,1013,1015,1018],{"className":1014},[807],[772,1016],{"className":1017,"style":845},[811],[772,1019,693],{"className":1020},[816,817],[857,1022,1023,1026,1029,1065,1142],{},[881,1024,1025],{},"II (SE)",[881,1027,1028],{},"90°–180°",[881,1030,1031,1032,926],{},"S ",[772,1033,1035,1050],{"className":1034},[775],[772,1036,1038],{"className":1037},[779],[781,1039,1040],{"xmlns":783},[785,1041,1042,1048],{},[788,1043,1044,1046],{},[791,1045,693],{},[791,1047,907],{"mathvariant":906},[795,1049,910],{"encoding":797},[772,1051,1053],{"className":1052,"ariaHidden":803},[802],[772,1054,1056,1059,1062],{"className":1055},[807],[772,1057],{"className":1058,"style":812},[811],[772,1060,693],{"className":1061},[816,817],[772,1063,907],{"className":1064},[816],[881,1066,1031,1067,926],{},[772,1068,1070,1100],{"className":1069},[775],[772,1071,1073],{"className":1072},[779],[781,1074,1075],{"xmlns":783},[785,1076,1077,1097],{},[788,1078,1079,1083,1087,1089,1092,1094],{},[980,1080,1082],{"stretchy":1081},"false","(",[1084,1085,1086],"mn",{},"180",[791,1088,907],{"mathvariant":906},[980,1090,1091],{},"−",[791,1093,793],{},[980,1095,1096],{"stretchy":1081},")",[795,1098,1099],{"encoding":797},"(180° - \\theta)",[772,1101,1103,1129],{"className":1102,"ariaHidden":803},[802],[772,1104,1106,1110,1114,1118,1122,1126],{"className":1105},[807],[772,1107],{"className":1108,"style":1109},[811],"height:1em;vertical-align:-0.25em;",[772,1111,1082],{"className":1112},[1113],"mopen",[772,1115,1117],{"className":1116},[816],"180°",[772,1119],{"className":1120,"style":1121},[1003],"margin-right:0.2222em;",[772,1123,1091],{"className":1124},[1125],"mbin",[772,1127],{"className":1128,"style":1121},[1003],[772,1130,1132,1135,1138],{"className":1131},[807],[772,1133],{"className":1134,"style":1109},[811],[772,1136,793],{"className":1137,"style":818},[816,817],[772,1139,1096],{"className":1140},[1141],"mclose",[881,1143,1144],{},[772,1145,1147,1171],{"className":1146},[775],[772,1148,1150],{"className":1149},[779],[781,1151,1152],{"xmlns":783},[785,1153,1154,1168],{},[788,1155,1156,1158,1160,1162,1164,1166],{},[791,1157,793],{},[980,1159,982],{},[1084,1161,1086],{},[791,1163,907],{"mathvariant":906},[980,1165,1091],{},[791,1167,693],{},[795,1169,1170],{"encoding":797},"\\theta = 180° - a",[772,1172,1174,1192,1211],{"className":1173,"ariaHidden":803},[802],[772,1175,1177,1180,1183,1186,1189],{"className":1176},[807],[772,1178],{"className":1179,"style":812},[811],[772,1181,793],{"className":1182,"style":818},[816,817],[772,1184],{"className":1185,"style":1004},[1003],[772,1187,982],{"className":1188},[1008],[772,1190],{"className":1191,"style":1004},[1003],[772,1193,1195,1199,1202,1205,1208],{"className":1194},[807],[772,1196],{"className":1197,"style":1198},[811],"height:0.7778em;vertical-align:-0.0833em;",[772,1200,1117],{"className":1201},[816],[772,1203],{"className":1204,"style":1121},[1003],[772,1206,1091],{"className":1207},[1125],[772,1209],{"className":1210,"style":1121},[1003],[772,1212,1214,1217],{"className":1213},[807],[772,1215],{"className":1216,"style":845},[811],[772,1218,693],{"className":1219},[816,817],[857,1221,1222,1225,1228,1264,1329],{},[881,1223,1224],{},"III (SW)",[881,1226,1227],{},"180°–270°",[881,1229,1031,1230,1263],{},[772,1231,1233,1248],{"className":1232},[775],[772,1234,1236],{"className":1235},[779],[781,1237,1238],{"xmlns":783},[785,1239,1240,1246],{},[788,1241,1242,1244],{},[791,1243,693],{},[791,1245,907],{"mathvariant":906},[795,1247,910],{"encoding":797},[772,1249,1251],{"className":1250,"ariaHidden":803},[802],[772,1252,1254,1257,1260],{"className":1253},[807],[772,1255],{"className":1256,"style":812},[811],[772,1258,693],{"className":1259},[816,817],[772,1261,907],{"className":1262},[816]," W",[881,1265,1031,1266,1263],{},[772,1267,1269,1293],{"className":1268},[775],[772,1270,1272],{"className":1271},[779],[781,1273,1274],{"xmlns":783},[785,1275,1276,1290],{},[788,1277,1278,1280,1282,1284,1286,1288],{},[980,1279,1082],{"stretchy":1081},[791,1281,793],{},[980,1283,1091],{},[1084,1285,1086],{},[791,1287,907],{"mathvariant":906},[980,1289,1096],{"stretchy":1081},[795,1291,1292],{"encoding":797},"(\\theta - 180°)",[772,1294,1296,1317],{"className":1295,"ariaHidden":803},[802],[772,1297,1299,1302,1305,1308,1311,1314],{"className":1298},[807],[772,1300],{"className":1301,"style":1109},[811],[772,1303,1082],{"className":1304},[1113],[772,1306,793],{"className":1307,"style":818},[816,817],[772,1309],{"className":1310,"style":1121},[1003],[772,1312,1091],{"className":1313},[1125],[772,1315],{"className":1316,"style":1121},[1003],[772,1318,1320,1323,1326],{"className":1319},[807],[772,1321],{"className":1322,"style":1109},[811],[772,1324,1117],{"className":1325},[816],[772,1327,1096],{"className":1328},[1141],[881,1330,1331],{},[772,1332,1334,1359],{"className":1333},[775],[772,1335,1337],{"className":1336},[779],[781,1338,1339],{"xmlns":783},[785,1340,1341,1356],{},[788,1342,1343,1345,1347,1349,1351,1354],{},[791,1344,793],{},[980,1346,982],{},[1084,1348,1086],{},[791,1350,907],{"mathvariant":906},[980,1352,1353],{},"+",[791,1355,693],{},[795,1357,1358],{"encoding":797},"\\theta = 180° + a",[772,1360,1362,1380,1398],{"className":1361,"ariaHidden":803},[802],[772,1363,1365,1368,1371,1374,1377],{"className":1364},[807],[772,1366],{"className":1367,"style":812},[811],[772,1369,793],{"className":1370,"style":818},[816,817],[772,1372],{"className":1373,"style":1004},[1003],[772,1375,982],{"className":1376},[1008],[772,1378],{"className":1379,"style":1004},[1003],[772,1381,1383,1386,1389,1392,1395],{"className":1382},[807],[772,1384],{"className":1385,"style":1198},[811],[772,1387,1117],{"className":1388},[816],[772,1390],{"className":1391,"style":1121},[1003],[772,1393,1353],{"className":1394},[1125],[772,1396],{"className":1397,"style":1121},[1003],[772,1399,1401,1404],{"className":1400},[807],[772,1402],{"className":1403,"style":845},[811],[772,1405,693],{"className":1406},[816,817],[857,1408,1409,1412,1415,1450,1517],{},[881,1410,1411],{},"IV (NW)",[881,1413,1414],{},"270°–360°",[881,1416,889,1417,1263],{},[772,1418,1420,1435],{"className":1419},[775],[772,1421,1423],{"className":1422},[779],[781,1424,1425],{"xmlns":783},[785,1426,1427,1433],{},[788,1428,1429,1431],{},[791,1430,693],{},[791,1432,907],{"mathvariant":906},[795,1434,910],{"encoding":797},[772,1436,1438],{"className":1437,"ariaHidden":803},[802],[772,1439,1441,1444,1447],{"className":1440},[807],[772,1442],{"className":1443,"style":812},[811],[772,1445,693],{"className":1446},[816,817],[772,1448,907],{"className":1449},[816],[881,1451,889,1452,1263],{},[772,1453,1455,1480],{"className":1454},[775],[772,1456,1458],{"className":1457},[779],[781,1459,1460],{"xmlns":783},[785,1461,1462,1477],{},[788,1463,1464,1466,1469,1471,1473,1475],{},[980,1465,1082],{"stretchy":1081},[1084,1467,1468],{},"360",[791,1470,907],{"mathvariant":906},[980,1472,1091],{},[791,1474,793],{},[980,1476,1096],{"stretchy":1081},[795,1478,1479],{"encoding":797},"(360° - \\theta)",[772,1481,1483,1505],{"className":1482,"ariaHidden":803},[802],[772,1484,1486,1489,1492,1496,1499,1502],{"className":1485},[807],[772,1487],{"className":1488,"style":1109},[811],[772,1490,1082],{"className":1491},[1113],[772,1493,1495],{"className":1494},[816],"360°",[772,1497],{"className":1498,"style":1121},[1003],[772,1500,1091],{"className":1501},[1125],[772,1503],{"className":1504,"style":1121},[1003],[772,1506,1508,1511,1514],{"className":1507},[807],[772,1509],{"className":1510,"style":1109},[811],[772,1512,793],{"className":1513,"style":818},[816,817],[772,1515,1096],{"className":1516},[1141],[881,1518,1519],{},[772,1520,1522,1546],{"className":1521},[775],[772,1523,1525],{"className":1524},[779],[781,1526,1527],{"xmlns":783},[785,1528,1529,1543],{},[788,1530,1531,1533,1535,1537,1539,1541],{},[791,1532,793],{},[980,1534,982],{},[1084,1536,1468],{},[791,1538,907],{"mathvariant":906},[980,1540,1091],{},[791,1542,693],{},[795,1544,1545],{"encoding":797},"\\theta = 360° - a",[772,1547,1549,1567,1585],{"className":1548,"ariaHidden":803},[802],[772,1550,1552,1555,1558,1561,1564],{"className":1551},[807],[772,1553],{"className":1554,"style":812},[811],[772,1556,793],{"className":1557,"style":818},[816,817],[772,1559],{"className":1560,"style":1004},[1003],[772,1562,982],{"className":1563},[1008],[772,1565],{"className":1566,"style":1004},[1003],[772,1568,1570,1573,1576,1579,1582],{"className":1569},[807],[772,1571],{"className":1572,"style":1198},[811],[772,1574,1495],{"className":1575},[816],[772,1577],{"className":1578,"style":1121},[1003],[772,1580,1091],{"className":1581},[1125],[772,1583],{"className":1584,"style":1121},[1003],[772,1586,1588,1591],{"className":1587},[807],[772,1589],{"className":1590,"style":845},[811],[772,1592,693],{"className":1593},[816,817],[706,1595,1597],{"id":1596},"practical-conversion-examples","Practical Conversion Examples",[11,1599,1600],{},"To demonstrate these conversions, consider three scenarios:",[1602,1603,1604],"ol",{},[338,1605,1606,1609],{},[318,1607,1608],{},"Southeast Quadrant (II)",": A whole-circle bearing of 143°26′ lies between 90° and 180°. The quadrant bearing is computed as:",[11,1611,1612],{},[772,1613,1615,1696],{"className":1614},[775],[772,1616,1618],{"className":1617},[779],[781,1619,1620],{"xmlns":783},[785,1621,1622,1693],{},[788,1623,1624,1628,1630,1633,1635,1643,1652,1654,1661,1668,1670,1673,1675,1677,1684,1691],{},[1625,1626,1627],"mtext",{},"QB",[980,1629,982],{},[1625,1631,1632],{},"S ",[980,1634,1082],{"stretchy":1081},[1636,1637,1638,1640],"msup",{},[1084,1639,1086],{},[980,1641,1642],{},"∘",[1636,1644,1645,1648],{},[1084,1646,1647],{},"00",[980,1649,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},"0em","′",[980,1653,1091],{},[1636,1655,1656,1659],{},[1084,1657,1658],{},"143",[980,1660,1642],{},[1636,1662,1663,1666],{},[1084,1664,1665],{},"26",[980,1667,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,1669,1096],{"stretchy":1081},[1625,1671,1672],{}," E",[980,1674,982],{},[1625,1676,1632],{},[1636,1678,1679,1682],{},[1084,1680,1681],{},"36",[980,1683,1642],{},[1636,1685,1686,1689],{},[1084,1687,1688],{},"34",[980,1690,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[1625,1692,1672],{},[795,1694,1695],{"encoding":797},"\\text{QB}=\\text{S }(180^\\circ00'-143^\\circ26')\\text{ E}=\\text{S }36^\\circ34'\\text{ E}",[772,1697,1699,1722,1829,1924],{"className":1698,"ariaHidden":803},[802],[772,1700,1702,1706,1713,1716,1719],{"className":1701},[807],[772,1703],{"className":1704,"style":1705},[811],"height:0.8778em;vertical-align:-0.1944em;",[772,1707,1710],{"className":1708},[816,1709],"text",[772,1711,1627],{"className":1712},[816],[772,1714],{"className":1715,"style":1004},[1003],[772,1717,982],{"className":1718},[1008],[772,1720],{"className":1721,"style":1004},[1003],[772,1723,1725,1729,1735,1738,1742,1784,1787,1820,1823,1826],{"className":1724},[807],[772,1726],{"className":1727,"style":1728},[811],"height:1.0019em;vertical-align:-0.25em;",[772,1730,1732],{"className":1731},[816,1709],[772,1733,1632],{"className":1734},[816],[772,1736,1082],{"className":1737},[1113],[772,1739,1741],{"className":1740},[816],"18",[772,1743,1745,1749],{"className":1744},[816],[772,1746,1748],{"className":1747},[816],"0",[772,1750,1753],{"className":1751},[1752],"msupsub",[772,1754,1757],{"className":1755},[1756],"vlist-t",[772,1758,1761],{"className":1759},[1760],"vlist-r",[772,1762,1766],{"className":1763,"style":1765},[1764],"vlist","height:0.6741em;",[772,1767,1769,1774],{"style":1768},"top:-3.063em;margin-right:0.05em;",[772,1770],{"className":1771,"style":1773},[1772],"pstrut","height:2.7em;",[772,1775,1781],{"className":1776},[1777,1778,1779,1780],"sizing","reset-size6","size3","mtight",[772,1782,1642],{"className":1783},[1125,1780],[772,1785,1748],{"className":1786},[816],[772,1788,1790,1793],{"className":1789},[816],[772,1791,1748],{"className":1792},[816],[772,1794,1796],{"className":1795},[1752],[772,1797,1799],{"className":1798},[1756],[772,1800,1802],{"className":1801},[1760],[772,1803,1806],{"className":1804,"style":1805},[1764],"height:0.7519em;",[772,1807,1808,1811],{"style":1768},[772,1809],{"className":1810,"style":1773},[1772],[772,1812,1814],{"className":1813},[1777,1778,1779,1780],[772,1815,1817],{"className":1816},[816,1780],[772,1818,1651],{"className":1819},[816,1780],[772,1821],{"className":1822,"style":1121},[1003],[772,1824,1091],{"className":1825},[1125],[772,1827],{"className":1828,"style":1121},[1003],[772,1830,1832,1835,1839,1869,1873,1906,1909,1915,1918,1921],{"className":1831},[807],[772,1833],{"className":1834,"style":1728},[811],[772,1836,1838],{"className":1837},[816],"14",[772,1840,1842,1846],{"className":1841},[816],[772,1843,1845],{"className":1844},[816],"3",[772,1847,1849],{"className":1848},[1752],[772,1850,1852],{"className":1851},[1756],[772,1853,1855],{"className":1854},[1760],[772,1856,1858],{"className":1857,"style":1765},[1764],[772,1859,1860,1863],{"style":1768},[772,1861],{"className":1862,"style":1773},[1772],[772,1864,1866],{"className":1865},[1777,1778,1779,1780],[772,1867,1642],{"className":1868},[1125,1780],[772,1870,1872],{"className":1871},[816],"2",[772,1874,1876,1880],{"className":1875},[816],[772,1877,1879],{"className":1878},[816],"6",[772,1881,1883],{"className":1882},[1752],[772,1884,1886],{"className":1885},[1756],[772,1887,1889],{"className":1888},[1760],[772,1890,1892],{"className":1891,"style":1805},[1764],[772,1893,1894,1897],{"style":1768},[772,1895],{"className":1896,"style":1773},[1772],[772,1898,1900],{"className":1899},[1777,1778,1779,1780],[772,1901,1903],{"className":1902},[816,1780],[772,1904,1651],{"className":1905},[816,1780],[772,1907,1096],{"className":1908},[1141],[772,1910,1912],{"className":1911},[816,1709],[772,1913,1672],{"className":1914},[816],[772,1916],{"className":1917,"style":1004},[1003],[772,1919,982],{"className":1920},[1008],[772,1922],{"className":1923,"style":1004},[1003],[772,1925,1927,1930,1936,1939,1968,1971,2004],{"className":1926},[807],[772,1928],{"className":1929,"style":1805},[811],[772,1931,1933],{"className":1932},[816,1709],[772,1934,1632],{"className":1935},[816],[772,1937,1845],{"className":1938},[816],[772,1940,1942,1945],{"className":1941},[816],[772,1943,1879],{"className":1944},[816],[772,1946,1948],{"className":1947},[1752],[772,1949,1951],{"className":1950},[1756],[772,1952,1954],{"className":1953},[1760],[772,1955,1957],{"className":1956,"style":1765},[1764],[772,1958,1959,1962],{"style":1768},[772,1960],{"className":1961,"style":1773},[1772],[772,1963,1965],{"className":1964},[1777,1778,1779,1780],[772,1966,1642],{"className":1967},[1125,1780],[772,1969,1845],{"className":1970},[816],[772,1972,1974,1978],{"className":1973},[816],[772,1975,1977],{"className":1976},[816],"4",[772,1979,1981],{"className":1980},[1752],[772,1982,1984],{"className":1983},[1756],[772,1985,1987],{"className":1986},[1760],[772,1988,1990],{"className":1989,"style":1805},[1764],[772,1991,1992,1995],{"style":1768},[772,1993],{"className":1994,"style":1773},[1772],[772,1996,1998],{"className":1997},[1777,1778,1779,1780],[772,1999,2001],{"className":2000},[816,1780],[772,2002,1651],{"className":2003},[816,1780],[772,2005,2007],{"className":2006},[816,1709],[772,2008,1672],{"className":2009},[816],[1602,2011,2012],{"start":16},[338,2013,2014,2017],{},[318,2015,2016],{},"Southwest Quadrant (III)",": A whole-circle bearing of 234°15′ lies between 180° and 270°. The quadrant bearing is computed as:",[11,2019,2020],{},[772,2021,2023,2095],{"className":2022},[775],[772,2024,2026],{"className":2025},[779],[781,2027,2028],{"xmlns":783},[785,2029,2030,2092],{},[788,2031,2032,2034,2036,2038,2040,2047,2054,2056,2062,2068,2070,2073,2075,2077,2084,2090],{},[1625,2033,1627],{},[980,2035,982],{},[1625,2037,1632],{},[980,2039,1082],{"stretchy":1081},[1636,2041,2042,2045],{},[1084,2043,2044],{},"234",[980,2046,1642],{},[1636,2048,2049,2052],{},[1084,2050,2051],{},"15",[980,2053,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,2055,1091],{},[1636,2057,2058,2060],{},[1084,2059,1086],{},[980,2061,1642],{},[1636,2063,2064,2066],{},[1084,2065,1647],{},[980,2067,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,2069,1096],{"stretchy":1081},[1625,2071,2072],{}," W",[980,2074,982],{},[1625,2076,1632],{},[1636,2078,2079,2082],{},[1084,2080,2081],{},"54",[980,2083,1642],{},[1636,2085,2086,2088],{},[1084,2087,2051],{},[980,2089,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[1625,2091,2072],{},[795,2093,2094],{"encoding":797},"\\text{QB}=\\text{S }(234^\\circ15'-180^\\circ00')\\text{ W}=\\text{S }54^\\circ15'\\text{ W}",[772,2096,2098,2119,2213,2304],{"className":2097,"ariaHidden":803},[802],[772,2099,2101,2104,2110,2113,2116],{"className":2100},[807],[772,2102],{"className":2103,"style":1705},[811],[772,2105,2107],{"className":2106},[816,1709],[772,2108,1627],{"className":2109},[816],[772,2111],{"className":2112,"style":1004},[1003],[772,2114,982],{"className":2115},[1008],[772,2117],{"className":2118,"style":1004},[1003],[772,2120,2122,2125,2131,2134,2138,2167,2171,2204,2207,2210],{"className":2121},[807],[772,2123],{"className":2124,"style":1728},[811],[772,2126,2128],{"className":2127},[816,1709],[772,2129,1632],{"className":2130},[816],[772,2132,1082],{"className":2133},[1113],[772,2135,2137],{"className":2136},[816],"23",[772,2139,2141,2144],{"className":2140},[816],[772,2142,1977],{"className":2143},[816],[772,2145,2147],{"className":2146},[1752],[772,2148,2150],{"className":2149},[1756],[772,2151,2153],{"className":2152},[1760],[772,2154,2156],{"className":2155,"style":1765},[1764],[772,2157,2158,2161],{"style":1768},[772,2159],{"className":2160,"style":1773},[1772],[772,2162,2164],{"className":2163},[1777,1778,1779,1780],[772,2165,1642],{"className":2166},[1125,1780],[772,2168,2170],{"className":2169},[816],"1",[772,2172,2174,2178],{"className":2173},[816],[772,2175,2177],{"className":2176},[816],"5",[772,2179,2181],{"className":2180},[1752],[772,2182,2184],{"className":2183},[1756],[772,2185,2187],{"className":2186},[1760],[772,2188,2190],{"className":2189,"style":1805},[1764],[772,2191,2192,2195],{"style":1768},[772,2193],{"className":2194,"style":1773},[1772],[772,2196,2198],{"className":2197},[1777,1778,1779,1780],[772,2199,2201],{"className":2200},[816,1780],[772,2202,1651],{"className":2203},[816,1780],[772,2205],{"className":2206,"style":1121},[1003],[772,2208,1091],{"className":2209},[1125],[772,2211],{"className":2212,"style":1121},[1003],[772,2214,2216,2219,2222,2251,2254,2286,2289,2295,2298,2301],{"className":2215},[807],[772,2217],{"className":2218,"style":1728},[811],[772,2220,1741],{"className":2221},[816],[772,2223,2225,2228],{"className":2224},[816],[772,2226,1748],{"className":2227},[816],[772,2229,2231],{"className":2230},[1752],[772,2232,2234],{"className":2233},[1756],[772,2235,2237],{"className":2236},[1760],[772,2238,2240],{"className":2239,"style":1765},[1764],[772,2241,2242,2245],{"style":1768},[772,2243],{"className":2244,"style":1773},[1772],[772,2246,2248],{"className":2247},[1777,1778,1779,1780],[772,2249,1642],{"className":2250},[1125,1780],[772,2252,1748],{"className":2253},[816],[772,2255,2257,2260],{"className":2256},[816],[772,2258,1748],{"className":2259},[816],[772,2261,2263],{"className":2262},[1752],[772,2264,2266],{"className":2265},[1756],[772,2267,2269],{"className":2268},[1760],[772,2270,2272],{"className":2271,"style":1805},[1764],[772,2273,2274,2277],{"style":1768},[772,2275],{"className":2276,"style":1773},[1772],[772,2278,2280],{"className":2279},[1777,1778,1779,1780],[772,2281,2283],{"className":2282},[816,1780],[772,2284,1651],{"className":2285},[816,1780],[772,2287,1096],{"className":2288},[1141],[772,2290,2292],{"className":2291},[816,1709],[772,2293,2072],{"className":2294},[816],[772,2296],{"className":2297,"style":1004},[1003],[772,2299,982],{"className":2300},[1008],[772,2302],{"className":2303,"style":1004},[1003],[772,2305,2307,2310,2316,2319,2348,2351,2383],{"className":2306},[807],[772,2308],{"className":2309,"style":1805},[811],[772,2311,2313],{"className":2312},[816,1709],[772,2314,1632],{"className":2315},[816],[772,2317,2177],{"className":2318},[816],[772,2320,2322,2325],{"className":2321},[816],[772,2323,1977],{"className":2324},[816],[772,2326,2328],{"className":2327},[1752],[772,2329,2331],{"className":2330},[1756],[772,2332,2334],{"className":2333},[1760],[772,2335,2337],{"className":2336,"style":1765},[1764],[772,2338,2339,2342],{"style":1768},[772,2340],{"className":2341,"style":1773},[1772],[772,2343,2345],{"className":2344},[1777,1778,1779,1780],[772,2346,1642],{"className":2347},[1125,1780],[772,2349,2170],{"className":2350},[816],[772,2352,2354,2357],{"className":2353},[816],[772,2355,2177],{"className":2356},[816],[772,2358,2360],{"className":2359},[1752],[772,2361,2363],{"className":2362},[1756],[772,2364,2366],{"className":2365},[1760],[772,2367,2369],{"className":2368,"style":1805},[1764],[772,2370,2371,2374],{"style":1768},[772,2372],{"className":2373,"style":1773},[1772],[772,2375,2377],{"className":2376},[1777,1778,1779,1780],[772,2378,2380],{"className":2379},[816,1780],[772,2381,1651],{"className":2382},[816,1780],[772,2384,2386],{"className":2385},[816,1709],[772,2387,2072],{"className":2388},[816],[1602,2390,2391],{"start":73},[338,2392,2393,2396],{},[318,2394,2395],{},"Northwest Quadrant (IV)",": A quadrant bearing of N 23°28′ W lies in the fourth quadrant. The whole-circle bearing is computed as:",[11,2398,2399],{},[772,2400,2402,2461],{"className":2401},[775],[772,2403,2405],{"className":2404},[779],[781,2406,2407],{"xmlns":783},[785,2408,2409,2458],{},[788,2410,2411,2413,2415,2421,2427,2429,2435,2442,2444,2451],{},[791,2412,793],{},[980,2414,982],{},[1636,2416,2417,2419],{},[1084,2418,1468],{},[980,2420,1642],{},[1636,2422,2423,2425],{},[1084,2424,1647],{},[980,2426,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,2428,1091],{},[1636,2430,2431,2433],{},[1084,2432,2137],{},[980,2434,1642],{},[1636,2436,2437,2440],{},[1084,2438,2439],{},"28",[980,2441,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,2443,982],{},[1636,2445,2446,2449],{},[1084,2447,2448],{},"336",[980,2450,1642],{},[1636,2452,2453,2456],{},[1084,2454,2455],{},"32",[980,2457,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,2459,2460],{"encoding":797},"\\theta=360^\\circ00'-23^\\circ28'=336^\\circ32'",[772,2462,2464,2482,2565,2648],{"className":2463,"ariaHidden":803},[802],[772,2465,2467,2470,2473,2476,2479],{"className":2466},[807],[772,2468],{"className":2469,"style":812},[811],[772,2471,793],{"className":2472,"style":818},[816,817],[772,2474],{"className":2475,"style":1004},[1003],[772,2477,982],{"className":2478},[1008],[772,2480],{"className":2481,"style":1004},[1003],[772,2483,2485,2489,2492,2521,2524,2556,2559,2562],{"className":2484},[807],[772,2486],{"className":2487,"style":2488},[811],"height:0.8352em;vertical-align:-0.0833em;",[772,2490,1681],{"className":2491},[816],[772,2493,2495,2498],{"className":2494},[816],[772,2496,1748],{"className":2497},[816],[772,2499,2501],{"className":2500},[1752],[772,2502,2504],{"className":2503},[1756],[772,2505,2507],{"className":2506},[1760],[772,2508,2510],{"className":2509,"style":1765},[1764],[772,2511,2512,2515],{"style":1768},[772,2513],{"className":2514,"style":1773},[1772],[772,2516,2518],{"className":2517},[1777,1778,1779,1780],[772,2519,1642],{"className":2520},[1125,1780],[772,2522,1748],{"className":2523},[816],[772,2525,2527,2530],{"className":2526},[816],[772,2528,1748],{"className":2529},[816],[772,2531,2533],{"className":2532},[1752],[772,2534,2536],{"className":2535},[1756],[772,2537,2539],{"className":2538},[1760],[772,2540,2542],{"className":2541,"style":1805},[1764],[772,2543,2544,2547],{"style":1768},[772,2545],{"className":2546,"style":1773},[1772],[772,2548,2550],{"className":2549},[1777,1778,1779,1780],[772,2551,2553],{"className":2552},[816,1780],[772,2554,1651],{"className":2555},[816,1780],[772,2557],{"className":2558,"style":1121},[1003],[772,2560,1091],{"className":2561},[1125],[772,2563],{"className":2564,"style":1121},[1003],[772,2566,2568,2571,2574,2603,2606,2639,2642,2645],{"className":2567},[807],[772,2569],{"className":2570,"style":1805},[811],[772,2572,1872],{"className":2573},[816],[772,2575,2577,2580],{"className":2576},[816],[772,2578,1845],{"className":2579},[816],[772,2581,2583],{"className":2582},[1752],[772,2584,2586],{"className":2585},[1756],[772,2587,2589],{"className":2588},[1760],[772,2590,2592],{"className":2591,"style":1765},[1764],[772,2593,2594,2597],{"style":1768},[772,2595],{"className":2596,"style":1773},[1772],[772,2598,2600],{"className":2599},[1777,1778,1779,1780],[772,2601,1642],{"className":2602},[1125,1780],[772,2604,1872],{"className":2605},[816],[772,2607,2609,2613],{"className":2608},[816],[772,2610,2612],{"className":2611},[816],"8",[772,2614,2616],{"className":2615},[1752],[772,2617,2619],{"className":2618},[1756],[772,2620,2622],{"className":2621},[1760],[772,2623,2625],{"className":2624,"style":1805},[1764],[772,2626,2627,2630],{"style":1768},[772,2628],{"className":2629,"style":1773},[1772],[772,2631,2633],{"className":2632},[1777,1778,1779,1780],[772,2634,2636],{"className":2635},[816,1780],[772,2637,1651],{"className":2638},[816,1780],[772,2640],{"className":2641,"style":1004},[1003],[772,2643,982],{"className":2644},[1008],[772,2646],{"className":2647,"style":1004},[1003],[772,2649,2651,2654,2658,2687,2690],{"className":2650},[807],[772,2652],{"className":2653,"style":1805},[811],[772,2655,2657],{"className":2656},[816],"33",[772,2659,2661,2664],{"className":2660},[816],[772,2662,1879],{"className":2663},[816],[772,2665,2667],{"className":2666},[1752],[772,2668,2670],{"className":2669},[1756],[772,2671,2673],{"className":2672},[1760],[772,2674,2676],{"className":2675,"style":1765},[1764],[772,2677,2678,2681],{"style":1768},[772,2679],{"className":2680,"style":1773},[1772],[772,2682,2684],{"className":2683},[1777,1778,1779,1780],[772,2685,1642],{"className":2686},[1125,1780],[772,2688,1845],{"className":2689},[816],[772,2691,2693,2696],{"className":2692},[816],[772,2694,1872],{"className":2695},[816],[772,2697,2699],{"className":2698},[1752],[772,2700,2702],{"className":2701},[1756],[772,2703,2705],{"className":2704},[1760],[772,2706,2708],{"className":2707,"style":1805},[1764],[772,2709,2710,2713],{"style":1768},[772,2711],{"className":2712,"style":1773},[1772],[772,2714,2716],{"className":2715},[1777,1778,1779,1780],[772,2717,2719],{"className":2718},[816,1780],[772,2720,1651],{"className":2721},[816,1780],[323,2723,2725],{"id":2724},"easting-and-northing","Easting and Northing",[11,2727,2728],{},"A common source of confusion for students and technicians is the difference between mathematical Cartesian coordinate systems and plane surveying grid networks.",[11,2730,2731,2732,2763,2764,2793,2794,2824,2825,2855],{},"In pure mathematics, the coordinate axes are defined as horizontal (",[772,2733,2735,2749],{"className":2734},[775],[772,2736,2738],{"className":2737},[779],[781,2739,2740],{"xmlns":783},[785,2741,2742,2747],{},[788,2743,2744],{},[791,2745,2746],{},"X",[795,2748,2746],{"encoding":797},[772,2750,2752],{"className":2751,"ariaHidden":803},[802],[772,2753,2755,2759],{"className":2754},[807],[772,2756],{"className":2757,"style":2758},[811],"height:0.6833em;",[772,2760,2746],{"className":2761,"style":2762},[816,817],"margin-right:0.0785em;",") and vertical (",[772,2765,2767,2781],{"className":2766},[775],[772,2768,2770],{"className":2769},[779],[781,2771,2772],{"xmlns":783},[785,2773,2774,2779],{},[788,2775,2776],{},[791,2777,2778],{},"Y",[795,2780,2778],{"encoding":797},[772,2782,2784],{"className":2783,"ariaHidden":803},[802],[772,2785,2787,2790],{"className":2786},[807],[772,2788],{"className":2789,"style":2758},[811],[772,2791,2778],{"className":2792,"style":1121},[816,817],"), with angles measured counterclockwise starting from the positive horizontal axis. In plane surveying, the horizontal grid axis represents Easting (",[772,2795,2797,2811],{"className":2796},[775],[772,2798,2800],{"className":2799},[779],[781,2801,2802],{"xmlns":783},[785,2803,2804,2809],{},[788,2805,2806],{},[791,2807,2808],{},"E",[795,2810,2808],{"encoding":797},[772,2812,2814],{"className":2813,"ariaHidden":803},[802],[772,2815,2817,2820],{"className":2816},[807],[772,2818],{"className":2819,"style":2758},[811],[772,2821,2808],{"className":2822,"style":2823},[816,817],"margin-right:0.0576em;",") and the vertical axis represents Northing (",[772,2826,2828,2842],{"className":2827},[775],[772,2829,2831],{"className":2830},[779],[781,2832,2833],{"xmlns":783},[785,2834,2835,2840],{},[788,2836,2837],{},[791,2838,2839],{},"N",[795,2841,2839],{"encoding":797},[772,2843,2845],{"className":2844,"ariaHidden":803},[802],[772,2846,2848,2851],{"className":2847},[807],[772,2849],{"className":2850,"style":2758},[811],[772,2852,2839],{"className":2853,"style":2854},[816,817],"margin-right:0.109em;","). Crucially, the reference direction of 0° is aligned with the vertical axis (North), and angles increase in a clockwise direction.",[2857,2858,2862],"pre",{"className":2859,"code":2861,"language":1709},[2860],"language-text","Mathematical Cartesian Grid            Plane Surveying Grid\n+Y (90°)                           +N (0° \u002F 360°)\n\n|                                  |\n|                                  |\n180° ---------+--------- 0° \u002F 360°     270° ---------+--------- 90°\n(-X) | (+X)                         (-E) | (+E)\n\n|                                  |\n-Y (270°)                              -N (180°)\n",[2863,2864,2861],"code",{"__ignoreMap":15},[11,2866,2867,2868,2896,2897,2925],{},"This 90° rotation and reversal of direction swap the standard trigonometric functions used to compute coordinate offsets. When writing and recording coordinates, plane surveying maintains the Cartesian convention of listing the horizontal component before the vertical component, resulting in the standard coordinate pair format: (",[772,2869,2871,2884],{"className":2870},[775],[772,2872,2874],{"className":2873},[779],[781,2875,2876],{"xmlns":783},[785,2877,2878,2882],{},[788,2879,2880],{},[791,2881,2808],{},[795,2883,2808],{"encoding":797},[772,2885,2887],{"className":2886,"ariaHidden":803},[802],[772,2888,2890,2893],{"className":2889},[807],[772,2891],{"className":2892,"style":2758},[811],[772,2894,2808],{"className":2895,"style":2823},[816,817],", ",[772,2898,2900,2913],{"className":2899},[775],[772,2901,2903],{"className":2902},[779],[781,2904,2905],{"xmlns":783},[785,2906,2907,2911],{},[788,2908,2909],{},[791,2910,2839],{},[795,2912,2839],{"encoding":797},[772,2914,2916],{"className":2915,"ariaHidden":803},[802],[772,2917,2919,2922],{"className":2918},[807],[772,2920],{"className":2921,"style":2758},[811],[772,2923,2839],{"className":2924,"style":2854},[816,817],").",[11,2927,2928],{},"The sign conventions of these grid coordinates are tied to the directional quadrant of the bearing vector. Eastings increase to the East and decrease to the West, while Northings increase to the North and decrease to the South. These relationships are summarized in the table below:",[851,2930,2931,3020],{},[854,2932,2933],{},[857,2934,2935,2937,2940,2943,2946,2984],{},[860,2936,862],{},[860,2938,2939],{},"General direction",[860,2941,2942],{},"Change in easting",[860,2944,2945],{},"Change in northing",[860,2947,2948,2949],{},"Sign of ",[772,2950,2952,2969],{"className":2951},[775],[772,2953,2955],{"className":2954},[779],[781,2956,2957],{"xmlns":783},[785,2958,2959,2966],{},[788,2960,2961,2964],{},[791,2962,2963],{"mathvariant":906},"Δ",[791,2965,2808],{},[795,2967,2968],{"encoding":797},"\\Delta E",[772,2970,2972],{"className":2971,"ariaHidden":803},[802],[772,2973,2975,2978,2981],{"className":2974},[807],[772,2976],{"className":2977,"style":2758},[811],[772,2979,2963],{"className":2980},[816],[772,2982,2808],{"className":2983,"style":2823},[816,817],[860,2985,2948,2986],{},[772,2987,2989,3005],{"className":2988},[775],[772,2990,2992],{"className":2991},[779],[781,2993,2994],{"xmlns":783},[785,2995,2996,3002],{},[788,2997,2998,3000],{},[791,2999,2963],{"mathvariant":906},[791,3001,2839],{},[795,3003,3004],{"encoding":797},"\\Delta N",[772,3006,3008],{"className":3007,"ariaHidden":803},[802],[772,3009,3011,3014,3017],{"className":3010},[807],[772,3012],{"className":3013,"style":2758},[811],[772,3015,2963],{"className":3016},[816],[772,3018,2839],{"className":3019,"style":2854},[816,817],[876,3021,3022,3041,3059,3076],{},[857,3023,3024,3027,3030,3033,3036,3039],{},[881,3025,3026],{},"I",[881,3028,3029],{},"Northeast (NE)",[881,3031,3032],{},"Eastward (departure)",[881,3034,3035],{},"Northward (latitude)",[881,3037,3038],{},"Positive (+)",[881,3040,3038],{},[857,3042,3043,3046,3049,3051,3054,3056],{},[881,3044,3045],{},"II",[881,3047,3048],{},"Southeast (SE)",[881,3050,3032],{},[881,3052,3053],{},"Southward (latitude)",[881,3055,3038],{},[881,3057,3058],{},"Negative (−)",[857,3060,3061,3064,3067,3070,3072,3074],{},[881,3062,3063],{},"III",[881,3065,3066],{},"Southwest (SW)",[881,3068,3069],{},"Westward (departure)",[881,3071,3053],{},[881,3073,3058],{},[881,3075,3058],{},[857,3077,3078,3081,3084,3086,3088,3090],{},[881,3079,3080],{},"IV",[881,3082,3083],{},"Northwest (NW)",[881,3085,3069],{},[881,3087,3035],{},[881,3089,3058],{},[881,3091,3038],{},[323,3093,3095],{"id":3094},"polar-to-rectangular-conversion","Polar to Rectangular Conversion",[11,3097,3098,3099,2896,3132,3165,3166,3195,3196,3224],{},"The forward coordinate calculation, known as a \"polar to rectangular\" conversion, computes the coordinate differences (",[772,3100,3102,3117],{"className":3101},[775],[772,3103,3105],{"className":3104},[779],[781,3106,3107],{"xmlns":783},[785,3108,3109,3115],{},[788,3110,3111,3113],{},[791,3112,2963],{"mathvariant":906},[791,3114,2808],{},[795,3116,2968],{"encoding":797},[772,3118,3120],{"className":3119,"ariaHidden":803},[802],[772,3121,3123,3126,3129],{"className":3122},[807],[772,3124],{"className":3125,"style":2758},[811],[772,3127,2963],{"className":3128},[816],[772,3130,2808],{"className":3131,"style":2823},[816,817],[772,3133,3135,3150],{"className":3134},[775],[772,3136,3138],{"className":3137},[779],[781,3139,3140],{"xmlns":783},[785,3141,3142,3148],{},[788,3143,3144,3146],{},[791,3145,2963],{"mathvariant":906},[791,3147,2839],{},[795,3149,3004],{"encoding":797},[772,3151,3153],{"className":3152,"ariaHidden":803},[802],[772,3154,3156,3159,3162],{"className":3155},[807],[772,3157],{"className":3158,"style":2758},[811],[772,3160,2963],{"className":3161},[816],[772,3163,2839],{"className":3164,"style":2854},[816,817],") from a known horizontal distance (",[772,3167,3169,3183],{"className":3168},[775],[772,3170,3172],{"className":3171},[779],[781,3173,3174],{"xmlns":783},[785,3175,3176,3181],{},[788,3177,3178],{},[791,3179,3180],{},"D",[795,3182,3180],{"encoding":797},[772,3184,3186],{"className":3185,"ariaHidden":803},[802],[772,3187,3189,3192],{"className":3188},[807],[772,3190],{"className":3191,"style":2758},[811],[772,3193,3180],{"className":3194,"style":818},[816,817],") and a whole-circle bearing (",[772,3197,3199,3212],{"className":3198},[775],[772,3200,3202],{"className":3201},[779],[781,3203,3204],{"xmlns":783},[785,3205,3206,3210],{},[788,3207,3208],{},[791,3209,793],{},[795,3211,798],{"encoding":797},[772,3213,3215],{"className":3214,"ariaHidden":803},[802],[772,3216,3218,3221],{"className":3217},[807],[772,3219],{"className":3220,"style":812},[811],[772,3222,793],{"className":3223,"style":818},[816,817],"). These coordinate offsets are also referred to as \"departures\" (change in Easting) and \"latitudes\" (change in Northing).",[11,3226,3227,3305],{},[772,3228,3230,3258],{"className":3229},[775],[772,3231,3233],{"className":3232},[779],[781,3234,3235],{"xmlns":783},[785,3236,3237,3255],{},[788,3238,3239,3241,3243,3245,3247,3250,3253],{},[791,3240,2963],{"mathvariant":906},[791,3242,2808],{},[980,3244,982],{},[791,3246,3180],{},[791,3248,3249],{},"sin",[980,3251,3252],{},"⁡",[791,3254,793],{},[795,3256,3257],{"encoding":797},"\\Delta E=D\\sin\\theta",[772,3259,3261,3282],{"className":3260,"ariaHidden":803},[802],[772,3262,3264,3267,3270,3273,3276,3279],{"className":3263},[807],[772,3265],{"className":3266,"style":2758},[811],[772,3268,2963],{"className":3269},[816],[772,3271,2808],{"className":3272,"style":2823},[816,817],[772,3274],{"className":3275,"style":1004},[1003],[772,3277,982],{"className":3278},[1008],[772,3280],{"className":3281,"style":1004},[1003],[772,3283,3285,3288,3291,3295,3299,3302],{"className":3284},[807],[772,3286],{"className":3287,"style":812},[811],[772,3289,3180],{"className":3290,"style":818},[816,817],[772,3292],{"className":3293,"style":3294},[1003],"margin-right:0.1667em;",[772,3296,3249],{"className":3297},[3298],"mop",[772,3300],{"className":3301,"style":3294},[1003],[772,3303,793],{"className":3304,"style":818},[816,817],[772,3306,3308,3335],{"className":3307},[775],[772,3309,3311],{"className":3310},[779],[781,3312,3313],{"xmlns":783},[785,3314,3315,3332],{},[788,3316,3317,3319,3321,3323,3325,3328,3330],{},[791,3318,2963],{"mathvariant":906},[791,3320,2839],{},[980,3322,982],{},[791,3324,3180],{},[791,3326,3327],{},"cos",[980,3329,3252],{},[791,3331,793],{},[795,3333,3334],{"encoding":797},"\\Delta N=D\\cos\\theta",[772,3336,3338,3359],{"className":3337,"ariaHidden":803},[802],[772,3339,3341,3344,3347,3350,3353,3356],{"className":3340},[807],[772,3342],{"className":3343,"style":2758},[811],[772,3345,2963],{"className":3346},[816],[772,3348,2839],{"className":3349,"style":2854},[816,817],[772,3351],{"className":3352,"style":1004},[1003],[772,3354,982],{"className":3355},[1008],[772,3357],{"className":3358,"style":1004},[1003],[772,3360,3362,3365,3368,3371,3374,3377],{"className":3361},[807],[772,3363],{"className":3364,"style":812},[811],[772,3366,3180],{"className":3367,"style":818},[816,817],[772,3369],{"className":3370,"style":3294},[1003],[772,3372,3327],{"className":3373},[3298],[772,3375],{"className":3376,"style":3294},[1003],[772,3378,793],{"className":3379,"style":818},[816,817],[11,3381,3382,3383,3411,3412,3440,3441,3474,3475,3508],{},"These equations use standard trigonometric functions where the whole-circle bearing (",[772,3384,3386,3399],{"className":3385},[775],[772,3387,3389],{"className":3388},[779],[781,3390,3391],{"xmlns":783},[785,3392,3393,3397],{},[788,3394,3395],{},[791,3396,793],{},[795,3398,798],{"encoding":797},[772,3400,3402],{"className":3401,"ariaHidden":803},[802],[772,3403,3405,3408],{"className":3404},[807],[772,3406],{"className":3407,"style":812},[811],[772,3409,793],{"className":3410,"style":818},[816,817],") is measured clockwise from North. If ",[772,3413,3415,3428],{"className":3414},[775],[772,3416,3418],{"className":3417},[779],[781,3419,3420],{"xmlns":783},[785,3421,3422,3426],{},[788,3423,3424],{},[791,3425,793],{},[795,3427,798],{"encoding":797},[772,3429,3431],{"className":3430,"ariaHidden":803},[802],[772,3432,3434,3437],{"className":3433},[807],[772,3435],{"className":3436,"style":812},[811],[772,3438,793],{"className":3439,"style":818},[816,817]," is input directly as a whole-circle bearing, the trigonometric functions automatically output the correct signs (positive or negative) for ",[772,3442,3444,3459],{"className":3443},[775],[772,3445,3447],{"className":3446},[779],[781,3448,3449],{"xmlns":783},[785,3450,3451,3457],{},[788,3452,3453,3455],{},[791,3454,2963],{"mathvariant":906},[791,3456,2808],{},[795,3458,2968],{"encoding":797},[772,3460,3462],{"className":3461,"ariaHidden":803},[802],[772,3463,3465,3468,3471],{"className":3464},[807],[772,3466],{"className":3467,"style":2758},[811],[772,3469,2963],{"className":3470},[816],[772,3472,2808],{"className":3473,"style":2823},[816,817]," and ",[772,3476,3478,3493],{"className":3477},[775],[772,3479,3481],{"className":3480},[779],[781,3482,3483],{"xmlns":783},[785,3484,3485,3491],{},[788,3486,3487,3489],{},[791,3488,2963],{"mathvariant":906},[791,3490,2839],{},[795,3492,3004],{"encoding":797},[772,3494,3496],{"className":3495,"ariaHidden":803},[802],[772,3497,3499,3502,3505],{"className":3498},[807],[772,3500],{"className":3501,"style":2758},[811],[772,3503,2963],{"className":3504},[816],[772,3506,2839],{"className":3507,"style":2854},[816,817],", regardless of which quadrant the line falls in.",[11,3510,3511,3512,2896,3590,3661,3662,2896,3732,3802],{},"Once these coordinate differences are calculated, the coordinates of the target point (",[772,3513,3515,3534],{"className":3514},[775],[772,3516,3518],{"className":3517},[779],[781,3519,3520],{"xmlns":783},[785,3521,3522,3531],{},[788,3523,3524],{},[3525,3526,3527,3529],"msub",{},[791,3528,2808],{},[1084,3530,1872],{},[795,3532,3533],{"encoding":797},"E_2",[772,3535,3537],{"className":3536,"ariaHidden":803},[802],[772,3538,3540,3544],{"className":3539},[807],[772,3541],{"className":3542,"style":3543},[811],"height:0.8333em;vertical-align:-0.15em;",[772,3545,3547,3550],{"className":3546},[816],[772,3548,2808],{"className":3549,"style":2823},[816,817],[772,3551,3553],{"className":3552},[1752],[772,3554,3557,3581],{"className":3555},[1756,3556],"vlist-t2",[772,3558,3560,3576],{"className":3559},[1760],[772,3561,3564],{"className":3562,"style":3563},[1764],"height:0.3011em;",[772,3565,3567,3570],{"style":3566},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[772,3568],{"className":3569,"style":1773},[1772],[772,3571,3573],{"className":3572},[1777,1778,1779,1780],[772,3574,1872],{"className":3575},[816,1780],[772,3577,3580],{"className":3578},[3579],"vlist-s","​",[772,3582,3584],{"className":3583},[1760],[772,3585,3588],{"className":3586,"style":3587},[1764],"height:0.15em;",[772,3589],{},[772,3591,3593,3611],{"className":3592},[775],[772,3594,3596],{"className":3595},[779],[781,3597,3598],{"xmlns":783},[785,3599,3600,3608],{},[788,3601,3602],{},[3525,3603,3604,3606],{},[791,3605,2839],{},[1084,3607,1872],{},[795,3609,3610],{"encoding":797},"N_2",[772,3612,3614],{"className":3613,"ariaHidden":803},[802],[772,3615,3617,3620],{"className":3616},[807],[772,3618],{"className":3619,"style":3543},[811],[772,3621,3623,3626],{"className":3622},[816],[772,3624,2839],{"className":3625,"style":2854},[816,817],[772,3627,3629],{"className":3628},[1752],[772,3630,3632,3653],{"className":3631},[1756,3556],[772,3633,3635,3650],{"className":3634},[1760],[772,3636,3638],{"className":3637,"style":3563},[1764],[772,3639,3641,3644],{"style":3640},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[772,3642],{"className":3643,"style":1773},[1772],[772,3645,3647],{"className":3646},[1777,1778,1779,1780],[772,3648,1872],{"className":3649},[816,1780],[772,3651,3580],{"className":3652},[3579],[772,3654,3656],{"className":3655},[1760],[772,3657,3659],{"className":3658,"style":3587},[1764],[772,3660],{},") are determined by adding the offsets to the starting point's coordinates (",[772,3663,3665,3683],{"className":3664},[775],[772,3666,3668],{"className":3667},[779],[781,3669,3670],{"xmlns":783},[785,3671,3672,3680],{},[788,3673,3674],{},[3525,3675,3676,3678],{},[791,3677,2808],{},[1084,3679,2170],{},[795,3681,3682],{"encoding":797},"E_1",[772,3684,3686],{"className":3685,"ariaHidden":803},[802],[772,3687,3689,3692],{"className":3688},[807],[772,3690],{"className":3691,"style":3543},[811],[772,3693,3695,3698],{"className":3694},[816],[772,3696,2808],{"className":3697,"style":2823},[816,817],[772,3699,3701],{"className":3700},[1752],[772,3702,3704,3724],{"className":3703},[1756,3556],[772,3705,3707,3721],{"className":3706},[1760],[772,3708,3710],{"className":3709,"style":3563},[1764],[772,3711,3712,3715],{"style":3566},[772,3713],{"className":3714,"style":1773},[1772],[772,3716,3718],{"className":3717},[1777,1778,1779,1780],[772,3719,2170],{"className":3720},[816,1780],[772,3722,3580],{"className":3723},[3579],[772,3725,3727],{"className":3726},[1760],[772,3728,3730],{"className":3729,"style":3587},[1764],[772,3731],{},[772,3733,3735,3753],{"className":3734},[775],[772,3736,3738],{"className":3737},[779],[781,3739,3740],{"xmlns":783},[785,3741,3742,3750],{},[788,3743,3744],{},[3525,3745,3746,3748],{},[791,3747,2839],{},[1084,3749,2170],{},[795,3751,3752],{"encoding":797},"N_1",[772,3754,3756],{"className":3755,"ariaHidden":803},[802],[772,3757,3759,3762],{"className":3758},[807],[772,3760],{"className":3761,"style":3543},[811],[772,3763,3765,3768],{"className":3764},[816],[772,3766,2839],{"className":3767,"style":2854},[816,817],[772,3769,3771],{"className":3770},[1752],[772,3772,3774,3794],{"className":3773},[1756,3556],[772,3775,3777,3791],{"className":3776},[1760],[772,3778,3780],{"className":3779,"style":3563},[1764],[772,3781,3782,3785],{"style":3640},[772,3783],{"className":3784,"style":1773},[1772],[772,3786,3788],{"className":3787},[1777,1778,1779,1780],[772,3789,2170],{"className":3790},[816,1780],[772,3792,3580],{"className":3793},[3579],[772,3795,3797],{"className":3796},[1760],[772,3798,3800],{"className":3799,"style":3587},[1764],[772,3801],{},"):",[11,3804,3805,3965],{},[772,3806,3808,3840],{"className":3807},[775],[772,3809,3811],{"className":3810},[779],[781,3812,3813],{"xmlns":783},[785,3814,3815,3837],{},[788,3816,3817,3823,3825,3831,3833,3835],{},[3525,3818,3819,3821],{},[791,3820,2808],{},[1084,3822,1872],{},[980,3824,982],{},[3525,3826,3827,3829],{},[791,3828,2808],{},[1084,3830,2170],{},[980,3832,1353],{},[791,3834,2963],{"mathvariant":906},[791,3836,2808],{},[795,3838,3839],{"encoding":797},"E_2=E_1+\\Delta E",[772,3841,3843,3898,3953],{"className":3842,"ariaHidden":803},[802],[772,3844,3846,3849,3889,3892,3895],{"className":3845},[807],[772,3847],{"className":3848,"style":3543},[811],[772,3850,3852,3855],{"className":3851},[816],[772,3853,2808],{"className":3854,"style":2823},[816,817],[772,3856,3858],{"className":3857},[1752],[772,3859,3861,3881],{"className":3860},[1756,3556],[772,3862,3864,3878],{"className":3863},[1760],[772,3865,3867],{"className":3866,"style":3563},[1764],[772,3868,3869,3872],{"style":3566},[772,3870],{"className":3871,"style":1773},[1772],[772,3873,3875],{"className":3874},[1777,1778,1779,1780],[772,3876,1872],{"className":3877},[816,1780],[772,3879,3580],{"className":3880},[3579],[772,3882,3884],{"className":3883},[1760],[772,3885,3887],{"className":3886,"style":3587},[1764],[772,3888],{},[772,3890],{"className":3891,"style":1004},[1003],[772,3893,982],{"className":3894},[1008],[772,3896],{"className":3897,"style":1004},[1003],[772,3899,3901,3904,3944,3947,3950],{"className":3900},[807],[772,3902],{"className":3903,"style":3543},[811],[772,3905,3907,3910],{"className":3906},[816],[772,3908,2808],{"className":3909,"style":2823},[816,817],[772,3911,3913],{"className":3912},[1752],[772,3914,3916,3936],{"className":3915},[1756,3556],[772,3917,3919,3933],{"className":3918},[1760],[772,3920,3922],{"className":3921,"style":3563},[1764],[772,3923,3924,3927],{"style":3566},[772,3925],{"className":3926,"style":1773},[1772],[772,3928,3930],{"className":3929},[1777,1778,1779,1780],[772,3931,2170],{"className":3932},[816,1780],[772,3934,3580],{"className":3935},[3579],[772,3937,3939],{"className":3938},[1760],[772,3940,3942],{"className":3941,"style":3587},[1764],[772,3943],{},[772,3945],{"className":3946,"style":1121},[1003],[772,3948,1353],{"className":3949},[1125],[772,3951],{"className":3952,"style":1121},[1003],[772,3954,3956,3959,3962],{"className":3955},[807],[772,3957],{"className":3958,"style":2758},[811],[772,3960,2963],{"className":3961},[816],[772,3963,2808],{"className":3964,"style":2823},[816,817],[772,3966,3968,4000],{"className":3967},[775],[772,3969,3971],{"className":3970},[779],[781,3972,3973],{"xmlns":783},[785,3974,3975,3997],{},[788,3976,3977,3983,3985,3991,3993,3995],{},[3525,3978,3979,3981],{},[791,3980,2839],{},[1084,3982,1872],{},[980,3984,982],{},[3525,3986,3987,3989],{},[791,3988,2839],{},[1084,3990,2170],{},[980,3992,1353],{},[791,3994,2963],{"mathvariant":906},[791,3996,2839],{},[795,3998,3999],{"encoding":797},"N_2=N_1+\\Delta N",[772,4001,4003,4058,4113],{"className":4002,"ariaHidden":803},[802],[772,4004,4006,4009,4049,4052,4055],{"className":4005},[807],[772,4007],{"className":4008,"style":3543},[811],[772,4010,4012,4015],{"className":4011},[816],[772,4013,2839],{"className":4014,"style":2854},[816,817],[772,4016,4018],{"className":4017},[1752],[772,4019,4021,4041],{"className":4020},[1756,3556],[772,4022,4024,4038],{"className":4023},[1760],[772,4025,4027],{"className":4026,"style":3563},[1764],[772,4028,4029,4032],{"style":3640},[772,4030],{"className":4031,"style":1773},[1772],[772,4033,4035],{"className":4034},[1777,1778,1779,1780],[772,4036,1872],{"className":4037},[816,1780],[772,4039,3580],{"className":4040},[3579],[772,4042,4044],{"className":4043},[1760],[772,4045,4047],{"className":4046,"style":3587},[1764],[772,4048],{},[772,4050],{"className":4051,"style":1004},[1003],[772,4053,982],{"className":4054},[1008],[772,4056],{"className":4057,"style":1004},[1003],[772,4059,4061,4064,4104,4107,4110],{"className":4060},[807],[772,4062],{"className":4063,"style":3543},[811],[772,4065,4067,4070],{"className":4066},[816],[772,4068,2839],{"className":4069,"style":2854},[816,817],[772,4071,4073],{"className":4072},[1752],[772,4074,4076,4096],{"className":4075},[1756,3556],[772,4077,4079,4093],{"className":4078},[1760],[772,4080,4082],{"className":4081,"style":3563},[1764],[772,4083,4084,4087],{"style":3640},[772,4085],{"className":4086,"style":1773},[1772],[772,4088,4090],{"className":4089},[1777,1778,1779,1780],[772,4091,2170],{"className":4092},[816,1780],[772,4094,3580],{"className":4095},[3579],[772,4097,4099],{"className":4098},[1760],[772,4100,4102],{"className":4101,"style":3587},[1764],[772,4103],{},[772,4105],{"className":4106,"style":1121},[1003],[772,4108,1353],{"className":4109},[1125],[772,4111],{"className":4112,"style":1121},[1003],[772,4114,4116,4119,4122],{"className":4115},[807],[772,4117],{"className":4118,"style":2758},[811],[772,4120,2963],{"className":4121},[816],[772,4123,2839],{"className":4124,"style":2854},[816,817],[11,4126,4127],{},"These forward formulas can be summarized as follows:",[335,4129,4130],{},[338,4131,4132,4135],{},[318,4133,4134],{},"Easting offset (departure)",":",[11,4137,4138],{},[772,4139,4141,4166],{"className":4140},[775],[772,4142,4144],{"className":4143},[779],[781,4145,4146],{"xmlns":783},[785,4147,4148,4164],{},[788,4149,4150,4152,4154,4156,4158,4160,4162],{},[791,4151,2963],{"mathvariant":906},[791,4153,2808],{},[980,4155,982],{},[791,4157,3180],{},[791,4159,3249],{},[980,4161,3252],{},[791,4163,793],{},[795,4165,3257],{"encoding":797},[772,4167,4169,4190],{"className":4168,"ariaHidden":803},[802],[772,4170,4172,4175,4178,4181,4184,4187],{"className":4171},[807],[772,4173],{"className":4174,"style":2758},[811],[772,4176,2963],{"className":4177},[816],[772,4179,2808],{"className":4180,"style":2823},[816,817],[772,4182],{"className":4183,"style":1004},[1003],[772,4185,982],{"className":4186},[1008],[772,4188],{"className":4189,"style":1004},[1003],[772,4191,4193,4196,4199,4202,4205,4208],{"className":4192},[807],[772,4194],{"className":4195,"style":812},[811],[772,4197,3180],{"className":4198,"style":818},[816,817],[772,4200],{"className":4201,"style":3294},[1003],[772,4203,3249],{"className":4204},[3298],[772,4206],{"className":4207,"style":3294},[1003],[772,4209,793],{"className":4210,"style":818},[816,817],[335,4212,4213],{},[338,4214,4215,4135],{},[318,4216,4217],{},"Northing offset (latitude)",[11,4219,4220],{},[772,4221,4223,4248],{"className":4222},[775],[772,4224,4226],{"className":4225},[779],[781,4227,4228],{"xmlns":783},[785,4229,4230,4246],{},[788,4231,4232,4234,4236,4238,4240,4242,4244],{},[791,4233,2963],{"mathvariant":906},[791,4235,2839],{},[980,4237,982],{},[791,4239,3180],{},[791,4241,3327],{},[980,4243,3252],{},[791,4245,793],{},[795,4247,3334],{"encoding":797},[772,4249,4251,4272],{"className":4250,"ariaHidden":803},[802],[772,4252,4254,4257,4260,4263,4266,4269],{"className":4253},[807],[772,4255],{"className":4256,"style":2758},[811],[772,4258,2963],{"className":4259},[816],[772,4261,2839],{"className":4262,"style":2854},[816,817],[772,4264],{"className":4265,"style":1004},[1003],[772,4267,982],{"className":4268},[1008],[772,4270],{"className":4271,"style":1004},[1003],[772,4273,4275,4278,4281,4284,4287,4290],{"className":4274},[807],[772,4276],{"className":4277,"style":812},[811],[772,4279,3180],{"className":4280,"style":818},[816,817],[772,4282],{"className":4283,"style":3294},[1003],[772,4285,3327],{"className":4286},[3298],[772,4288],{"className":4289,"style":3294},[1003],[772,4291,793],{"className":4292,"style":818},[816,817],[335,4294,4295],{},[338,4296,4297,4135],{},[318,4298,4299],{},"Target easting coordinate",[11,4301,4302],{},[772,4303,4305,4336],{"className":4304},[775],[772,4306,4308],{"className":4307},[779],[781,4309,4310],{"xmlns":783},[785,4311,4312,4334],{},[788,4313,4314,4320,4322,4328,4330,4332],{},[3525,4315,4316,4318],{},[791,4317,2808],{},[1084,4319,1872],{},[980,4321,982],{},[3525,4323,4324,4326],{},[791,4325,2808],{},[1084,4327,2170],{},[980,4329,1353],{},[791,4331,2963],{"mathvariant":906},[791,4333,2808],{},[795,4335,3839],{"encoding":797},[772,4337,4339,4394,4449],{"className":4338,"ariaHidden":803},[802],[772,4340,4342,4345,4385,4388,4391],{"className":4341},[807],[772,4343],{"className":4344,"style":3543},[811],[772,4346,4348,4351],{"className":4347},[816],[772,4349,2808],{"className":4350,"style":2823},[816,817],[772,4352,4354],{"className":4353},[1752],[772,4355,4357,4377],{"className":4356},[1756,3556],[772,4358,4360,4374],{"className":4359},[1760],[772,4361,4363],{"className":4362,"style":3563},[1764],[772,4364,4365,4368],{"style":3566},[772,4366],{"className":4367,"style":1773},[1772],[772,4369,4371],{"className":4370},[1777,1778,1779,1780],[772,4372,1872],{"className":4373},[816,1780],[772,4375,3580],{"className":4376},[3579],[772,4378,4380],{"className":4379},[1760],[772,4381,4383],{"className":4382,"style":3587},[1764],[772,4384],{},[772,4386],{"className":4387,"style":1004},[1003],[772,4389,982],{"className":4390},[1008],[772,4392],{"className":4393,"style":1004},[1003],[772,4395,4397,4400,4440,4443,4446],{"className":4396},[807],[772,4398],{"className":4399,"style":3543},[811],[772,4401,4403,4406],{"className":4402},[816],[772,4404,2808],{"className":4405,"style":2823},[816,817],[772,4407,4409],{"className":4408},[1752],[772,4410,4412,4432],{"className":4411},[1756,3556],[772,4413,4415,4429],{"className":4414},[1760],[772,4416,4418],{"className":4417,"style":3563},[1764],[772,4419,4420,4423],{"style":3566},[772,4421],{"className":4422,"style":1773},[1772],[772,4424,4426],{"className":4425},[1777,1778,1779,1780],[772,4427,2170],{"className":4428},[816,1780],[772,4430,3580],{"className":4431},[3579],[772,4433,4435],{"className":4434},[1760],[772,4436,4438],{"className":4437,"style":3587},[1764],[772,4439],{},[772,4441],{"className":4442,"style":1121},[1003],[772,4444,1353],{"className":4445},[1125],[772,4447],{"className":4448,"style":1121},[1003],[772,4450,4452,4455,4458],{"className":4451},[807],[772,4453],{"className":4454,"style":2758},[811],[772,4456,2963],{"className":4457},[816],[772,4459,2808],{"className":4460,"style":2823},[816,817],[335,4462,4463],{},[338,4464,4465,4135],{},[318,4466,4467],{},"Target northing coordinate",[11,4469,4470],{},[772,4471,4473,4504],{"className":4472},[775],[772,4474,4476],{"className":4475},[779],[781,4477,4478],{"xmlns":783},[785,4479,4480,4502],{},[788,4481,4482,4488,4490,4496,4498,4500],{},[3525,4483,4484,4486],{},[791,4485,2839],{},[1084,4487,1872],{},[980,4489,982],{},[3525,4491,4492,4494],{},[791,4493,2839],{},[1084,4495,2170],{},[980,4497,1353],{},[791,4499,2963],{"mathvariant":906},[791,4501,2839],{},[795,4503,3999],{"encoding":797},[772,4505,4507,4562,4617],{"className":4506,"ariaHidden":803},[802],[772,4508,4510,4513,4553,4556,4559],{"className":4509},[807],[772,4511],{"className":4512,"style":3543},[811],[772,4514,4516,4519],{"className":4515},[816],[772,4517,2839],{"className":4518,"style":2854},[816,817],[772,4520,4522],{"className":4521},[1752],[772,4523,4525,4545],{"className":4524},[1756,3556],[772,4526,4528,4542],{"className":4527},[1760],[772,4529,4531],{"className":4530,"style":3563},[1764],[772,4532,4533,4536],{"style":3640},[772,4534],{"className":4535,"style":1773},[1772],[772,4537,4539],{"className":4538},[1777,1778,1779,1780],[772,4540,1872],{"className":4541},[816,1780],[772,4543,3580],{"className":4544},[3579],[772,4546,4548],{"className":4547},[1760],[772,4549,4551],{"className":4550,"style":3587},[1764],[772,4552],{},[772,4554],{"className":4555,"style":1004},[1003],[772,4557,982],{"className":4558},[1008],[772,4560],{"className":4561,"style":1004},[1003],[772,4563,4565,4568,4608,4611,4614],{"className":4564},[807],[772,4566],{"className":4567,"style":3543},[811],[772,4569,4571,4574],{"className":4570},[816],[772,4572,2839],{"className":4573,"style":2854},[816,817],[772,4575,4577],{"className":4576},[1752],[772,4578,4580,4600],{"className":4579},[1756,3556],[772,4581,4583,4597],{"className":4582},[1760],[772,4584,4586],{"className":4585,"style":3563},[1764],[772,4587,4588,4591],{"style":3640},[772,4589],{"className":4590,"style":1773},[1772],[772,4592,4594],{"className":4593},[1777,1778,1779,1780],[772,4595,2170],{"className":4596},[816,1780],[772,4598,3580],{"className":4599},[3579],[772,4601,4603],{"className":4602},[1760],[772,4604,4606],{"className":4605,"style":3587},[1764],[772,4607],{},[772,4609],{"className":4610,"style":1121},[1003],[772,4612,1353],{"className":4613},[1125],[772,4615],{"className":4616,"style":1121},[1003],[772,4618,4620,4623,4626],{"className":4619},[807],[772,4621],{"className":4622,"style":2758},[811],[772,4624,2963],{"className":4625},[816],[772,4627,2839],{"className":4628,"style":2854},[816,817],[323,4630,4632],{"id":4631},"rectangular-to-polar-conversion","Rectangular to Polar Conversion",[11,4634,4635,4636,4664,4665,4693,4694,2896,4763,4832,4833,2896,4902,2925],{},"The inverse coordinate calculation, or \"rectangular to polar\" conversion, determines the horizontal distance (",[772,4637,4639,4652],{"className":4638},[775],[772,4640,4642],{"className":4641},[779],[781,4643,4644],{"xmlns":783},[785,4645,4646,4650],{},[788,4647,4648],{},[791,4649,3180],{},[795,4651,3180],{"encoding":797},[772,4653,4655],{"className":4654,"ariaHidden":803},[802],[772,4656,4658,4661],{"className":4657},[807],[772,4659],{"className":4660,"style":2758},[811],[772,4662,3180],{"className":4663,"style":818},[816,817],") and whole-circle bearing (",[772,4666,4668,4681],{"className":4667},[775],[772,4669,4671],{"className":4670},[779],[781,4672,4673],{"xmlns":783},[785,4674,4675,4679],{},[788,4676,4677],{},[791,4678,793],{},[795,4680,798],{"encoding":797},[772,4682,4684],{"className":4683,"ariaHidden":803},[802],[772,4685,4687,4690],{"className":4686},[807],[772,4688],{"className":4689,"style":812},[811],[772,4691,793],{"className":4692,"style":818},[816,817],") between two points with known coordinates: Station 1 (",[772,4695,4697,4714],{"className":4696},[775],[772,4698,4700],{"className":4699},[779],[781,4701,4702],{"xmlns":783},[785,4703,4704,4712],{},[788,4705,4706],{},[3525,4707,4708,4710],{},[791,4709,2808],{},[1084,4711,2170],{},[795,4713,3682],{"encoding":797},[772,4715,4717],{"className":4716,"ariaHidden":803},[802],[772,4718,4720,4723],{"className":4719},[807],[772,4721],{"className":4722,"style":3543},[811],[772,4724,4726,4729],{"className":4725},[816],[772,4727,2808],{"className":4728,"style":2823},[816,817],[772,4730,4732],{"className":4731},[1752],[772,4733,4735,4755],{"className":4734},[1756,3556],[772,4736,4738,4752],{"className":4737},[1760],[772,4739,4741],{"className":4740,"style":3563},[1764],[772,4742,4743,4746],{"style":3566},[772,4744],{"className":4745,"style":1773},[1772],[772,4747,4749],{"className":4748},[1777,1778,1779,1780],[772,4750,2170],{"className":4751},[816,1780],[772,4753,3580],{"className":4754},[3579],[772,4756,4758],{"className":4757},[1760],[772,4759,4761],{"className":4760,"style":3587},[1764],[772,4762],{},[772,4764,4766,4783],{"className":4765},[775],[772,4767,4769],{"className":4768},[779],[781,4770,4771],{"xmlns":783},[785,4772,4773,4781],{},[788,4774,4775],{},[3525,4776,4777,4779],{},[791,4778,2839],{},[1084,4780,2170],{},[795,4782,3752],{"encoding":797},[772,4784,4786],{"className":4785,"ariaHidden":803},[802],[772,4787,4789,4792],{"className":4788},[807],[772,4790],{"className":4791,"style":3543},[811],[772,4793,4795,4798],{"className":4794},[816],[772,4796,2839],{"className":4797,"style":2854},[816,817],[772,4799,4801],{"className":4800},[1752],[772,4802,4804,4824],{"className":4803},[1756,3556],[772,4805,4807,4821],{"className":4806},[1760],[772,4808,4810],{"className":4809,"style":3563},[1764],[772,4811,4812,4815],{"style":3640},[772,4813],{"className":4814,"style":1773},[1772],[772,4816,4818],{"className":4817},[1777,1778,1779,1780],[772,4819,2170],{"className":4820},[816,1780],[772,4822,3580],{"className":4823},[3579],[772,4825,4827],{"className":4826},[1760],[772,4828,4830],{"className":4829,"style":3587},[1764],[772,4831],{},") and Station 2 (",[772,4834,4836,4853],{"className":4835},[775],[772,4837,4839],{"className":4838},[779],[781,4840,4841],{"xmlns":783},[785,4842,4843,4851],{},[788,4844,4845],{},[3525,4846,4847,4849],{},[791,4848,2808],{},[1084,4850,1872],{},[795,4852,3533],{"encoding":797},[772,4854,4856],{"className":4855,"ariaHidden":803},[802],[772,4857,4859,4862],{"className":4858},[807],[772,4860],{"className":4861,"style":3543},[811],[772,4863,4865,4868],{"className":4864},[816],[772,4866,2808],{"className":4867,"style":2823},[816,817],[772,4869,4871],{"className":4870},[1752],[772,4872,4874,4894],{"className":4873},[1756,3556],[772,4875,4877,4891],{"className":4876},[1760],[772,4878,4880],{"className":4879,"style":3563},[1764],[772,4881,4882,4885],{"style":3566},[772,4883],{"className":4884,"style":1773},[1772],[772,4886,4888],{"className":4887},[1777,1778,1779,1780],[772,4889,1872],{"className":4890},[816,1780],[772,4892,3580],{"className":4893},[3579],[772,4895,4897],{"className":4896},[1760],[772,4898,4900],{"className":4899,"style":3587},[1764],[772,4901],{},[772,4903,4905,4922],{"className":4904},[775],[772,4906,4908],{"className":4907},[779],[781,4909,4910],{"xmlns":783},[785,4911,4912,4920],{},[788,4913,4914],{},[3525,4915,4916,4918],{},[791,4917,2839],{},[1084,4919,1872],{},[795,4921,3610],{"encoding":797},[772,4923,4925],{"className":4924,"ariaHidden":803},[802],[772,4926,4928,4931],{"className":4927},[807],[772,4929],{"className":4930,"style":3543},[811],[772,4932,4934,4937],{"className":4933},[816],[772,4935,2839],{"className":4936,"style":2854},[816,817],[772,4938,4940],{"className":4939},[1752],[772,4941,4943,4963],{"className":4942},[1756,3556],[772,4944,4946,4960],{"className":4945},[1760],[772,4947,4949],{"className":4948,"style":3563},[1764],[772,4950,4951,4954],{"style":3640},[772,4952],{"className":4953,"style":1773},[1772],[772,4955,4957],{"className":4956},[1777,1778,1779,1780],[772,4958,1872],{"className":4959},[816,1780],[772,4961,3580],{"className":4962},[3579],[772,4964,4966],{"className":4965},[1760],[772,4967,4969],{"className":4968,"style":3587},[1764],[772,4970],{},[11,4972,4973],{},"First, calculate the coordinate differences:",[11,4975,4976,5136],{},[772,4977,4979,5011],{"className":4978},[775],[772,4980,4982],{"className":4981},[779],[781,4983,4984],{"xmlns":783},[785,4985,4986,5008],{},[788,4987,4988,4990,4992,4994,5000,5002],{},[791,4989,2963],{"mathvariant":906},[791,4991,2808],{},[980,4993,982],{},[3525,4995,4996,4998],{},[791,4997,2808],{},[1084,4999,1872],{},[980,5001,1091],{},[3525,5003,5004,5006],{},[791,5005,2808],{},[1084,5007,2170],{},[795,5009,5010],{"encoding":797},"\\Delta E=E_2-E_1",[772,5012,5014,5035,5090],{"className":5013,"ariaHidden":803},[802],[772,5015,5017,5020,5023,5026,5029,5032],{"className":5016},[807],[772,5018],{"className":5019,"style":2758},[811],[772,5021,2963],{"className":5022},[816],[772,5024,2808],{"className":5025,"style":2823},[816,817],[772,5027],{"className":5028,"style":1004},[1003],[772,5030,982],{"className":5031},[1008],[772,5033],{"className":5034,"style":1004},[1003],[772,5036,5038,5041,5081,5084,5087],{"className":5037},[807],[772,5039],{"className":5040,"style":3543},[811],[772,5042,5044,5047],{"className":5043},[816],[772,5045,2808],{"className":5046,"style":2823},[816,817],[772,5048,5050],{"className":5049},[1752],[772,5051,5053,5073],{"className":5052},[1756,3556],[772,5054,5056,5070],{"className":5055},[1760],[772,5057,5059],{"className":5058,"style":3563},[1764],[772,5060,5061,5064],{"style":3566},[772,5062],{"className":5063,"style":1773},[1772],[772,5065,5067],{"className":5066},[1777,1778,1779,1780],[772,5068,1872],{"className":5069},[816,1780],[772,5071,3580],{"className":5072},[3579],[772,5074,5076],{"className":5075},[1760],[772,5077,5079],{"className":5078,"style":3587},[1764],[772,5080],{},[772,5082],{"className":5083,"style":1121},[1003],[772,5085,1091],{"className":5086},[1125],[772,5088],{"className":5089,"style":1121},[1003],[772,5091,5093,5096],{"className":5092},[807],[772,5094],{"className":5095,"style":3543},[811],[772,5097,5099,5102],{"className":5098},[816],[772,5100,2808],{"className":5101,"style":2823},[816,817],[772,5103,5105],{"className":5104},[1752],[772,5106,5108,5128],{"className":5107},[1756,3556],[772,5109,5111,5125],{"className":5110},[1760],[772,5112,5114],{"className":5113,"style":3563},[1764],[772,5115,5116,5119],{"style":3566},[772,5117],{"className":5118,"style":1773},[1772],[772,5120,5122],{"className":5121},[1777,1778,1779,1780],[772,5123,2170],{"className":5124},[816,1780],[772,5126,3580],{"className":5127},[3579],[772,5129,5131],{"className":5130},[1760],[772,5132,5134],{"className":5133,"style":3587},[1764],[772,5135],{},[772,5137,5139,5171],{"className":5138},[775],[772,5140,5142],{"className":5141},[779],[781,5143,5144],{"xmlns":783},[785,5145,5146,5168],{},[788,5147,5148,5150,5152,5154,5160,5162],{},[791,5149,2963],{"mathvariant":906},[791,5151,2839],{},[980,5153,982],{},[3525,5155,5156,5158],{},[791,5157,2839],{},[1084,5159,1872],{},[980,5161,1091],{},[3525,5163,5164,5166],{},[791,5165,2839],{},[1084,5167,2170],{},[795,5169,5170],{"encoding":797},"\\Delta N=N_2-N_1",[772,5172,5174,5195,5250],{"className":5173,"ariaHidden":803},[802],[772,5175,5177,5180,5183,5186,5189,5192],{"className":5176},[807],[772,5178],{"className":5179,"style":2758},[811],[772,5181,2963],{"className":5182},[816],[772,5184,2839],{"className":5185,"style":2854},[816,817],[772,5187],{"className":5188,"style":1004},[1003],[772,5190,982],{"className":5191},[1008],[772,5193],{"className":5194,"style":1004},[1003],[772,5196,5198,5201,5241,5244,5247],{"className":5197},[807],[772,5199],{"className":5200,"style":3543},[811],[772,5202,5204,5207],{"className":5203},[816],[772,5205,2839],{"className":5206,"style":2854},[816,817],[772,5208,5210],{"className":5209},[1752],[772,5211,5213,5233],{"className":5212},[1756,3556],[772,5214,5216,5230],{"className":5215},[1760],[772,5217,5219],{"className":5218,"style":3563},[1764],[772,5220,5221,5224],{"style":3640},[772,5222],{"className":5223,"style":1773},[1772],[772,5225,5227],{"className":5226},[1777,1778,1779,1780],[772,5228,1872],{"className":5229},[816,1780],[772,5231,3580],{"className":5232},[3579],[772,5234,5236],{"className":5235},[1760],[772,5237,5239],{"className":5238,"style":3587},[1764],[772,5240],{},[772,5242],{"className":5243,"style":1121},[1003],[772,5245,1091],{"className":5246},[1125],[772,5248],{"className":5249,"style":1121},[1003],[772,5251,5253,5256],{"className":5252},[807],[772,5254],{"className":5255,"style":3543},[811],[772,5257,5259,5262],{"className":5258},[816],[772,5260,2839],{"className":5261,"style":2854},[816,817],[772,5263,5265],{"className":5264},[1752],[772,5266,5268,5288],{"className":5267},[1756,3556],[772,5269,5271,5285],{"className":5270},[1760],[772,5272,5274],{"className":5273,"style":3563},[1764],[772,5275,5276,5279],{"style":3640},[772,5277],{"className":5278,"style":1773},[1772],[772,5280,5282],{"className":5281},[1777,1778,1779,1780],[772,5283,2170],{"className":5284},[816,1780],[772,5286,3580],{"className":5287},[3579],[772,5289,5291],{"className":5290},[1760],[772,5292,5294],{"className":5293,"style":3587},[1764],[772,5295],{},[11,5297,5298,5299,5327],{},"Using the Pythagorean theorem, the horizontal distance (",[772,5300,5302,5315],{"className":5301},[775],[772,5303,5305],{"className":5304},[779],[781,5306,5307],{"xmlns":783},[785,5308,5309,5313],{},[788,5310,5311],{},[791,5312,3180],{},[795,5314,3180],{"encoding":797},[772,5316,5318],{"className":5317,"ariaHidden":803},[802],[772,5319,5321,5324],{"className":5320},[807],[772,5322],{"className":5323,"style":2758},[811],[772,5325,3180],{"className":5326,"style":818},[816,817],") is computed as:",[11,5329,5330],{},[772,5331,5333,5372],{"className":5332},[775],[772,5334,5336],{"className":5335},[779],[781,5337,5338],{"xmlns":783},[785,5339,5340,5369],{},[788,5341,5342,5344,5346],{},[791,5343,3180],{},[980,5345,982],{},[5347,5348,5349],"msqrt",{},[788,5350,5351,5353,5359,5361,5363],{},[791,5352,2963],{"mathvariant":906},[1636,5354,5355,5357],{},[791,5356,2808],{},[1084,5358,1872],{},[980,5360,1353],{},[791,5362,2963],{"mathvariant":906},[1636,5364,5365,5367],{},[791,5366,2839],{},[1084,5368,1872],{},[795,5370,5371],{"encoding":797},"D=\\sqrt{\\Delta E^2+\\Delta N^2}",[772,5373,5375,5393],{"className":5374,"ariaHidden":803},[802],[772,5376,5378,5381,5384,5387,5390],{"className":5377},[807],[772,5379],{"className":5380,"style":2758},[811],[772,5382,3180],{"className":5383,"style":818},[816,817],[772,5385],{"className":5386,"style":1004},[1003],[772,5388,982],{"className":5389},[1008],[772,5391],{"className":5392,"style":1004},[1003],[772,5394,5396,5400],{"className":5395},[807],[772,5397],{"className":5398,"style":5399},[811],"height:1.04em;vertical-align:-0.1266em;",[772,5401,5404],{"className":5402},[816,5403],"sqrt",[772,5405,5407,5528],{"className":5406},[1756,3556],[772,5408,5410,5525],{"className":5409},[1760],[772,5411,5414,5502],{"className":5412,"style":5413},[1764],"height:0.9134em;",[772,5415,5419,5423],{"className":5416,"style":5418},[5417],"svg-align","top:-3em;",[772,5420],{"className":5421,"style":5422},[1772],"height:3em;",[772,5424,5427,5430,5461,5464,5467,5470,5473],{"className":5425,"style":5426},[816],"padding-left:0.833em;",[772,5428,2963],{"className":5429},[816],[772,5431,5433,5436],{"className":5432},[816],[772,5434,2808],{"className":5435,"style":2823},[816,817],[772,5437,5439],{"className":5438},[1752],[772,5440,5442],{"className":5441},[1756],[772,5443,5445],{"className":5444},[1760],[772,5446,5449],{"className":5447,"style":5448},[1764],"height:0.7401em;",[772,5450,5452,5455],{"style":5451},"top:-2.989em;margin-right:0.05em;",[772,5453],{"className":5454,"style":1773},[1772],[772,5456,5458],{"className":5457},[1777,1778,1779,1780],[772,5459,1872],{"className":5460},[816,1780],[772,5462],{"className":5463,"style":1121},[1003],[772,5465,1353],{"className":5466},[1125],[772,5468],{"className":5469,"style":1121},[1003],[772,5471,2963],{"className":5472},[816],[772,5474,5476,5479],{"className":5475},[816],[772,5477,2839],{"className":5478,"style":2854},[816,817],[772,5480,5482],{"className":5481},[1752],[772,5483,5485],{"className":5484},[1756],[772,5486,5488],{"className":5487},[1760],[772,5489,5491],{"className":5490,"style":5448},[1764],[772,5492,5493,5496],{"style":5451},[772,5494],{"className":5495,"style":1773},[1772],[772,5497,5499],{"className":5498},[1777,1778,1779,1780],[772,5500,1872],{"className":5501},[816,1780],[772,5503,5505,5508],{"style":5504},"top:-2.8734em;",[772,5506],{"className":5507,"style":5422},[1772],[772,5509,5513],{"className":5510,"style":5512},[5511],"hide-tail","min-width:0.853em;height:1.08em;",[5514,5515,5521],"svg",{"xmlns":5516,"width":5517,"height":5518,"viewBox":5519,"preserveAspectRatio":5520},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[5522,5523],"path",{"d":5524},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[772,5526,3580],{"className":5527},[3579],[772,5529,5531],{"className":5530},[1760],[772,5532,5535],{"className":5533,"style":5534},[1764],"height:0.1266em;",[772,5536],{},[11,5538,5539,5540,3802],{},"Determining the direction requires the arctangent function. If using a standard calculator, computing the inverse tangent of the absolute ratio of the coordinates yields an acute angle relative to the meridian, known as the reference quadrant angle (",[772,5541,5543,5558],{"className":5542},[775],[772,5544,5546],{"className":5545},[779],[781,5547,5548],{"xmlns":783},[785,5549,5550,5555],{},[788,5551,5552],{},[791,5553,5554],{},"α",[795,5556,5557],{"encoding":797},"\\alpha",[772,5559,5561],{"className":5560,"ariaHidden":803},[802],[772,5562,5564,5567],{"className":5563},[807],[772,5565],{"className":5566,"style":845},[811],[772,5568,5554],{"className":5569,"style":5570},[816,817],"margin-right:0.0037em;",[11,5572,5573],{},[772,5574,5576,5627],{"className":5575},[775],[772,5577,5579],{"className":5578},[779],[781,5580,5581],{"xmlns":783},[785,5582,5583,5624],{},[788,5584,5585,5587,5589,5592,5594],{},[791,5586,5554],{},[980,5588,982],{},[791,5590,5591],{},"arctan",[980,5593,3252],{},[788,5595,5596,5598,5622],{},[980,5597,1082],{"fence":803},[5599,5600,5601,5612],"mfrac",{},[788,5602,5603,5606,5608,5610],{},[791,5604,5605],{"mathvariant":906},"∣",[791,5607,2963],{"mathvariant":906},[791,5609,2808],{},[791,5611,5605],{"mathvariant":906},[788,5613,5614,5616,5618,5620],{},[791,5615,5605],{"mathvariant":906},[791,5617,2963],{"mathvariant":906},[791,5619,2839],{},[791,5621,5605],{"mathvariant":906},[980,5623,1096],{"fence":803},[795,5625,5626],{"encoding":797},"\\alpha=\\arctan\\left(\\frac{|\\Delta E|}{|\\Delta N|}\\right)",[772,5628,5630,5648],{"className":5629,"ariaHidden":803},[802],[772,5631,5633,5636,5639,5642,5645],{"className":5632},[807],[772,5634],{"className":5635,"style":845},[811],[772,5637,5554],{"className":5638,"style":5570},[816,817],[772,5640],{"className":5641,"style":1004},[1003],[772,5643,982],{"className":5644},[1008],[772,5646],{"className":5647,"style":1004},[1003],[772,5649,5651,5655,5658,5661],{"className":5650},[807],[772,5652],{"className":5653,"style":5654},[811],"height:1.8em;vertical-align:-0.65em;",[772,5656,5591],{"className":5657},[3298],[772,5659],{"className":5660,"style":3294},[1003],[772,5662,5665,5675,5764],{"className":5663},[5664],"minner",[772,5666,5670],{"className":5667,"style":5669},[1113,5668],"delimcenter","top:0em;",[772,5671,1082],{"className":5672},[5673,5674],"delimsizing","size2",[772,5676,5678,5682,5761],{"className":5677},[816],[772,5679],{"className":5680},[1113,5681],"nulldelimiter",[772,5683,5685],{"className":5684},[5599],[772,5686,5688,5752],{"className":5687},[1756,3556],[772,5689,5691,5749],{"className":5690},[1760],[772,5692,5695,5717,5728],{"className":5693,"style":5694},[1764],"height:1.01em;",[772,5696,5698,5701],{"style":5697},"top:-2.655em;",[772,5699],{"className":5700,"style":5422},[1772],[772,5702,5704],{"className":5703},[1777,1778,1779,1780],[772,5705,5707,5711,5714],{"className":5706},[816,1780],[772,5708,5710],{"className":5709},[816,1780],"∣Δ",[772,5712,2839],{"className":5713,"style":2854},[816,817,1780],[772,5715,5605],{"className":5716},[816,1780],[772,5718,5720,5723],{"style":5719},"top:-3.23em;",[772,5721],{"className":5722,"style":5422},[1772],[772,5724],{"className":5725,"style":5727},[5726],"frac-line","border-bottom-width:0.04em;",[772,5729,5731,5734],{"style":5730},"top:-3.485em;",[772,5732],{"className":5733,"style":5422},[1772],[772,5735,5737],{"className":5736},[1777,1778,1779,1780],[772,5738,5740,5743,5746],{"className":5739},[816,1780],[772,5741,5710],{"className":5742},[816,1780],[772,5744,2808],{"className":5745,"style":2823},[816,817,1780],[772,5747,5605],{"className":5748},[816,1780],[772,5750,3580],{"className":5751},[3579],[772,5753,5755],{"className":5754},[1760],[772,5756,5759],{"className":5757,"style":5758},[1764],"height:0.52em;",[772,5760],{},[772,5762],{"className":5763},[1141,5681],[772,5765,5767],{"className":5766,"style":5669},[1141,5668],[772,5768,1096],{"className":5769},[5673,5674],[11,5771,5772,5773,5801,5802,3474,5835,5868],{},"The whole-circle bearing (",[772,5774,5776,5789],{"className":5775},[775],[772,5777,5779],{"className":5778},[779],[781,5780,5781],{"xmlns":783},[785,5782,5783,5787],{},[788,5784,5785],{},[791,5786,793],{},[795,5788,798],{"encoding":797},[772,5790,5792],{"className":5791,"ariaHidden":803},[802],[772,5793,5795,5798],{"className":5794},[807],[772,5796],{"className":5797,"style":812},[811],[772,5799,793],{"className":5800,"style":818},[816,817],") is then determined by applying the appropriate quadrant rules based on the signs of ",[772,5803,5805,5820],{"className":5804},[775],[772,5806,5808],{"className":5807},[779],[781,5809,5810],{"xmlns":783},[785,5811,5812,5818],{},[788,5813,5814,5816],{},[791,5815,2963],{"mathvariant":906},[791,5817,2808],{},[795,5819,2968],{"encoding":797},[772,5821,5823],{"className":5822,"ariaHidden":803},[802],[772,5824,5826,5829,5832],{"className":5825},[807],[772,5827],{"className":5828,"style":2758},[811],[772,5830,2963],{"className":5831},[816],[772,5833,2808],{"className":5834,"style":2823},[816,817],[772,5836,5838,5853],{"className":5837},[775],[772,5839,5841],{"className":5840},[779],[781,5842,5843],{"xmlns":783},[785,5844,5845,5851],{},[788,5846,5847,5849],{},[791,5848,2963],{"mathvariant":906},[791,5850,2839],{},[795,5852,3004],{"encoding":797},[772,5854,5856],{"className":5855,"ariaHidden":803},[802],[772,5857,5859,5862,5865],{"className":5858},[807],[772,5860],{"className":5861,"style":2758},[811],[772,5863,2963],{"className":5864},[816],[772,5866,2839],{"className":5867,"style":2854},[816,817],".",[11,5870,5871,5872,5906],{},"In programmatic applications and surveying software, the dual-argument arctangent function, ",[772,5873,5875,5890],{"className":5874},[775],[772,5876,5878],{"className":5877},[779],[781,5879,5880],{"xmlns":783},[785,5881,5882,5887],{},[788,5883,5884],{},[1625,5885,5886],{},"atan2",[795,5888,5889],{"encoding":797},"\\text{atan2}",[772,5891,5893],{"className":5892,"ariaHidden":803},[802],[772,5894,5896,5900],{"className":5895},[807],[772,5897],{"className":5898,"style":5899},[811],"height:0.6444em;",[772,5901,5903],{"className":5902},[816,1709],[772,5904,5886],{"className":5905},[816],", is used instead. This function automatically evaluates the signs of both arguments to place the resulting angle in the correct quadrant:",[11,5908,5909],{},[772,5910,5912,5950],{"className":5911},[775],[772,5913,5915],{"className":5914},[779],[781,5916,5917],{"xmlns":783},[785,5918,5919,5947],{},[788,5920,5921,5928,5930,5932,5934,5936,5938,5941,5943,5945],{},[3525,5922,5923,5925],{},[791,5924,793],{},[1625,5926,5927],{},"rad",[980,5929,982],{},[1625,5931,5886],{},[980,5933,1082],{"stretchy":1081},[791,5935,2963],{"mathvariant":906},[791,5937,2808],{},[980,5939,5940],{"separator":803},",",[791,5942,2963],{"mathvariant":906},[791,5944,2839],{},[980,5946,1096],{"stretchy":1081},[795,5948,5949],{"encoding":797},"\\theta_{\\text{rad}}=\\text{atan2}(\\Delta E,\\Delta N)",[772,5951,5953,6017],{"className":5952,"ariaHidden":803},[802],[772,5954,5956,5960,6008,6011,6014],{"className":5955},[807],[772,5957],{"className":5958,"style":5959},[811],"height:0.8444em;vertical-align:-0.15em;",[772,5961,5963,5966],{"className":5962},[816],[772,5964,793],{"className":5965,"style":818},[816,817],[772,5967,5969],{"className":5968},[1752],[772,5970,5972,6000],{"className":5971},[1756,3556],[772,5973,5975,5997],{"className":5974},[1760],[772,5976,5979],{"className":5977,"style":5978},[1764],"height:0.3361em;",[772,5980,5982,5985],{"style":5981},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[772,5983],{"className":5984,"style":1773},[1772],[772,5986,5988],{"className":5987},[1777,1778,1779,1780],[772,5989,5991],{"className":5990},[816,1780],[772,5992,5994],{"className":5993},[816,1709,1780],[772,5995,5927],{"className":5996},[816,1780],[772,5998,3580],{"className":5999},[3579],[772,6001,6003],{"className":6002},[1760],[772,6004,6006],{"className":6005,"style":3587},[1764],[772,6007],{},[772,6009],{"className":6010,"style":1004},[1003],[772,6012,982],{"className":6013},[1008],[772,6015],{"className":6016,"style":1004},[1003],[772,6018,6020,6023,6029,6032,6035,6038,6042,6045,6048,6051],{"className":6019},[807],[772,6021],{"className":6022,"style":1109},[811],[772,6024,6026],{"className":6025},[816,1709],[772,6027,5886],{"className":6028},[816],[772,6030,1082],{"className":6031},[1113],[772,6033,2963],{"className":6034},[816],[772,6036,2808],{"className":6037,"style":2823},[816,817],[772,6039,5940],{"className":6040},[6041],"mpunct",[772,6043],{"className":6044,"style":3294},[1003],[772,6046,2963],{"className":6047},[816],[772,6049,2839],{"className":6050,"style":2854},[816,817],[772,6052,1096],{"className":6053},[1141],[11,6055,6056,6057,6088,6089,6126,6127,6161,6162,6351],{},"The output of ",[772,6058,6060,6073],{"className":6059},[775],[772,6061,6063],{"className":6062},[779],[781,6064,6065],{"xmlns":783},[785,6066,6067,6071],{},[788,6068,6069],{},[1625,6070,5886],{},[795,6072,5889],{"encoding":797},[772,6074,6076],{"className":6075,"ariaHidden":803},[802],[772,6077,6079,6082],{"className":6078},[807],[772,6080],{"className":6081,"style":5899},[811],[772,6083,6085],{"className":6084},[816,1709],[772,6086,5886],{"className":6087},[816]," is typically returned in radians ranging from ",[772,6090,6092,6109],{"className":6091},[775],[772,6093,6095],{"className":6094},[779],[781,6096,6097],{"xmlns":783},[785,6098,6099,6106],{},[788,6100,6101,6103],{},[980,6102,1091],{},[791,6104,6105],{},"π",[795,6107,6108],{"encoding":797},"-\\pi",[772,6110,6112],{"className":6111,"ariaHidden":803},[802],[772,6113,6115,6119,6122],{"className":6114},[807],[772,6116],{"className":6117,"style":6118},[811],"height:0.6667em;vertical-align:-0.0833em;",[772,6120,1091],{"className":6121},[816],[772,6123,6105],{"className":6124,"style":6125},[816,817],"margin-right:0.0359em;"," to ",[772,6128,6130,6146],{"className":6129},[775],[772,6131,6133],{"className":6132},[779],[781,6134,6135],{"xmlns":783},[785,6136,6137,6143],{},[788,6138,6139,6141],{},[980,6140,1353],{},[791,6142,6105],{},[795,6144,6145],{"encoding":797},"+\\pi",[772,6147,6149],{"className":6148,"ariaHidden":803},[802],[772,6150,6152,6155,6158],{"className":6151},[807],[772,6153],{"className":6154,"style":6118},[811],[772,6156,1353],{"className":6157},[816],[772,6159,6105],{"className":6160,"style":6125},[816,817],". To convert this value to decimal degrees and normalize it to a standard surveying azimuth range (",[772,6163,6165,6202],{"className":6164},[775],[772,6166,6168],{"className":6167},[779],[781,6169,6170],{"xmlns":783},[785,6171,6172,6199],{},[788,6173,6174,6180,6183,6190,6193],{},[1636,6175,6176,6178],{},[1084,6177,1748],{},[980,6179,1642],{},[980,6181,6182],{},"≤",[3525,6184,6185,6187],{},[791,6186,793],{},[1625,6188,6189],{},"deg",[980,6191,6192],{},"\u003C",[1636,6194,6195,6197],{},[1084,6196,1468],{},[980,6198,1642],{},[795,6200,6201],{"encoding":797},"0^\\circ \\le \\theta_{\\text{deg}} \u003C 360^\\circ",[772,6203,6205,6250,6313],{"className":6204,"ariaHidden":803},[802],[772,6206,6208,6212,6241,6244,6247],{"className":6207},[807],[772,6209],{"className":6210,"style":6211},[811],"height:0.8101em;vertical-align:-0.136em;",[772,6213,6215,6218],{"className":6214},[816],[772,6216,1748],{"className":6217},[816],[772,6219,6221],{"className":6220},[1752],[772,6222,6224],{"className":6223},[1756],[772,6225,6227],{"className":6226},[1760],[772,6228,6230],{"className":6229,"style":1765},[1764],[772,6231,6232,6235],{"style":1768},[772,6233],{"className":6234,"style":1773},[1772],[772,6236,6238],{"className":6237},[1777,1778,1779,1780],[772,6239,1642],{"className":6240},[1125,1780],[772,6242],{"className":6243,"style":1004},[1003],[772,6245,6182],{"className":6246},[1008],[772,6248],{"className":6249,"style":1004},[1003],[772,6251,6253,6257,6304,6307,6310],{"className":6252},[807],[772,6254],{"className":6255,"style":6256},[811],"height:0.9805em;vertical-align:-0.2861em;",[772,6258,6260,6263],{"className":6259},[816],[772,6261,793],{"className":6262,"style":818},[816,817],[772,6264,6266],{"className":6265},[1752],[772,6267,6269,6295],{"className":6268},[1756,3556],[772,6270,6272,6292],{"className":6271},[1760],[772,6273,6275],{"className":6274,"style":5978},[1764],[772,6276,6277,6280],{"style":5981},[772,6278],{"className":6279,"style":1773},[1772],[772,6281,6283],{"className":6282},[1777,1778,1779,1780],[772,6284,6286],{"className":6285},[816,1780],[772,6287,6289],{"className":6288},[816,1709,1780],[772,6290,6189],{"className":6291},[816,1780],[772,6293,3580],{"className":6294},[3579],[772,6296,6298],{"className":6297},[1760],[772,6299,6302],{"className":6300,"style":6301},[1764],"height:0.2861em;",[772,6303],{},[772,6305],{"className":6306,"style":1004},[1003],[772,6308,6192],{"className":6309},[1008],[772,6311],{"className":6312,"style":1004},[1003],[772,6314,6316,6319,6322],{"className":6315},[807],[772,6317],{"className":6318,"style":1765},[811],[772,6320,1681],{"className":6321},[816],[772,6323,6325,6328],{"className":6324},[816],[772,6326,1748],{"className":6327},[816],[772,6329,6331],{"className":6330},[1752],[772,6332,6334],{"className":6333},[1756],[772,6335,6337],{"className":6336},[1760],[772,6338,6340],{"className":6339,"style":1765},[1764],[772,6341,6342,6345],{"style":1768},[772,6343],{"className":6344,"style":1773},[1772],[772,6346,6348],{"className":6347},[1777,1778,1779,1780],[772,6349,1642],{"className":6350},[1125,1780],"), apply the following steps:",[11,6353,6354,6617],{},[772,6355,6357,6398],{"className":6356},[775],[772,6358,6360],{"className":6359},[779],[781,6361,6362],{"xmlns":783},[785,6363,6364,6395],{},[788,6365,6366,6372,6374,6380,6383],{},[3525,6367,6368,6370],{},[791,6369,793],{},[1625,6371,6189],{},[980,6373,982],{},[3525,6375,6376,6378],{},[791,6377,793],{},[1625,6379,5927],{},[980,6381,6382],{},"×",[788,6384,6385,6387,6393],{},[980,6386,1082],{"fence":803},[5599,6388,6389,6391],{},[1084,6390,1086],{},[791,6392,6105],{},[980,6394,1096],{"fence":803},[795,6396,6397],{"encoding":797},"\\theta_{\\text{deg}}=\\theta_{\\text{rad}}\\times\\left(\\frac{180}{\\pi}\\right)",[772,6399,6401,6462,6523],{"className":6400,"ariaHidden":803},[802],[772,6402,6404,6407,6453,6456,6459],{"className":6403},[807],[772,6405],{"className":6406,"style":6256},[811],[772,6408,6410,6413],{"className":6409},[816],[772,6411,793],{"className":6412,"style":818},[816,817],[772,6414,6416],{"className":6415},[1752],[772,6417,6419,6445],{"className":6418},[1756,3556],[772,6420,6422,6442],{"className":6421},[1760],[772,6423,6425],{"className":6424,"style":5978},[1764],[772,6426,6427,6430],{"style":5981},[772,6428],{"className":6429,"style":1773},[1772],[772,6431,6433],{"className":6432},[1777,1778,1779,1780],[772,6434,6436],{"className":6435},[816,1780],[772,6437,6439],{"className":6438},[816,1709,1780],[772,6440,6189],{"className":6441},[816,1780],[772,6443,3580],{"className":6444},[3579],[772,6446,6448],{"className":6447},[1760],[772,6449,6451],{"className":6450,"style":6301},[1764],[772,6452],{},[772,6454],{"className":6455,"style":1004},[1003],[772,6457,982],{"className":6458},[1008],[772,6460],{"className":6461,"style":1004},[1003],[772,6463,6465,6468,6514,6517,6520],{"className":6464},[807],[772,6466],{"className":6467,"style":5959},[811],[772,6469,6471,6474],{"className":6470},[816],[772,6472,793],{"className":6473,"style":818},[816,817],[772,6475,6477],{"className":6476},[1752],[772,6478,6480,6506],{"className":6479},[1756,3556],[772,6481,6483,6503],{"className":6482},[1760],[772,6484,6486],{"className":6485,"style":5978},[1764],[772,6487,6488,6491],{"style":5981},[772,6489],{"className":6490,"style":1773},[1772],[772,6492,6494],{"className":6493},[1777,1778,1779,1780],[772,6495,6497],{"className":6496},[816,1780],[772,6498,6500],{"className":6499},[816,1709,1780],[772,6501,5927],{"className":6502},[816,1780],[772,6504,3580],{"className":6505},[3579],[772,6507,6509],{"className":6508},[1760],[772,6510,6512],{"className":6511,"style":3587},[1764],[772,6513],{},[772,6515],{"className":6516,"style":1121},[1003],[772,6518,6382],{"className":6519},[1125],[772,6521],{"className":6522,"style":1121},[1003],[772,6524,6526,6530],{"className":6525},[807],[772,6527],{"className":6528,"style":6529},[811],"height:1.2em;vertical-align:-0.35em;",[772,6531,6533,6540,6611],{"className":6532},[5664],[772,6534,6536],{"className":6535,"style":5669},[1113,5668],[772,6537,1082],{"className":6538},[5673,6539],"size1",[772,6541,6543,6546,6608],{"className":6542},[816],[772,6544],{"className":6545},[1113,5681],[772,6547,6549],{"className":6548},[5599],[772,6550,6552,6599],{"className":6551},[1756,3556],[772,6553,6555,6596],{"className":6554},[1760],[772,6556,6559,6573,6581],{"className":6557,"style":6558},[1764],"height:0.8451em;",[772,6560,6561,6564],{"style":5697},[772,6562],{"className":6563,"style":5422},[1772],[772,6565,6567],{"className":6566},[1777,1778,1779,1780],[772,6568,6570],{"className":6569},[816,1780],[772,6571,6105],{"className":6572,"style":6125},[816,817,1780],[772,6574,6575,6578],{"style":5719},[772,6576],{"className":6577,"style":5422},[1772],[772,6579],{"className":6580,"style":5727},[5726],[772,6582,6584,6587],{"style":6583},"top:-3.394em;",[772,6585],{"className":6586,"style":5422},[1772],[772,6588,6590],{"className":6589},[1777,1778,1779,1780],[772,6591,6593],{"className":6592},[816,1780],[772,6594,1086],{"className":6595},[816,1780],[772,6597,3580],{"className":6598},[3579],[772,6600,6602],{"className":6601},[1760],[772,6603,6606],{"className":6604,"style":6605},[1764],"height:0.345em;",[772,6607],{},[772,6609],{"className":6610},[1141,5681],[772,6612,6614],{"className":6613,"style":5669},[1141,5668],[772,6615,1096],{"className":6616},[5673,6539],[772,6618,6620,6674],{"className":6619},[775],[772,6621,6623],{"className":6622},[779],[781,6624,6625],{"xmlns":783},[785,6626,6627,6671],{},[788,6628,6629,6632,6638,6640,6646,6649,6655,6657,6663,6665],{},[1625,6630,6631],{},"If ",[3525,6633,6634,6636],{},[791,6635,793],{},[1625,6637,6189],{},[980,6639,6192],{},[1636,6641,6642,6644],{},[1084,6643,1748],{},[980,6645,1642],{},[1625,6647,6648],{},", then ",[3525,6650,6651,6653],{},[791,6652,793],{},[1625,6654,6189],{},[980,6656,982],{},[3525,6658,6659,6661],{},[791,6660,793],{},[1625,6662,6189],{},[980,6664,1353],{},[1636,6666,6667,6669],{},[1084,6668,1468],{},[980,6670,1642],{},[795,6672,6673],{"encoding":797},"\\text{If }\\theta_{\\text{deg}}\u003C0^\\circ\\text{, then }\\theta_{\\text{deg}}=\\theta_{\\text{deg}}+360^\\circ",[772,6675,6677,6744,6840,6901],{"className":6676,"ariaHidden":803},[802],[772,6678,6680,6683,6689,6735,6738,6741],{"className":6679},[807],[772,6681],{"className":6682,"style":6256},[811],[772,6684,6686],{"className":6685},[816,1709],[772,6687,6631],{"className":6688},[816],[772,6690,6692,6695],{"className":6691},[816],[772,6693,793],{"className":6694,"style":818},[816,817],[772,6696,6698],{"className":6697},[1752],[772,6699,6701,6727],{"className":6700},[1756,3556],[772,6702,6704,6724],{"className":6703},[1760],[772,6705,6707],{"className":6706,"style":5978},[1764],[772,6708,6709,6712],{"style":5981},[772,6710],{"className":6711,"style":1773},[1772],[772,6713,6715],{"className":6714},[1777,1778,1779,1780],[772,6716,6718],{"className":6717},[816,1780],[772,6719,6721],{"className":6720},[816,1709,1780],[772,6722,6189],{"className":6723},[816,1780],[772,6725,3580],{"className":6726},[3579],[772,6728,6730],{"className":6729},[1760],[772,6731,6733],{"className":6732,"style":6301},[1764],[772,6734],{},[772,6736],{"className":6737,"style":1004},[1003],[772,6739,6192],{"className":6740},[1008],[772,6742],{"className":6743,"style":1004},[1003],[772,6745,6747,6750,6779,6785,6831,6834,6837],{"className":6746},[807],[772,6748],{"className":6749,"style":6256},[811],[772,6751,6753,6756],{"className":6752},[816],[772,6754,1748],{"className":6755},[816],[772,6757,6759],{"className":6758},[1752],[772,6760,6762],{"className":6761},[1756],[772,6763,6765],{"className":6764},[1760],[772,6766,6768],{"className":6767,"style":1765},[1764],[772,6769,6770,6773],{"style":1768},[772,6771],{"className":6772,"style":1773},[1772],[772,6774,6776],{"className":6775},[1777,1778,1779,1780],[772,6777,1642],{"className":6778},[1125,1780],[772,6780,6782],{"className":6781},[816,1709],[772,6783,6648],{"className":6784},[816],[772,6786,6788,6791],{"className":6787},[816],[772,6789,793],{"className":6790,"style":818},[816,817],[772,6792,6794],{"className":6793},[1752],[772,6795,6797,6823],{"className":6796},[1756,3556],[772,6798,6800,6820],{"className":6799},[1760],[772,6801,6803],{"className":6802,"style":5978},[1764],[772,6804,6805,6808],{"style":5981},[772,6806],{"className":6807,"style":1773},[1772],[772,6809,6811],{"className":6810},[1777,1778,1779,1780],[772,6812,6814],{"className":6813},[816,1780],[772,6815,6817],{"className":6816},[816,1709,1780],[772,6818,6189],{"className":6819},[816,1780],[772,6821,3580],{"className":6822},[3579],[772,6824,6826],{"className":6825},[1760],[772,6827,6829],{"className":6828,"style":6301},[1764],[772,6830],{},[772,6832],{"className":6833,"style":1004},[1003],[772,6835,982],{"className":6836},[1008],[772,6838],{"className":6839,"style":1004},[1003],[772,6841,6843,6846,6892,6895,6898],{"className":6842},[807],[772,6844],{"className":6845,"style":6256},[811],[772,6847,6849,6852],{"className":6848},[816],[772,6850,793],{"className":6851,"style":818},[816,817],[772,6853,6855],{"className":6854},[1752],[772,6856,6858,6884],{"className":6857},[1756,3556],[772,6859,6861,6881],{"className":6860},[1760],[772,6862,6864],{"className":6863,"style":5978},[1764],[772,6865,6866,6869],{"style":5981},[772,6867],{"className":6868,"style":1773},[1772],[772,6870,6872],{"className":6871},[1777,1778,1779,1780],[772,6873,6875],{"className":6874},[816,1780],[772,6876,6878],{"className":6877},[816,1709,1780],[772,6879,6189],{"className":6880},[816,1780],[772,6882,3580],{"className":6883},[3579],[772,6885,6887],{"className":6886},[1760],[772,6888,6890],{"className":6889,"style":6301},[1764],[772,6891],{},[772,6893],{"className":6894,"style":1121},[1003],[772,6896,1353],{"className":6897},[1125],[772,6899],{"className":6900,"style":1121},[1003],[772,6902,6904,6907,6910],{"className":6903},[807],[772,6905],{"className":6906,"style":1765},[811],[772,6908,1681],{"className":6909},[816],[772,6911,6913,6916],{"className":6912},[816],[772,6914,1748],{"className":6915},[816],[772,6917,6919],{"className":6918},[1752],[772,6920,6922],{"className":6921},[1756],[772,6923,6925],{"className":6924},[1760],[772,6926,6928],{"className":6927,"style":1765},[1764],[772,6929,6930,6933],{"style":1768},[772,6931],{"className":6932,"style":1773},[1772],[772,6934,6936],{"className":6935},[1777,1778,1779,1780],[772,6937,1642],{"className":6938},[1125,1780],[11,6940,6941,6942,7012,7013,5868],{},"Surveying Core and most programming environments use ",[772,6943,6945,6973],{"className":6944},[775],[772,6946,6948],{"className":6947},[779],[781,6949,6950],{"xmlns":783},[785,6951,6952,6970],{},[788,6953,6954,6956,6958,6960,6962,6964,6966,6968],{},[1625,6955,5886],{},[980,6957,1082],{"stretchy":1081},[791,6959,2963],{"mathvariant":906},[791,6961,2808],{},[980,6963,5940],{"separator":803},[791,6965,2963],{"mathvariant":906},[791,6967,2839],{},[980,6969,1096],{"stretchy":1081},[795,6971,6972],{"encoding":797},"\\text{atan2}(\\Delta E, \\Delta N)",[772,6974,6976],{"className":6975,"ariaHidden":803},[802],[772,6977,6979,6982,6988,6991,6994,6997,7000,7003,7006,7009],{"className":6978},[807],[772,6980],{"className":6981,"style":1109},[811],[772,6983,6985],{"className":6984},[816,1709],[772,6986,5886],{"className":6987},[816],[772,6989,1082],{"className":6990},[1113],[772,6992,2963],{"className":6993},[816],[772,6995,2808],{"className":6996,"style":2823},[816,817],[772,6998,5940],{"className":6999},[6041],[772,7001],{"className":7002,"style":3294},[1003],[772,7004,2963],{"className":7005},[816],[772,7007,2839],{"className":7008,"style":2854},[816,817],[772,7010,1096],{"className":7011},[1141]," — the same order as JavaScript ",[2863,7014,7015],{},"Math.atan2(dE, dN)",[323,7017,7019],{"id":7018},"worked-engineering-examples","Worked Engineering Examples",[706,7021,7023],{"id":7022},"worked-example-1-polar-to-rectangular-forward-calculation","Worked Example 1: Polar to Rectangular (Forward Calculation)",[11,7025,7026],{},"A field surveying crew occupies a control monument (Station A) with known coordinates:",[335,7028,7029,7102],{},[338,7030,7031,7032,7101],{},"Easting of Station A (",[772,7033,7035,7052],{"className":7034},[775],[772,7036,7038],{"className":7037},[779],[781,7039,7040],{"xmlns":783},[785,7041,7042,7050],{},[788,7043,7044],{},[3525,7045,7046,7048],{},[791,7047,2808],{},[1084,7049,2170],{},[795,7051,3682],{"encoding":797},[772,7053,7055],{"className":7054,"ariaHidden":803},[802],[772,7056,7058,7061],{"className":7057},[807],[772,7059],{"className":7060,"style":3543},[811],[772,7062,7064,7067],{"className":7063},[816],[772,7065,2808],{"className":7066,"style":2823},[816,817],[772,7068,7070],{"className":7069},[1752],[772,7071,7073,7093],{"className":7072},[1756,3556],[772,7074,7076,7090],{"className":7075},[1760],[772,7077,7079],{"className":7078,"style":3563},[1764],[772,7080,7081,7084],{"style":3566},[772,7082],{"className":7083,"style":1773},[1772],[772,7085,7087],{"className":7086},[1777,1778,1779,1780],[772,7088,2170],{"className":7089},[816,1780],[772,7091,3580],{"className":7092},[3579],[772,7094,7096],{"className":7095},[1760],[772,7097,7099],{"className":7098,"style":3587},[1764],[772,7100],{},"): 500000.000 m",[338,7103,7104,7105,7174],{},"Northing of Station A (",[772,7106,7108,7125],{"className":7107},[775],[772,7109,7111],{"className":7110},[779],[781,7112,7113],{"xmlns":783},[785,7114,7115,7123],{},[788,7116,7117],{},[3525,7118,7119,7121],{},[791,7120,2839],{},[1084,7122,2170],{},[795,7124,3752],{"encoding":797},[772,7126,7128],{"className":7127,"ariaHidden":803},[802],[772,7129,7131,7134],{"className":7130},[807],[772,7132],{"className":7133,"style":3543},[811],[772,7135,7137,7140],{"className":7136},[816],[772,7138,2839],{"className":7139,"style":2854},[816,817],[772,7141,7143],{"className":7142},[1752],[772,7144,7146,7166],{"className":7145},[1756,3556],[772,7147,7149,7163],{"className":7148},[1760],[772,7150,7152],{"className":7151,"style":3563},[1764],[772,7153,7154,7157],{"style":3640},[772,7155],{"className":7156,"style":1773},[1772],[772,7158,7160],{"className":7159},[1777,1778,1779,1780],[772,7161,2170],{"className":7162},[816,1780],[772,7164,3580],{"className":7165},[3579],[772,7167,7169],{"className":7168},[1760],[772,7170,7172],{"className":7171,"style":3587},[1764],[772,7173],{},"): 150000.000 m",[11,7176,7177,7178,7206,7207,7235],{},"A total station measures a horizontal distance (",[772,7179,7181,7194],{"className":7180},[775],[772,7182,7184],{"className":7183},[779],[781,7185,7186],{"xmlns":783},[785,7187,7188,7192],{},[788,7189,7190],{},[791,7191,3180],{},[795,7193,3180],{"encoding":797},[772,7195,7197],{"className":7196,"ariaHidden":803},[802],[772,7198,7200,7203],{"className":7199},[807],[772,7201],{"className":7202,"style":2758},[811],[772,7204,3180],{"className":7205,"style":818},[816,817],") and grid azimuth (",[772,7208,7210,7223],{"className":7209},[775],[772,7211,7213],{"className":7212},[779],[781,7214,7215],{"xmlns":783},[785,7216,7217,7221],{},[788,7218,7219],{},[791,7220,793],{},[795,7222,798],{"encoding":797},[772,7224,7226],{"className":7225,"ariaHidden":803},[802],[772,7227,7229,7232],{"className":7228},[807],[772,7230],{"className":7231,"style":812},[811],[772,7233,793],{"className":7234,"style":818},[816,817],") to a proposed property corner (Station B):",[335,7237,7238,7270],{},[338,7239,7240,7241,7269],{},"Distance (",[772,7242,7244,7257],{"className":7243},[775],[772,7245,7247],{"className":7246},[779],[781,7248,7249],{"xmlns":783},[785,7250,7251,7255],{},[788,7252,7253],{},[791,7254,3180],{},[795,7256,3180],{"encoding":797},[772,7258,7260],{"className":7259,"ariaHidden":803},[802],[772,7261,7263,7266],{"className":7262},[807],[772,7264],{"className":7265,"style":2758},[811],[772,7267,3180],{"className":7268,"style":818},[816,817],"): 83.472 m",[338,7271,7272,7273,7301,7302],{},"Azimuth (",[772,7274,7276,7289],{"className":7275},[775],[772,7277,7279],{"className":7278},[779],[781,7280,7281],{"xmlns":783},[785,7282,7283,7287],{},[788,7284,7285],{},[791,7286,793],{},[795,7288,798],{"encoding":797},[772,7290,7292],{"className":7291,"ariaHidden":803},[802],[772,7293,7295,7298],{"className":7294},[807],[772,7296],{"className":7297,"style":812},[811],[772,7299,793],{"className":7300,"style":818},[816,817],"): ",[772,7303,7305,7328],{"className":7304},[775],[772,7306,7308],{"className":7307},[779],[781,7309,7310],{"xmlns":783},[785,7311,7312,7325],{},[788,7313,7314,7316,7318],{},[791,7315,793],{},[980,7317,982],{},[1636,7319,7320,7323],{},[1084,7321,7322],{},"127.5430",[980,7324,1642],{},[795,7326,7327],{"encoding":797},"\\theta=127.5430^\\circ",[772,7329,7331,7349],{"className":7330,"ariaHidden":803},[802],[772,7332,7334,7337,7340,7343,7346],{"className":7333},[807],[772,7335],{"className":7336,"style":812},[811],[772,7338,793],{"className":7339,"style":818},[816,817],[772,7341],{"className":7342,"style":1004},[1003],[772,7344,982],{"className":7345},[1008],[772,7347],{"className":7348,"style":1004},[1003],[772,7350,7352,7355,7359],{"className":7351},[807],[772,7353],{"className":7354,"style":1765},[811],[772,7356,7358],{"className":7357},[816],"127.543",[772,7360,7362,7365],{"className":7361},[816],[772,7363,1748],{"className":7364},[816],[772,7366,7368],{"className":7367},[1752],[772,7369,7371],{"className":7370},[1756],[772,7372,7374],{"className":7373},[1760],[772,7375,7377],{"className":7376,"style":1765},[1764],[772,7378,7379,7382],{"style":1768},[772,7380],{"className":7381,"style":1773},[1772],[772,7383,7385],{"className":7384},[1777,1778,1779,1780],[772,7386,1642],{"className":7387},[1125,1780],[11,7389,7390],{},[318,7391,7392],{},"Step 1: Convert the azimuth from decimal degrees to radians",[11,7394,7395],{},[772,7396,7398,7444],{"className":7397},[775],[772,7399,7401],{"className":7400},[779],[781,7402,7403],{"xmlns":783},[785,7404,7405,7441],{},[788,7406,7407,7413,7415,7421,7423,7435,7438],{},[3525,7408,7409,7411],{},[791,7410,793],{},[1625,7412,5927],{},[980,7414,982],{},[1636,7416,7417,7419],{},[1084,7418,7322],{},[980,7420,1642],{},[980,7422,6382],{},[788,7424,7425,7427,7433],{},[980,7426,1082],{"fence":803},[5599,7428,7429,7431],{},[791,7430,6105],{},[1084,7432,1086],{},[980,7434,1096],{"fence":803},[980,7436,7437],{},"≈",[1084,7439,7440],{},"2.226045952",[795,7442,7443],{"encoding":797},"\\theta_{\\text{rad}}=127.5430^\\circ\\times\\left(\\frac{\\pi}{180}\\right)\\approx2.226045952",[772,7445,7447,7508,7556,7655],{"className":7446,"ariaHidden":803},[802],[772,7448,7450,7453,7499,7502,7505],{"className":7449},[807],[772,7451],{"className":7452,"style":5959},[811],[772,7454,7456,7459],{"className":7455},[816],[772,7457,793],{"className":7458,"style":818},[816,817],[772,7460,7462],{"className":7461},[1752],[772,7463,7465,7491],{"className":7464},[1756,3556],[772,7466,7468,7488],{"className":7467},[1760],[772,7469,7471],{"className":7470,"style":5978},[1764],[772,7472,7473,7476],{"style":5981},[772,7474],{"className":7475,"style":1773},[1772],[772,7477,7479],{"className":7478},[1777,1778,1779,1780],[772,7480,7482],{"className":7481},[816,1780],[772,7483,7485],{"className":7484},[816,1709,1780],[772,7486,5927],{"className":7487},[816,1780],[772,7489,3580],{"className":7490},[3579],[772,7492,7494],{"className":7493},[1760],[772,7495,7497],{"className":7496,"style":3587},[1764],[772,7498],{},[772,7500],{"className":7501,"style":1004},[1003],[772,7503,982],{"className":7504},[1008],[772,7506],{"className":7507,"style":1004},[1003],[772,7509,7511,7515,7518,7547,7550,7553],{"className":7510},[807],[772,7512],{"className":7513,"style":7514},[811],"height:0.7574em;vertical-align:-0.0833em;",[772,7516,7358],{"className":7517},[816],[772,7519,7521,7524],{"className":7520},[816],[772,7522,1748],{"className":7523},[816],[772,7525,7527],{"className":7526},[1752],[772,7528,7530],{"className":7529},[1756],[772,7531,7533],{"className":7532},[1760],[772,7534,7536],{"className":7535,"style":1765},[1764],[772,7537,7538,7541],{"style":1768},[772,7539],{"className":7540,"style":1773},[1772],[772,7542,7544],{"className":7543},[1777,1778,1779,1780],[772,7545,1642],{"className":7546},[1125,1780],[772,7548],{"className":7549,"style":1121},[1003],[772,7551,6382],{"className":7552},[1125],[772,7554],{"className":7555,"style":1121},[1003],[772,7557,7559,7562,7646,7649,7652],{"className":7558},[807],[772,7560],{"className":7561,"style":6529},[811],[772,7563,7565,7571,7640],{"className":7564},[5664],[772,7566,7568],{"className":7567,"style":5669},[1113,5668],[772,7569,1082],{"className":7570},[5673,6539],[772,7572,7574,7577,7637],{"className":7573},[816],[772,7575],{"className":7576},[1113,5681],[772,7578,7580],{"className":7579},[5599],[772,7581,7583,7629],{"className":7582},[1756,3556],[772,7584,7586,7626],{"className":7585},[1760],[772,7587,7590,7604,7612],{"className":7588,"style":7589},[1764],"height:0.6954em;",[772,7591,7592,7595],{"style":5697},[772,7593],{"className":7594,"style":5422},[1772],[772,7596,7598],{"className":7597},[1777,1778,1779,1780],[772,7599,7601],{"className":7600},[816,1780],[772,7602,1086],{"className":7603},[816,1780],[772,7605,7606,7609],{"style":5719},[772,7607],{"className":7608,"style":5422},[1772],[772,7610],{"className":7611,"style":5727},[5726],[772,7613,7614,7617],{"style":6583},[772,7615],{"className":7616,"style":5422},[1772],[772,7618,7620],{"className":7619},[1777,1778,1779,1780],[772,7621,7623],{"className":7622},[816,1780],[772,7624,6105],{"className":7625,"style":6125},[816,817,1780],[772,7627,3580],{"className":7628},[3579],[772,7630,7632],{"className":7631},[1760],[772,7633,7635],{"className":7634,"style":6605},[1764],[772,7636],{},[772,7638],{"className":7639},[1141,5681],[772,7641,7643],{"className":7642,"style":5669},[1141,5668],[772,7644,1096],{"className":7645},[5673,6539],[772,7647],{"className":7648,"style":1004},[1003],[772,7650,7437],{"className":7651},[1008],[772,7653],{"className":7654,"style":1004},[1003],[772,7656,7658,7661],{"className":7657},[807],[772,7659],{"className":7660,"style":5899},[811],[772,7662,7440],{"className":7663},[816],[11,7665,7666],{},"The converted angle is approximately 2.226045952 radians.",[11,7668,7669],{},[318,7670,7671,7672,3474,7705,1096],{},"Step 2: Calculate the coordinate offsets (",[772,7673,7675,7690],{"className":7674},[775],[772,7676,7678],{"className":7677},[779],[781,7679,7680],{"xmlns":783},[785,7681,7682,7688],{},[788,7683,7684,7686],{},[791,7685,2963],{"mathvariant":906},[791,7687,2808],{},[795,7689,2968],{"encoding":797},[772,7691,7693],{"className":7692,"ariaHidden":803},[802],[772,7694,7696,7699,7702],{"className":7695},[807],[772,7697],{"className":7698,"style":2758},[811],[772,7700,2963],{"className":7701},[816],[772,7703,2808],{"className":7704,"style":2823},[816,817],[772,7706,7708,7723],{"className":7707},[775],[772,7709,7711],{"className":7710},[779],[781,7712,7713],{"xmlns":783},[785,7714,7715,7721],{},[788,7716,7717,7719],{},[791,7718,2963],{"mathvariant":906},[791,7720,2839],{},[795,7722,3004],{"encoding":797},[772,7724,7726],{"className":7725,"ariaHidden":803},[802],[772,7727,7729,7732,7735],{"className":7728},[807],[772,7730],{"className":7731,"style":2758},[811],[772,7733,2963],{"className":7734},[816],[772,7736,2839],{"className":7737,"style":2854},[816,817],[11,7739,7740],{},[772,7741,7743,7791],{"className":7742},[775],[772,7744,7746],{"className":7745},[779],[781,7747,7748],{"xmlns":783},[785,7749,7750,7788],{},[788,7751,7752,7754,7756,7758,7760,7762,7764,7766,7768,7771,7773,7775,7777,7779,7781,7783,7785],{},[791,7753,2963],{"mathvariant":906},[791,7755,2808],{},[980,7757,982],{},[791,7759,3180],{},[791,7761,3249],{},[980,7763,3252],{},[791,7765,793],{},[980,7767,982],{},[1084,7769,7770],{},"83.472",[980,7772,6382],{},[791,7774,3249],{},[980,7776,3252],{},[980,7778,1082],{"stretchy":1081},[1084,7780,7440],{},[980,7782,1096],{"stretchy":1081},[980,7784,7437],{},[1084,7786,7787],{},"66.185987",[795,7789,7790],{"encoding":797},"\\Delta E=D\\sin\\theta=83.472\\times\\sin(2.226045952)\\approx66.185987",[772,7792,7794,7815,7845,7864,7891],{"className":7793,"ariaHidden":803},[802],[772,7795,7797,7800,7803,7806,7809,7812],{"className":7796},[807],[772,7798],{"className":7799,"style":2758},[811],[772,7801,2963],{"className":7802},[816],[772,7804,2808],{"className":7805,"style":2823},[816,817],[772,7807],{"className":7808,"style":1004},[1003],[772,7810,982],{"className":7811},[1008],[772,7813],{"className":7814,"style":1004},[1003],[772,7816,7818,7821,7824,7827,7830,7833,7836,7839,7842],{"className":7817},[807],[772,7819],{"className":7820,"style":812},[811],[772,7822,3180],{"className":7823,"style":818},[816,817],[772,7825],{"className":7826,"style":3294},[1003],[772,7828,3249],{"className":7829},[3298],[772,7831],{"className":7832,"style":3294},[1003],[772,7834,793],{"className":7835,"style":818},[816,817],[772,7837],{"className":7838,"style":1004},[1003],[772,7840,982],{"className":7841},[1008],[772,7843],{"className":7844,"style":1004},[1003],[772,7846,7848,7852,7855,7858,7861],{"className":7847},[807],[772,7849],{"className":7850,"style":7851},[811],"height:0.7278em;vertical-align:-0.0833em;",[772,7853,7770],{"className":7854},[816],[772,7856],{"className":7857,"style":1121},[1003],[772,7859,6382],{"className":7860},[1125],[772,7862],{"className":7863,"style":1121},[1003],[772,7865,7867,7870,7873,7876,7879,7882,7885,7888],{"className":7866},[807],[772,7868],{"className":7869,"style":1109},[811],[772,7871,3249],{"className":7872},[3298],[772,7874,1082],{"className":7875},[1113],[772,7877,7440],{"className":7878},[816],[772,7880,1096],{"className":7881},[1141],[772,7883],{"className":7884,"style":1004},[1003],[772,7886,7437],{"className":7887},[1008],[772,7889],{"className":7890,"style":1004},[1003],[772,7892,7894,7897],{"className":7893},[807],[772,7895],{"className":7896,"style":5899},[811],[772,7898,7787],{"className":7899},[816],[11,7901,7902],{},"The Easting offset is approximately 66.185987 m.",[11,7904,7905],{},[772,7906,7908,7957],{"className":7907},[775],[772,7909,7911],{"className":7910},[779],[781,7912,7913],{"xmlns":783},[785,7914,7915,7954],{},[788,7916,7917,7919,7921,7923,7925,7927,7929,7931,7933,7935,7937,7939,7941,7943,7945,7947,7949,7951],{},[791,7918,2963],{"mathvariant":906},[791,7920,2839],{},[980,7922,982],{},[791,7924,3180],{},[791,7926,3327],{},[980,7928,3252],{},[791,7930,793],{},[980,7932,982],{},[1084,7934,7770],{},[980,7936,6382],{},[791,7938,3327],{},[980,7940,3252],{},[980,7942,1082],{"stretchy":1081},[1084,7944,7440],{},[980,7946,1096],{"stretchy":1081},[980,7948,7437],{},[980,7950,1091],{},[1084,7952,7953],{},"50.861997",[795,7955,7956],{"encoding":797},"\\Delta N=D\\cos\\theta=83.472\\times\\cos(2.226045952)\\approx-50.861997",[772,7958,7960,7981,8011,8029,8056],{"className":7959,"ariaHidden":803},[802],[772,7961,7963,7966,7969,7972,7975,7978],{"className":7962},[807],[772,7964],{"className":7965,"style":2758},[811],[772,7967,2963],{"className":7968},[816],[772,7970,2839],{"className":7971,"style":2854},[816,817],[772,7973],{"className":7974,"style":1004},[1003],[772,7976,982],{"className":7977},[1008],[772,7979],{"className":7980,"style":1004},[1003],[772,7982,7984,7987,7990,7993,7996,7999,8002,8005,8008],{"className":7983},[807],[772,7985],{"className":7986,"style":812},[811],[772,7988,3180],{"className":7989,"style":818},[816,817],[772,7991],{"className":7992,"style":3294},[1003],[772,7994,3327],{"className":7995},[3298],[772,7997],{"className":7998,"style":3294},[1003],[772,8000,793],{"className":8001,"style":818},[816,817],[772,8003],{"className":8004,"style":1004},[1003],[772,8006,982],{"className":8007},[1008],[772,8009],{"className":8010,"style":1004},[1003],[772,8012,8014,8017,8020,8023,8026],{"className":8013},[807],[772,8015],{"className":8016,"style":7851},[811],[772,8018,7770],{"className":8019},[816],[772,8021],{"className":8022,"style":1121},[1003],[772,8024,6382],{"className":8025},[1125],[772,8027],{"className":8028,"style":1121},[1003],[772,8030,8032,8035,8038,8041,8044,8047,8050,8053],{"className":8031},[807],[772,8033],{"className":8034,"style":1109},[811],[772,8036,3327],{"className":8037},[3298],[772,8039,1082],{"className":8040},[1113],[772,8042,7440],{"className":8043},[816],[772,8045,1096],{"className":8046},[1141],[772,8048],{"className":8049,"style":1004},[1003],[772,8051,7437],{"className":8052},[1008],[772,8054],{"className":8055,"style":1004},[1003],[772,8057,8059,8062,8065],{"className":8058},[807],[772,8060],{"className":8061,"style":7851},[811],[772,8063,1091],{"className":8064},[816],[772,8066,7953],{"className":8067},[816],[11,8069,8070],{},"The Northing offset is approximately −50.861997 m.",[11,8072,8073],{},[318,8074,8075],{},"Step 3: Round the offsets to millimeter precision (3 decimal places)",[335,8077,8078,8138],{},[338,8079,8080,8137],{},[772,8081,8083,8104],{"className":8082},[775],[772,8084,8086],{"className":8085},[779],[781,8087,8088],{"xmlns":783},[785,8089,8090,8101],{},[788,8091,8092,8094,8096,8098],{},[791,8093,2963],{"mathvariant":906},[791,8095,2808],{},[980,8097,7437],{},[1084,8099,8100],{},"66.186",[795,8102,8103],{"encoding":797},"\\Delta E\\approx66.186",[772,8105,8107,8128],{"className":8106,"ariaHidden":803},[802],[772,8108,8110,8113,8116,8119,8122,8125],{"className":8109},[807],[772,8111],{"className":8112,"style":2758},[811],[772,8114,2963],{"className":8115},[816],[772,8117,2808],{"className":8118,"style":2823},[816,817],[772,8120],{"className":8121,"style":1004},[1003],[772,8123,7437],{"className":8124},[1008],[772,8126],{"className":8127,"style":1004},[1003],[772,8129,8131,8134],{"className":8130},[807],[772,8132],{"className":8133,"style":5899},[811],[772,8135,8100],{"className":8136},[816]," m",[338,8139,8140,8137],{},[772,8141,8143,8166],{"className":8142},[775],[772,8144,8146],{"className":8145},[779],[781,8147,8148],{"xmlns":783},[785,8149,8150,8163],{},[788,8151,8152,8154,8156,8158,8160],{},[791,8153,2963],{"mathvariant":906},[791,8155,2839],{},[980,8157,7437],{},[980,8159,1091],{},[1084,8161,8162],{},"50.862",[795,8164,8165],{"encoding":797},"\\Delta N\\approx-50.862",[772,8167,8169,8190],{"className":8168,"ariaHidden":803},[802],[772,8170,8172,8175,8178,8181,8184,8187],{"className":8171},[807],[772,8173],{"className":8174,"style":2758},[811],[772,8176,2963],{"className":8177},[816],[772,8179,2839],{"className":8180,"style":2854},[816,817],[772,8182],{"className":8183,"style":1004},[1003],[772,8185,7437],{"className":8186},[1008],[772,8188],{"className":8189,"style":1004},[1003],[772,8191,8193,8196,8199],{"className":8192},[807],[772,8194],{"className":8195,"style":7851},[811],[772,8197,1091],{"className":8198},[816],[772,8200,8162],{"className":8201},[816],[11,8203,8204],{},[318,8205,8206],{},"Step 4: Calculate the absolute coordinates of Station B",[11,8208,8209],{},[772,8210,8212,8258],{"className":8211},[775],[772,8213,8215],{"className":8214},[779],[781,8216,8217],{"xmlns":783},[785,8218,8219,8255],{},[788,8220,8221,8227,8229,8235,8237,8239,8241,8243,8246,8248,8250,8252],{},[3525,8222,8223,8225],{},[791,8224,2808],{},[1084,8226,1872],{},[980,8228,982],{},[3525,8230,8231,8233],{},[791,8232,2808],{},[1084,8234,2170],{},[980,8236,1353],{},[791,8238,2963],{"mathvariant":906},[791,8240,2808],{},[980,8242,982],{},[1084,8244,8245],{},"500000.000",[980,8247,1353],{},[1084,8249,8100],{},[980,8251,982],{},[1084,8253,8254],{},"500066.186",[795,8256,8257],{"encoding":797},"E_2=E_1+\\Delta E=500000.000+66.186=500066.186",[772,8259,8261,8316,8371,8392,8410,8428],{"className":8260,"ariaHidden":803},[802],[772,8262,8264,8267,8307,8310,8313],{"className":8263},[807],[772,8265],{"className":8266,"style":3543},[811],[772,8268,8270,8273],{"className":8269},[816],[772,8271,2808],{"className":8272,"style":2823},[816,817],[772,8274,8276],{"className":8275},[1752],[772,8277,8279,8299],{"className":8278},[1756,3556],[772,8280,8282,8296],{"className":8281},[1760],[772,8283,8285],{"className":8284,"style":3563},[1764],[772,8286,8287,8290],{"style":3566},[772,8288],{"className":8289,"style":1773},[1772],[772,8291,8293],{"className":8292},[1777,1778,1779,1780],[772,8294,1872],{"className":8295},[816,1780],[772,8297,3580],{"className":8298},[3579],[772,8300,8302],{"className":8301},[1760],[772,8303,8305],{"className":8304,"style":3587},[1764],[772,8306],{},[772,8308],{"className":8309,"style":1004},[1003],[772,8311,982],{"className":8312},[1008],[772,8314],{"className":8315,"style":1004},[1003],[772,8317,8319,8322,8362,8365,8368],{"className":8318},[807],[772,8320],{"className":8321,"style":3543},[811],[772,8323,8325,8328],{"className":8324},[816],[772,8326,2808],{"className":8327,"style":2823},[816,817],[772,8329,8331],{"className":8330},[1752],[772,8332,8334,8354],{"className":8333},[1756,3556],[772,8335,8337,8351],{"className":8336},[1760],[772,8338,8340],{"className":8339,"style":3563},[1764],[772,8341,8342,8345],{"style":3566},[772,8343],{"className":8344,"style":1773},[1772],[772,8346,8348],{"className":8347},[1777,1778,1779,1780],[772,8349,2170],{"className":8350},[816,1780],[772,8352,3580],{"className":8353},[3579],[772,8355,8357],{"className":8356},[1760],[772,8358,8360],{"className":8359,"style":3587},[1764],[772,8361],{},[772,8363],{"className":8364,"style":1121},[1003],[772,8366,1353],{"className":8367},[1125],[772,8369],{"className":8370,"style":1121},[1003],[772,8372,8374,8377,8380,8383,8386,8389],{"className":8373},[807],[772,8375],{"className":8376,"style":2758},[811],[772,8378,2963],{"className":8379},[816],[772,8381,2808],{"className":8382,"style":2823},[816,817],[772,8384],{"className":8385,"style":1004},[1003],[772,8387,982],{"className":8388},[1008],[772,8390],{"className":8391,"style":1004},[1003],[772,8393,8395,8398,8401,8404,8407],{"className":8394},[807],[772,8396],{"className":8397,"style":7851},[811],[772,8399,8245],{"className":8400},[816],[772,8402],{"className":8403,"style":1121},[1003],[772,8405,1353],{"className":8406},[1125],[772,8408],{"className":8409,"style":1121},[1003],[772,8411,8413,8416,8419,8422,8425],{"className":8412},[807],[772,8414],{"className":8415,"style":5899},[811],[772,8417,8100],{"className":8418},[816],[772,8420],{"className":8421,"style":1004},[1003],[772,8423,982],{"className":8424},[1008],[772,8426],{"className":8427,"style":1004},[1003],[772,8429,8431,8434],{"className":8430},[807],[772,8432],{"className":8433,"style":5899},[811],[772,8435,8254],{"className":8436},[816],[11,8438,8439],{},"The Easting of Station B is 500066.186 m.",[11,8441,8442],{},[772,8443,8445,8497],{"className":8444},[775],[772,8446,8448],{"className":8447},[779],[781,8449,8450],{"xmlns":783},[785,8451,8452,8494],{},[788,8453,8454,8460,8462,8468,8470,8472,8474,8476,8479,8481,8483,8485,8487,8489,8491],{},[3525,8455,8456,8458],{},[791,8457,2839],{},[1084,8459,1872],{},[980,8461,982],{},[3525,8463,8464,8466],{},[791,8465,2839],{},[1084,8467,2170],{},[980,8469,1353],{},[791,8471,2963],{"mathvariant":906},[791,8473,2839],{},[980,8475,982],{},[1084,8477,8478],{},"150000.000",[980,8480,1353],{},[980,8482,1082],{"stretchy":1081},[980,8484,1091],{},[1084,8486,8162],{},[980,8488,1096],{"stretchy":1081},[980,8490,982],{},[1084,8492,8493],{},"149949.138",[795,8495,8496],{"encoding":797},"N_2=N_1+\\Delta N=150000.000+(-50.862)=149949.138",[772,8498,8500,8555,8610,8631,8649,8676],{"className":8499,"ariaHidden":803},[802],[772,8501,8503,8506,8546,8549,8552],{"className":8502},[807],[772,8504],{"className":8505,"style":3543},[811],[772,8507,8509,8512],{"className":8508},[816],[772,8510,2839],{"className":8511,"style":2854},[816,817],[772,8513,8515],{"className":8514},[1752],[772,8516,8518,8538],{"className":8517},[1756,3556],[772,8519,8521,8535],{"className":8520},[1760],[772,8522,8524],{"className":8523,"style":3563},[1764],[772,8525,8526,8529],{"style":3640},[772,8527],{"className":8528,"style":1773},[1772],[772,8530,8532],{"className":8531},[1777,1778,1779,1780],[772,8533,1872],{"className":8534},[816,1780],[772,8536,3580],{"className":8537},[3579],[772,8539,8541],{"className":8540},[1760],[772,8542,8544],{"className":8543,"style":3587},[1764],[772,8545],{},[772,8547],{"className":8548,"style":1004},[1003],[772,8550,982],{"className":8551},[1008],[772,8553],{"className":8554,"style":1004},[1003],[772,8556,8558,8561,8601,8604,8607],{"className":8557},[807],[772,8559],{"className":8560,"style":3543},[811],[772,8562,8564,8567],{"className":8563},[816],[772,8565,2839],{"className":8566,"style":2854},[816,817],[772,8568,8570],{"className":8569},[1752],[772,8571,8573,8593],{"className":8572},[1756,3556],[772,8574,8576,8590],{"className":8575},[1760],[772,8577,8579],{"className":8578,"style":3563},[1764],[772,8580,8581,8584],{"style":3640},[772,8582],{"className":8583,"style":1773},[1772],[772,8585,8587],{"className":8586},[1777,1778,1779,1780],[772,8588,2170],{"className":8589},[816,1780],[772,8591,3580],{"className":8592},[3579],[772,8594,8596],{"className":8595},[1760],[772,8597,8599],{"className":8598,"style":3587},[1764],[772,8600],{},[772,8602],{"className":8603,"style":1121},[1003],[772,8605,1353],{"className":8606},[1125],[772,8608],{"className":8609,"style":1121},[1003],[772,8611,8613,8616,8619,8622,8625,8628],{"className":8612},[807],[772,8614],{"className":8615,"style":2758},[811],[772,8617,2963],{"className":8618},[816],[772,8620,2839],{"className":8621,"style":2854},[816,817],[772,8623],{"className":8624,"style":1004},[1003],[772,8626,982],{"className":8627},[1008],[772,8629],{"className":8630,"style":1004},[1003],[772,8632,8634,8637,8640,8643,8646],{"className":8633},[807],[772,8635],{"className":8636,"style":7851},[811],[772,8638,8478],{"className":8639},[816],[772,8641],{"className":8642,"style":1121},[1003],[772,8644,1353],{"className":8645},[1125],[772,8647],{"className":8648,"style":1121},[1003],[772,8650,8652,8655,8658,8661,8664,8667,8670,8673],{"className":8651},[807],[772,8653],{"className":8654,"style":1109},[811],[772,8656,1082],{"className":8657},[1113],[772,8659,1091],{"className":8660},[816],[772,8662,8162],{"className":8663},[816],[772,8665,1096],{"className":8666},[1141],[772,8668],{"className":8669,"style":1004},[1003],[772,8671,982],{"className":8672},[1008],[772,8674],{"className":8675,"style":1004},[1003],[772,8677,8679,8682],{"className":8678},[807],[772,8680],{"className":8681,"style":5899},[811],[772,8683,8493],{"className":8684},[816],[11,8686,8687],{},"The Northing of Station B is 149949.138 m.",[11,8689,8690,8691,5868],{},"The final coordinates of Station B are ",[318,8692,8693],{},"500066.186 E, 149949.138 N",[706,8695,8697],{"id":8696},"worked-example-2-rectangular-to-polar-check-back-verification","Worked Example 2: Rectangular to Polar (Check-Back Verification)",[11,8699,8700],{},"To verify the calculations, run the inverse process using the rounded coordinates of Station B (500066.186 E, 149949.138 N) to compute the distance and azimuth back to Station A (500000.000 E, 150000.000 N).",[11,8702,8703],{},[318,8704,8705],{},"Step 1: Calculate the coordinate differences",[11,8707,8708],{},[772,8709,8711,8755],{"className":8710},[775],[772,8712,8714],{"className":8713},[779],[781,8715,8716],{"xmlns":783},[785,8717,8718,8752],{},[788,8719,8720,8722,8724,8726,8732,8734,8740,8742,8744,8746,8748,8750],{},[791,8721,2963],{"mathvariant":906},[791,8723,2808],{},[980,8725,982],{},[3525,8727,8728,8730],{},[791,8729,2808],{},[1084,8731,1872],{},[980,8733,1091],{},[3525,8735,8736,8738],{},[791,8737,2808],{},[1084,8739,2170],{},[980,8741,982],{},[1084,8743,8254],{},[980,8745,1091],{},[1084,8747,8245],{},[980,8749,982],{},[1084,8751,8100],{},[795,8753,8754],{"encoding":797},"\\Delta E=E_2-E_1=500066.186-500000.000=66.186",[772,8756,8758,8779,8834,8889,8907,8925],{"className":8757,"ariaHidden":803},[802],[772,8759,8761,8764,8767,8770,8773,8776],{"className":8760},[807],[772,8762],{"className":8763,"style":2758},[811],[772,8765,2963],{"className":8766},[816],[772,8768,2808],{"className":8769,"style":2823},[816,817],[772,8771],{"className":8772,"style":1004},[1003],[772,8774,982],{"className":8775},[1008],[772,8777],{"className":8778,"style":1004},[1003],[772,8780,8782,8785,8825,8828,8831],{"className":8781},[807],[772,8783],{"className":8784,"style":3543},[811],[772,8786,8788,8791],{"className":8787},[816],[772,8789,2808],{"className":8790,"style":2823},[816,817],[772,8792,8794],{"className":8793},[1752],[772,8795,8797,8817],{"className":8796},[1756,3556],[772,8798,8800,8814],{"className":8799},[1760],[772,8801,8803],{"className":8802,"style":3563},[1764],[772,8804,8805,8808],{"style":3566},[772,8806],{"className":8807,"style":1773},[1772],[772,8809,8811],{"className":8810},[1777,1778,1779,1780],[772,8812,1872],{"className":8813},[816,1780],[772,8815,3580],{"className":8816},[3579],[772,8818,8820],{"className":8819},[1760],[772,8821,8823],{"className":8822,"style":3587},[1764],[772,8824],{},[772,8826],{"className":8827,"style":1121},[1003],[772,8829,1091],{"className":8830},[1125],[772,8832],{"className":8833,"style":1121},[1003],[772,8835,8837,8840,8880,8883,8886],{"className":8836},[807],[772,8838],{"className":8839,"style":3543},[811],[772,8841,8843,8846],{"className":8842},[816],[772,8844,2808],{"className":8845,"style":2823},[816,817],[772,8847,8849],{"className":8848},[1752],[772,8850,8852,8872],{"className":8851},[1756,3556],[772,8853,8855,8869],{"className":8854},[1760],[772,8856,8858],{"className":8857,"style":3563},[1764],[772,8859,8860,8863],{"style":3566},[772,8861],{"className":8862,"style":1773},[1772],[772,8864,8866],{"className":8865},[1777,1778,1779,1780],[772,8867,2170],{"className":8868},[816,1780],[772,8870,3580],{"className":8871},[3579],[772,8873,8875],{"className":8874},[1760],[772,8876,8878],{"className":8877,"style":3587},[1764],[772,8879],{},[772,8881],{"className":8882,"style":1004},[1003],[772,8884,982],{"className":8885},[1008],[772,8887],{"className":8888,"style":1004},[1003],[772,8890,8892,8895,8898,8901,8904],{"className":8891},[807],[772,8893],{"className":8894,"style":7851},[811],[772,8896,8254],{"className":8897},[816],[772,8899],{"className":8900,"style":1121},[1003],[772,8902,1091],{"className":8903},[1125],[772,8905],{"className":8906,"style":1121},[1003],[772,8908,8910,8913,8916,8919,8922],{"className":8909},[807],[772,8911],{"className":8912,"style":5899},[811],[772,8914,8245],{"className":8915},[816],[772,8917],{"className":8918,"style":1004},[1003],[772,8920,982],{"className":8921},[1008],[772,8923],{"className":8924,"style":1004},[1003],[772,8926,8928,8931],{"className":8927},[807],[772,8929],{"className":8930,"style":5899},[811],[772,8932,8100],{"className":8933},[816],[11,8935,8936],{},"The difference in Easting is 66.186 m.",[11,8938,8939],{},[772,8940,8942,8988],{"className":8941},[775],[772,8943,8945],{"className":8944},[779],[781,8946,8947],{"xmlns":783},[785,8948,8949,8985],{},[788,8950,8951,8953,8955,8957,8963,8965,8971,8973,8975,8977,8979,8981,8983],{},[791,8952,2963],{"mathvariant":906},[791,8954,2839],{},[980,8956,982],{},[3525,8958,8959,8961],{},[791,8960,2839],{},[1084,8962,1872],{},[980,8964,1091],{},[3525,8966,8967,8969],{},[791,8968,2839],{},[1084,8970,2170],{},[980,8972,982],{},[1084,8974,8493],{},[980,8976,1091],{},[1084,8978,8478],{},[980,8980,982],{},[980,8982,1091],{},[1084,8984,8162],{},[795,8986,8987],{"encoding":797},"\\Delta N=N_2-N_1=149949.138-150000.000=-50.862",[772,8989,8991,9012,9067,9122,9140,9158],{"className":8990,"ariaHidden":803},[802],[772,8992,8994,8997,9000,9003,9006,9009],{"className":8993},[807],[772,8995],{"className":8996,"style":2758},[811],[772,8998,2963],{"className":8999},[816],[772,9001,2839],{"className":9002,"style":2854},[816,817],[772,9004],{"className":9005,"style":1004},[1003],[772,9007,982],{"className":9008},[1008],[772,9010],{"className":9011,"style":1004},[1003],[772,9013,9015,9018,9058,9061,9064],{"className":9014},[807],[772,9016],{"className":9017,"style":3543},[811],[772,9019,9021,9024],{"className":9020},[816],[772,9022,2839],{"className":9023,"style":2854},[816,817],[772,9025,9027],{"className":9026},[1752],[772,9028,9030,9050],{"className":9029},[1756,3556],[772,9031,9033,9047],{"className":9032},[1760],[772,9034,9036],{"className":9035,"style":3563},[1764],[772,9037,9038,9041],{"style":3640},[772,9039],{"className":9040,"style":1773},[1772],[772,9042,9044],{"className":9043},[1777,1778,1779,1780],[772,9045,1872],{"className":9046},[816,1780],[772,9048,3580],{"className":9049},[3579],[772,9051,9053],{"className":9052},[1760],[772,9054,9056],{"className":9055,"style":3587},[1764],[772,9057],{},[772,9059],{"className":9060,"style":1121},[1003],[772,9062,1091],{"className":9063},[1125],[772,9065],{"className":9066,"style":1121},[1003],[772,9068,9070,9073,9113,9116,9119],{"className":9069},[807],[772,9071],{"className":9072,"style":3543},[811],[772,9074,9076,9079],{"className":9075},[816],[772,9077,2839],{"className":9078,"style":2854},[816,817],[772,9080,9082],{"className":9081},[1752],[772,9083,9085,9105],{"className":9084},[1756,3556],[772,9086,9088,9102],{"className":9087},[1760],[772,9089,9091],{"className":9090,"style":3563},[1764],[772,9092,9093,9096],{"style":3640},[772,9094],{"className":9095,"style":1773},[1772],[772,9097,9099],{"className":9098},[1777,1778,1779,1780],[772,9100,2170],{"className":9101},[816,1780],[772,9103,3580],{"className":9104},[3579],[772,9106,9108],{"className":9107},[1760],[772,9109,9111],{"className":9110,"style":3587},[1764],[772,9112],{},[772,9114],{"className":9115,"style":1004},[1003],[772,9117,982],{"className":9118},[1008],[772,9120],{"className":9121,"style":1004},[1003],[772,9123,9125,9128,9131,9134,9137],{"className":9124},[807],[772,9126],{"className":9127,"style":7851},[811],[772,9129,8493],{"className":9130},[816],[772,9132],{"className":9133,"style":1121},[1003],[772,9135,1091],{"className":9136},[1125],[772,9138],{"className":9139,"style":1121},[1003],[772,9141,9143,9146,9149,9152,9155],{"className":9142},[807],[772,9144],{"className":9145,"style":5899},[811],[772,9147,8478],{"className":9148},[816],[772,9150],{"className":9151,"style":1004},[1003],[772,9153,982],{"className":9154},[1008],[772,9156],{"className":9157,"style":1004},[1003],[772,9159,9161,9164,9167],{"className":9160},[807],[772,9162],{"className":9163,"style":7851},[811],[772,9165,1091],{"className":9166},[816],[772,9168,8162],{"className":9169},[816],[11,9171,9172],{},"The difference in Northing is −50.862 m.",[11,9174,9175],{},[318,9176,9177,9178,1096],{},"Step 2: Calculate the horizontal distance (",[772,9179,9181,9194],{"className":9180},[775],[772,9182,9184],{"className":9183},[779],[781,9185,9186],{"xmlns":783},[785,9187,9188,9192],{},[788,9189,9190],{},[791,9191,3180],{},[795,9193,3180],{"encoding":797},[772,9195,9197],{"className":9196,"ariaHidden":803},[802],[772,9198,9200,9203],{"className":9199},[807],[772,9201],{"className":9202,"style":2758},[811],[772,9204,3180],{"className":9205,"style":818},[816,817],[11,9207,9208],{},[772,9209,9211,9279],{"className":9210},[775],[772,9212,9214],{"className":9213},[779],[781,9215,9216],{"xmlns":783},[785,9217,9218,9276],{},[788,9219,9220,9222,9224,9246,9248],{},[791,9221,3180],{},[980,9223,982],{},[5347,9225,9226],{},[788,9227,9228,9230,9236,9238,9240],{},[791,9229,2963],{"mathvariant":906},[1636,9231,9232,9234],{},[791,9233,2808],{},[1084,9235,1872],{},[980,9237,1353],{},[791,9239,2963],{"mathvariant":906},[1636,9241,9242,9244],{},[791,9243,2839],{},[1084,9245,1872],{},[980,9247,982],{},[5347,9249,9250],{},[788,9251,9252,9254,9256,9262,9264,9266,9268,9270],{},[980,9253,1082],{"stretchy":1081},[1084,9255,8100],{},[1636,9257,9258,9260],{},[980,9259,1096],{"stretchy":1081},[1084,9261,1872],{},[980,9263,1353],{},[980,9265,1082],{"stretchy":1081},[980,9267,1091],{},[1084,9269,8162],{},[1636,9271,9272,9274],{},[980,9273,1096],{"stretchy":1081},[1084,9275,1872],{},[795,9277,9278],{"encoding":797},"D=\\sqrt{\\Delta E^2+\\Delta N^2}=\\sqrt{(66.186)^2+(-50.862)^2}",[772,9280,9282,9300,9432],{"className":9281,"ariaHidden":803},[802],[772,9283,9285,9288,9291,9294,9297],{"className":9284},[807],[772,9286],{"className":9287,"style":2758},[811],[772,9289,3180],{"className":9290,"style":818},[816,817],[772,9292],{"className":9293,"style":1004},[1003],[772,9295,982],{"className":9296},[1008],[772,9298],{"className":9299,"style":1004},[1003],[772,9301,9303,9306,9423,9426,9429],{"className":9302},[807],[772,9304],{"className":9305,"style":5399},[811],[772,9307,9309],{"className":9308},[816,5403],[772,9310,9312,9415],{"className":9311},[1756,3556],[772,9313,9315,9412],{"className":9314},[1760],[772,9316,9318,9400],{"className":9317,"style":5413},[1764],[772,9319,9321,9324],{"className":9320,"style":5418},[5417],[772,9322],{"className":9323,"style":5422},[1772],[772,9325,9327,9330,9359,9362,9365,9368,9371],{"className":9326,"style":5426},[816],[772,9328,2963],{"className":9329},[816],[772,9331,9333,9336],{"className":9332},[816],[772,9334,2808],{"className":9335,"style":2823},[816,817],[772,9337,9339],{"className":9338},[1752],[772,9340,9342],{"className":9341},[1756],[772,9343,9345],{"className":9344},[1760],[772,9346,9348],{"className":9347,"style":5448},[1764],[772,9349,9350,9353],{"style":5451},[772,9351],{"className":9352,"style":1773},[1772],[772,9354,9356],{"className":9355},[1777,1778,1779,1780],[772,9357,1872],{"className":9358},[816,1780],[772,9360],{"className":9361,"style":1121},[1003],[772,9363,1353],{"className":9364},[1125],[772,9366],{"className":9367,"style":1121},[1003],[772,9369,2963],{"className":9370},[816],[772,9372,9374,9377],{"className":9373},[816],[772,9375,2839],{"className":9376,"style":2854},[816,817],[772,9378,9380],{"className":9379},[1752],[772,9381,9383],{"className":9382},[1756],[772,9384,9386],{"className":9385},[1760],[772,9387,9389],{"className":9388,"style":5448},[1764],[772,9390,9391,9394],{"style":5451},[772,9392],{"className":9393,"style":1773},[1772],[772,9395,9397],{"className":9396},[1777,1778,1779,1780],[772,9398,1872],{"className":9399},[816,1780],[772,9401,9402,9405],{"style":5504},[772,9403],{"className":9404,"style":5422},[1772],[772,9406,9408],{"className":9407,"style":5512},[5511],[5514,9409,9410],{"xmlns":5516,"width":5517,"height":5518,"viewBox":5519,"preserveAspectRatio":5520},[5522,9411],{"d":5524},[772,9413,3580],{"className":9414},[3579],[772,9416,9418],{"className":9417},[1760],[772,9419,9421],{"className":9420,"style":5534},[1764],[772,9422],{},[772,9424],{"className":9425,"style":1004},[1003],[772,9427,982],{"className":9428},[1008],[772,9430],{"className":9431,"style":1004},[1003],[772,9433,9435,9439],{"className":9434},[807],[772,9436],{"className":9437,"style":9438},[811],"height:1.24em;vertical-align:-0.305em;",[772,9440,9442],{"className":9441},[816,5403],[772,9443,9445,9566],{"className":9444},[1756,3556],[772,9446,9448,9563],{"className":9447},[1760],[772,9449,9452,9546],{"className":9450,"style":9451},[1764],"height:0.935em;",[772,9453,9456,9460],{"className":9454,"style":9455},[5417],"top:-3.2em;",[772,9457],{"className":9458,"style":9459},[1772],"height:3.2em;",[772,9461,9464,9467,9470,9499,9502,9505,9508,9511,9514,9517],{"className":9462,"style":9463},[816],"padding-left:1em;",[772,9465,1082],{"className":9466},[1113],[772,9468,8100],{"className":9469},[816],[772,9471,9473,9476],{"className":9472},[1141],[772,9474,1096],{"className":9475},[1141],[772,9477,9479],{"className":9478},[1752],[772,9480,9482],{"className":9481},[1756],[772,9483,9485],{"className":9484},[1760],[772,9486,9488],{"className":9487,"style":5448},[1764],[772,9489,9490,9493],{"style":5451},[772,9491],{"className":9492,"style":1773},[1772],[772,9494,9496],{"className":9495},[1777,1778,1779,1780],[772,9497,1872],{"className":9498},[816,1780],[772,9500],{"className":9501,"style":1121},[1003],[772,9503,1353],{"className":9504},[1125],[772,9506],{"className":9507,"style":1121},[1003],[772,9509,1082],{"className":9510},[1113],[772,9512,1091],{"className":9513},[816],[772,9515,8162],{"className":9516},[816],[772,9518,9520,9523],{"className":9519},[1141],[772,9521,1096],{"className":9522},[1141],[772,9524,9526],{"className":9525},[1752],[772,9527,9529],{"className":9528},[1756],[772,9530,9532],{"className":9531},[1760],[772,9533,9535],{"className":9534,"style":5448},[1764],[772,9536,9537,9540],{"style":5451},[772,9538],{"className":9539,"style":1773},[1772],[772,9541,9543],{"className":9542},[1777,1778,1779,1780],[772,9544,1872],{"className":9545},[816,1780],[772,9547,9549,9552],{"style":9548},"top:-2.895em;",[772,9550],{"className":9551,"style":9459},[1772],[772,9553,9556],{"className":9554,"style":9555},[5511],"min-width:1.02em;height:1.28em;",[5514,9557,9560],{"xmlns":5516,"width":5517,"height":9558,"viewBox":9559,"preserveAspectRatio":5520},"1.28em","0 0 400000 1296",[5522,9561],{"d":9562},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[772,9564,3580],{"className":9565},[3579],[772,9567,9569],{"className":9568},[1760],[772,9570,9573],{"className":9571,"style":9572},[1764],"height:0.305em;",[772,9574],{},[11,9576,9577],{},[772,9578,9580,9620],{"className":9579},[775],[772,9581,9583],{"className":9582},[779],[781,9584,9585],{"xmlns":783},[785,9586,9587,9617],{},[788,9588,9589,9591,9593,9605,9607,9612,9614],{},[791,9590,3180],{},[980,9592,982],{},[5347,9594,9595],{},[788,9596,9597,9600,9602],{},[1084,9598,9599],{},"4380.586596",[980,9601,1353],{},[1084,9603,9604],{},"2586.943044",[980,9606,982],{},[5347,9608,9609],{},[1084,9610,9611],{},"6967.52964",[980,9613,7437],{},[1084,9615,9616],{},"83.4717296",[795,9618,9619],{"encoding":797},"D=\\sqrt{4380.586596+2586.943044}=\\sqrt{6967.52964}\\approx83.4717296",[772,9621,9623,9641,9719,9785],{"className":9622,"ariaHidden":803},[802],[772,9624,9626,9629,9632,9635,9638],{"className":9625},[807],[772,9627],{"className":9628,"style":2758},[811],[772,9630,3180],{"className":9631,"style":818},[816,817],[772,9633],{"className":9634,"style":1004},[1003],[772,9636,982],{"className":9637},[1008],[772,9639],{"className":9640,"style":1004},[1003],[772,9642,9644,9648,9710,9713,9716],{"className":9643},[807],[772,9645],{"className":9646,"style":9647},[811],"height:1.04em;vertical-align:-0.1744em;",[772,9649,9651],{"className":9650},[816,5403],[772,9652,9654,9701],{"className":9653},[1756,3556],[772,9655,9657,9698],{"className":9656},[1760],[772,9658,9661,9685],{"className":9659,"style":9660},[1764],"height:0.8656em;",[772,9662,9664,9667],{"className":9663,"style":5418},[5417],[772,9665],{"className":9666,"style":5422},[1772],[772,9668,9670,9673,9676,9679,9682],{"className":9669,"style":5426},[816],[772,9671,9599],{"className":9672},[816],[772,9674],{"className":9675,"style":1121},[1003],[772,9677,1353],{"className":9678},[1125],[772,9680],{"className":9681,"style":1121},[1003],[772,9683,9604],{"className":9684},[816],[772,9686,9688,9691],{"style":9687},"top:-2.8256em;",[772,9689],{"className":9690,"style":5422},[1772],[772,9692,9694],{"className":9693,"style":5512},[5511],[5514,9695,9696],{"xmlns":5516,"width":5517,"height":5518,"viewBox":5519,"preserveAspectRatio":5520},[5522,9697],{"d":5524},[772,9699,3580],{"className":9700},[3579],[772,9702,9704],{"className":9703},[1760],[772,9705,9708],{"className":9706,"style":9707},[1764],"height:0.1744em;",[772,9709],{},[772,9711],{"className":9712,"style":1004},[1003],[772,9714,982],{"className":9715},[1008],[772,9717],{"className":9718,"style":1004},[1003],[772,9720,9722,9726,9776,9779,9782],{"className":9721},[807],[772,9723],{"className":9724,"style":9725},[811],"height:1.04em;vertical-align:-0.1328em;",[772,9727,9729],{"className":9728},[816,5403],[772,9730,9732,9767],{"className":9731},[1756,3556],[772,9733,9735,9764],{"className":9734},[1760],[772,9736,9739,9751],{"className":9737,"style":9738},[1764],"height:0.9072em;",[772,9740,9742,9745],{"className":9741,"style":5418},[5417],[772,9743],{"className":9744,"style":5422},[1772],[772,9746,9748],{"className":9747,"style":5426},[816],[772,9749,9611],{"className":9750},[816],[772,9752,9754,9757],{"style":9753},"top:-2.8672em;",[772,9755],{"className":9756,"style":5422},[1772],[772,9758,9760],{"className":9759,"style":5512},[5511],[5514,9761,9762],{"xmlns":5516,"width":5517,"height":5518,"viewBox":5519,"preserveAspectRatio":5520},[5522,9763],{"d":5524},[772,9765,3580],{"className":9766},[3579],[772,9768,9770],{"className":9769},[1760],[772,9771,9774],{"className":9772,"style":9773},[1764],"height:0.1328em;",[772,9775],{},[772,9777],{"className":9778,"style":1004},[1003],[772,9780,7437],{"className":9781},[1008],[772,9783],{"className":9784,"style":1004},[1003],[772,9786,9788,9791],{"className":9787},[807],[772,9789],{"className":9790,"style":5899},[811],[772,9792,9616],{"className":9793},[816],[11,9795,9796,9797,9800],{},"Rounding to three decimal places yields ",[318,9798,9799],{},"83.472 m",", which matches the original measured distance.",[11,9802,9803],{},[318,9804,9805,9806,1096],{},"Step 3: Calculate the whole-circle bearing (",[772,9807,9809,9822],{"className":9808},[775],[772,9810,9812],{"className":9811},[779],[781,9813,9814],{"xmlns":783},[785,9815,9816,9820],{},[788,9817,9818],{},[791,9819,793],{},[795,9821,798],{"encoding":797},[772,9823,9825],{"className":9824,"ariaHidden":803},[802],[772,9826,9828,9831],{"className":9827},[807],[772,9829],{"className":9830,"style":812},[811],[772,9832,793],{"className":9833,"style":818},[816,817],[11,9835,9836,9837,9839],{},"Using the programmatic ",[2863,9838,5886],{}," function, substitute the coordinate differences:",[11,9841,9842],{},[772,9843,9845,9884],{"className":9844},[775],[772,9846,9848],{"className":9847},[779],[781,9849,9850],{"xmlns":783},[785,9851,9852,9881],{},[788,9853,9854,9860,9862,9864,9866,9868,9870,9872,9874,9876,9878],{},[3525,9855,9856,9858],{},[791,9857,793],{},[1625,9859,5927],{},[980,9861,982],{},[1625,9863,5886],{},[980,9865,1082],{"stretchy":1081},[1084,9867,8100],{},[980,9869,5940],{"separator":803},[980,9871,1091],{},[1084,9873,8162],{},[980,9875,1096],{"stretchy":1081},[980,9877,7437],{},[1084,9879,9880],{},"2.226011974",[795,9882,9883],{"encoding":797},"\\theta_{\\text{rad}}=\\text{atan2}(66.186,-50.862)\\approx2.226011974",[772,9885,9887,9948,9990],{"className":9886,"ariaHidden":803},[802],[772,9888,9890,9893,9939,9942,9945],{"className":9889},[807],[772,9891],{"className":9892,"style":5959},[811],[772,9894,9896,9899],{"className":9895},[816],[772,9897,793],{"className":9898,"style":818},[816,817],[772,9900,9902],{"className":9901},[1752],[772,9903,9905,9931],{"className":9904},[1756,3556],[772,9906,9908,9928],{"className":9907},[1760],[772,9909,9911],{"className":9910,"style":5978},[1764],[772,9912,9913,9916],{"style":5981},[772,9914],{"className":9915,"style":1773},[1772],[772,9917,9919],{"className":9918},[1777,1778,1779,1780],[772,9920,9922],{"className":9921},[816,1780],[772,9923,9925],{"className":9924},[816,1709,1780],[772,9926,5927],{"className":9927},[816,1780],[772,9929,3580],{"className":9930},[3579],[772,9932,9934],{"className":9933},[1760],[772,9935,9937],{"className":9936,"style":3587},[1764],[772,9938],{},[772,9940],{"className":9941,"style":1004},[1003],[772,9943,982],{"className":9944},[1008],[772,9946],{"className":9947,"style":1004},[1003],[772,9949,9951,9954,9960,9963,9966,9969,9972,9975,9978,9981,9984,9987],{"className":9950},[807],[772,9952],{"className":9953,"style":1109},[811],[772,9955,9957],{"className":9956},[816,1709],[772,9958,5886],{"className":9959},[816],[772,9961,1082],{"className":9962},[1113],[772,9964,8100],{"className":9965},[816],[772,9967,5940],{"className":9968},[6041],[772,9970],{"className":9971,"style":3294},[1003],[772,9973,1091],{"className":9974},[816],[772,9976,8162],{"className":9977},[816],[772,9979,1096],{"className":9980},[1141],[772,9982],{"className":9983,"style":1004},[1003],[772,9985,7437],{"className":9986},[1008],[772,9988],{"className":9989,"style":1004},[1003],[772,9991,9993,9996],{"className":9992},[807],[772,9994],{"className":9995,"style":5899},[811],[772,9997,9880],{"className":9998},[816],[11,10000,10001],{},"Convert the radian value to decimal degrees:",[11,10003,10004],{},[772,10005,10007,10052],{"className":10006},[775],[772,10008,10010],{"className":10009},[779],[781,10011,10012],{"xmlns":783},[785,10013,10014,10049],{},[788,10015,10016,10022,10024,10026,10028,10040,10042],{},[3525,10017,10018,10020],{},[791,10019,793],{},[1625,10021,6189],{},[980,10023,982],{},[1084,10025,9880],{},[980,10027,6382],{},[788,10029,10030,10032,10038],{},[980,10031,1082],{"fence":803},[5599,10033,10034,10036],{},[1084,10035,1086],{},[791,10037,6105],{},[980,10039,1096],{"fence":803},[980,10041,7437],{},[1636,10043,10044,10047],{},[1084,10045,10046],{},"127.539818",[980,10048,1642],{},[795,10050,10051],{"encoding":797},"\\theta_{\\text{deg}}=2.226011974\\times\\left(\\frac{180}{\\pi}\\right)\\approx127.539818^\\circ",[772,10053,10055,10116,10134,10232],{"className":10054,"ariaHidden":803},[802],[772,10056,10058,10061,10107,10110,10113],{"className":10057},[807],[772,10059],{"className":10060,"style":6256},[811],[772,10062,10064,10067],{"className":10063},[816],[772,10065,793],{"className":10066,"style":818},[816,817],[772,10068,10070],{"className":10069},[1752],[772,10071,10073,10099],{"className":10072},[1756,3556],[772,10074,10076,10096],{"className":10075},[1760],[772,10077,10079],{"className":10078,"style":5978},[1764],[772,10080,10081,10084],{"style":5981},[772,10082],{"className":10083,"style":1773},[1772],[772,10085,10087],{"className":10086},[1777,1778,1779,1780],[772,10088,10090],{"className":10089},[816,1780],[772,10091,10093],{"className":10092},[816,1709,1780],[772,10094,6189],{"className":10095},[816,1780],[772,10097,3580],{"className":10098},[3579],[772,10100,10102],{"className":10101},[1760],[772,10103,10105],{"className":10104,"style":6301},[1764],[772,10106],{},[772,10108],{"className":10109,"style":1004},[1003],[772,10111,982],{"className":10112},[1008],[772,10114],{"className":10115,"style":1004},[1003],[772,10117,10119,10122,10125,10128,10131],{"className":10118},[807],[772,10120],{"className":10121,"style":7851},[811],[772,10123,9880],{"className":10124},[816],[772,10126],{"className":10127,"style":1121},[1003],[772,10129,6382],{"className":10130},[1125],[772,10132],{"className":10133,"style":1121},[1003],[772,10135,10137,10140,10223,10226,10229],{"className":10136},[807],[772,10138],{"className":10139,"style":6529},[811],[772,10141,10143,10149,10217],{"className":10142},[5664],[772,10144,10146],{"className":10145,"style":5669},[1113,5668],[772,10147,1082],{"className":10148},[5673,6539],[772,10150,10152,10155,10214],{"className":10151},[816],[772,10153],{"className":10154},[1113,5681],[772,10156,10158],{"className":10157},[5599],[772,10159,10161,10206],{"className":10160},[1756,3556],[772,10162,10164,10203],{"className":10163},[1760],[772,10165,10167,10181,10189],{"className":10166,"style":6558},[1764],[772,10168,10169,10172],{"style":5697},[772,10170],{"className":10171,"style":5422},[1772],[772,10173,10175],{"className":10174},[1777,1778,1779,1780],[772,10176,10178],{"className":10177},[816,1780],[772,10179,6105],{"className":10180,"style":6125},[816,817,1780],[772,10182,10183,10186],{"style":5719},[772,10184],{"className":10185,"style":5422},[1772],[772,10187],{"className":10188,"style":5727},[5726],[772,10190,10191,10194],{"style":6583},[772,10192],{"className":10193,"style":5422},[1772],[772,10195,10197],{"className":10196},[1777,1778,1779,1780],[772,10198,10200],{"className":10199},[816,1780],[772,10201,1086],{"className":10202},[816,1780],[772,10204,3580],{"className":10205},[3579],[772,10207,10209],{"className":10208},[1760],[772,10210,10212],{"className":10211,"style":6605},[1764],[772,10213],{},[772,10215],{"className":10216},[1141,5681],[772,10218,10220],{"className":10219,"style":5669},[1141,5668],[772,10221,1096],{"className":10222},[5673,6539],[772,10224],{"className":10225,"style":1004},[1003],[772,10227,7437],{"className":10228},[1008],[772,10230],{"className":10231,"style":1004},[1003],[772,10233,10235,10238,10242],{"className":10234},[807],[772,10236],{"className":10237,"style":1765},[811],[772,10239,10241],{"className":10240},[816],"127.53981",[772,10243,10245,10248],{"className":10244},[816],[772,10246,2612],{"className":10247},[816],[772,10249,10251],{"className":10250},[1752],[772,10252,10254],{"className":10253},[1756],[772,10255,10257],{"className":10256},[1760],[772,10258,10260],{"className":10259,"style":1765},[1764],[772,10261,10262,10265],{"style":1768},[772,10263],{"className":10264,"style":1773},[1772],[772,10266,10268],{"className":10267},[1777,1778,1779,1780],[772,10269,1642],{"className":10270},[1125,1780],[11,10272,10273,10274,10277],{},"Rounding the azimuth to four decimal places yields ",[318,10275,10276],{},"127.5401°"," (or 127°32′24″ in DMS format).",[11,10279,10280],{},[318,10281,10282],{},"Step 4: Analyze the rounding effect",[11,10284,10285],{},"Comparing the original azimuth (127.5430°) with the inversed azimuth (127.5401°) reveals a slight difference of 0.0029° (or 10.4″). This difference is a natural mathematical consequence of rounding the coordinates of Station B to millimeter precision (3 decimal places).",[11,10287,10288],{},"A coordinate rounding shift of just 1 mm over a distance of 83.472 m introduces an angular variation of:",[11,10290,10291],{},[772,10292,10294,10340],{"className":10293},[775],[772,10295,10297],{"className":10296},[779],[781,10298,10299],{"xmlns":783},[785,10300,10301,10337],{},[788,10302,10303,10306,10308,10310,10313,10315,10328,10330],{},[791,10304,10305],{},"δ",[791,10307,793],{},[980,10309,982],{},[791,10311,10312],{},"arcsin",[980,10314,3252],{},[788,10316,10317,10319,10326],{},[980,10318,1082],{"fence":803},[5599,10320,10321,10324],{},[1084,10322,10323],{},"0.001",[1084,10325,7770],{},[980,10327,1096],{"fence":803},[980,10329,7437],{},[1636,10331,10332,10335],{},[1084,10333,10334],{},"0.000687",[980,10336,1642],{},[795,10338,10339],{"encoding":797},"\\delta\\theta=\\arcsin\\left(\\frac{0.001}{83.472}\\right)\\approx0.000687^\\circ",[772,10341,10343,10365,10469],{"className":10342,"ariaHidden":803},[802],[772,10344,10346,10349,10353,10356,10359,10362],{"className":10345},[807],[772,10347],{"className":10348,"style":812},[811],[772,10350,10305],{"className":10351,"style":10352},[816,817],"margin-right:0.0379em;",[772,10354,793],{"className":10355,"style":818},[816,817],[772,10357],{"className":10358,"style":1004},[1003],[772,10360,982],{"className":10361},[1008],[772,10363],{"className":10364,"style":1004},[1003],[772,10366,10368,10371,10374,10377,10460,10463,10466],{"className":10367},[807],[772,10369],{"className":10370,"style":6529},[811],[772,10372,10312],{"className":10373},[3298],[772,10375],{"className":10376,"style":3294},[1003],[772,10378,10380,10386,10454],{"className":10379},[5664],[772,10381,10383],{"className":10382,"style":5669},[1113,5668],[772,10384,1082],{"className":10385},[5673,6539],[772,10387,10389,10392,10451],{"className":10388},[816],[772,10390],{"className":10391},[1113,5681],[772,10393,10395],{"className":10394},[5599],[772,10396,10398,10443],{"className":10397},[1756,3556],[772,10399,10401,10440],{"className":10400},[1760],[772,10402,10404,10418,10426],{"className":10403,"style":6558},[1764],[772,10405,10406,10409],{"style":5697},[772,10407],{"className":10408,"style":5422},[1772],[772,10410,10412],{"className":10411},[1777,1778,1779,1780],[772,10413,10415],{"className":10414},[816,1780],[772,10416,7770],{"className":10417},[816,1780],[772,10419,10420,10423],{"style":5719},[772,10421],{"className":10422,"style":5422},[1772],[772,10424],{"className":10425,"style":5727},[5726],[772,10427,10428,10431],{"style":6583},[772,10429],{"className":10430,"style":5422},[1772],[772,10432,10434],{"className":10433},[1777,1778,1779,1780],[772,10435,10437],{"className":10436},[816,1780],[772,10438,10323],{"className":10439},[816,1780],[772,10441,3580],{"className":10442},[3579],[772,10444,10446],{"className":10445},[1760],[772,10447,10449],{"className":10448,"style":6605},[1764],[772,10450],{},[772,10452],{"className":10453},[1141,5681],[772,10455,10457],{"className":10456,"style":5669},[1141,5668],[772,10458,1096],{"className":10459},[5673,6539],[772,10461],{"className":10462,"style":1004},[1003],[772,10464,7437],{"className":10465},[1008],[772,10467],{"className":10468,"style":1004},[1003],[772,10470,10472,10475,10479],{"className":10471},[807],[772,10473],{"className":10474,"style":1765},[811],[772,10476,10478],{"className":10477},[816],"0.00068",[772,10480,10482,10486],{"className":10481},[816],[772,10483,10485],{"className":10484},[816],"7",[772,10487,10489],{"className":10488},[1752],[772,10490,10492],{"className":10491},[1756],[772,10493,10495],{"className":10494},[1760],[772,10496,10498],{"className":10497,"style":1765},[1764],[772,10499,10500,10503],{"style":1768},[772,10501],{"className":10502,"style":1773},[1772],[772,10504,10506],{"className":10505},[1777,1778,1779,1780],[772,10507,1642],{"className":10508},[1125,1780],[11,10510,10511],{},"This variation is approximately equal to 0.000687 degrees (or 2.5 seconds).",[11,10513,10514],{},"This minor discrepancy demonstrates why high-precision control networks require coordinate records to be maintained to four or five decimal places, while general construction layout tasks are completed to three decimal places.",[323,10516,10518],{"id":10517},"common-computational-errors-in-coordinate-geometry","Common Computational Errors in Coordinate Geometry",[11,10520,10521],{},"When setting up custom spreadsheets, writing automation scripts, or performing manual calculations, civil engineering technicians and surveyors frequently encounter standard computational errors.",[706,10523,10525],{"id":10524},"swapping-trigonometric-functions-eastingnorthing-vs-xy","Swapping trigonometric functions (Easting\u002FNorthing vs. X\u002FY)",[11,10527,10528,10529,10557,10558,10632,10633,3474,10707,10781],{},"Because standard mathematics defines the horizontal axis as ",[772,10530,10532,10545],{"className":10531},[775],[772,10533,10535],{"className":10534},[779],[781,10536,10537],{"xmlns":783},[785,10538,10539,10543],{},[788,10540,10541],{},[791,10542,2746],{},[795,10544,2746],{"encoding":797},[772,10546,10548],{"className":10547,"ariaHidden":803},[802],[772,10549,10551,10554],{"className":10550},[807],[772,10552],{"className":10553,"style":2758},[811],[772,10555,2746],{"className":10556,"style":2762},[816,817]," and uses the cosine function for horizontal projections, beginners often compute ",[772,10559,10561,10587],{"className":10560},[775],[772,10562,10564],{"className":10563},[779],[781,10565,10566],{"xmlns":783},[785,10567,10568,10584],{},[788,10569,10570,10572,10574,10576,10578,10580,10582],{},[791,10571,2963],{"mathvariant":906},[791,10573,2808],{},[980,10575,982],{},[791,10577,3180],{},[791,10579,3327],{},[980,10581,3252],{},[791,10583,793],{},[795,10585,10586],{"encoding":797},"\\Delta E = D \\cos\\theta",[772,10588,10590,10611],{"className":10589,"ariaHidden":803},[802],[772,10591,10593,10596,10599,10602,10605,10608],{"className":10592},[807],[772,10594],{"className":10595,"style":2758},[811],[772,10597,2963],{"className":10598},[816],[772,10600,2808],{"className":10601,"style":2823},[816,817],[772,10603],{"className":10604,"style":1004},[1003],[772,10606,982],{"className":10607},[1008],[772,10609],{"className":10610,"style":1004},[1003],[772,10612,10614,10617,10620,10623,10626,10629],{"className":10613},[807],[772,10615],{"className":10616,"style":812},[811],[772,10618,3180],{"className":10619,"style":818},[816,817],[772,10621],{"className":10622,"style":3294},[1003],[772,10624,3327],{"className":10625},[3298],[772,10627],{"className":10628,"style":3294},[1003],[772,10630,793],{"className":10631,"style":818},[816,817],". In plane surveying, because the angle is measured from the vertical axis (North), the trigonometric functions are reversed: ",[772,10634,10636,10662],{"className":10635},[775],[772,10637,10639],{"className":10638},[779],[781,10640,10641],{"xmlns":783},[785,10642,10643,10659],{},[788,10644,10645,10647,10649,10651,10653,10655,10657],{},[791,10646,2963],{"mathvariant":906},[791,10648,2808],{},[980,10650,982],{},[791,10652,3180],{},[791,10654,3249],{},[980,10656,3252],{},[791,10658,793],{},[795,10660,10661],{"encoding":797},"\\Delta E = D \\sin\\theta",[772,10663,10665,10686],{"className":10664,"ariaHidden":803},[802],[772,10666,10668,10671,10674,10677,10680,10683],{"className":10667},[807],[772,10669],{"className":10670,"style":2758},[811],[772,10672,2963],{"className":10673},[816],[772,10675,2808],{"className":10676,"style":2823},[816,817],[772,10678],{"className":10679,"style":1004},[1003],[772,10681,982],{"className":10682},[1008],[772,10684],{"className":10685,"style":1004},[1003],[772,10687,10689,10692,10695,10698,10701,10704],{"className":10688},[807],[772,10690],{"className":10691,"style":812},[811],[772,10693,3180],{"className":10694,"style":818},[816,817],[772,10696],{"className":10697,"style":3294},[1003],[772,10699,3249],{"className":10700},[3298],[772,10702],{"className":10703,"style":3294},[1003],[772,10705,793],{"className":10706,"style":818},[816,817],[772,10708,10710,10736],{"className":10709},[775],[772,10711,10713],{"className":10712},[779],[781,10714,10715],{"xmlns":783},[785,10716,10717,10733],{},[788,10718,10719,10721,10723,10725,10727,10729,10731],{},[791,10720,2963],{"mathvariant":906},[791,10722,2839],{},[980,10724,982],{},[791,10726,3180],{},[791,10728,3327],{},[980,10730,3252],{},[791,10732,793],{},[795,10734,10735],{"encoding":797},"\\Delta N = D \\cos\\theta",[772,10737,10739,10760],{"className":10738,"ariaHidden":803},[802],[772,10740,10742,10745,10748,10751,10754,10757],{"className":10741},[807],[772,10743],{"className":10744,"style":2758},[811],[772,10746,2963],{"className":10747},[816],[772,10749,2839],{"className":10750,"style":2854},[816,817],[772,10752],{"className":10753,"style":1004},[1003],[772,10755,982],{"className":10756},[1008],[772,10758],{"className":10759,"style":1004},[1003],[772,10761,10763,10766,10769,10772,10775,10778],{"className":10762},[807],[772,10764],{"className":10765,"style":812},[811],[772,10767,3180],{"className":10768,"style":818},[816,817],[772,10770],{"className":10771,"style":3294},[1003],[772,10773,3327],{"className":10774},[3298],[772,10776],{"className":10777,"style":3294},[1003],[772,10779,793],{"className":10780,"style":818},[816,817],". Swapping these functions mirrors the calculated coordinates across the 45° diagonal of the quadrant, resulting in incorrect point locations.",[706,10783,10785],{"id":10784},"radian-vs-degree-discrepancies","Radian vs. degree discrepancies",[11,10787,10788,10789,11026],{},"Calculator and programming environments (such as Microsoft Excel, Python, and JavaScript) default to radians for trigonometric functions. Passing a decimal degree angle directly into a sine or cosine function without first converting it to radians (",[772,10790,10792,10826],{"className":10791},[775],[772,10793,10795],{"className":10794},[779],[781,10796,10797],{"xmlns":783},[785,10798,10799,10823],{},[788,10800,10801,10807,10809,10815,10817],{},[3525,10802,10803,10805],{},[791,10804,793],{},[1625,10806,5927],{},[980,10808,982],{},[3525,10810,10811,10813],{},[791,10812,793],{},[1625,10814,6189],{},[980,10816,6382],{},[5599,10818,10819,10821],{},[791,10820,6105],{},[1084,10822,1086],{},[795,10824,10825],{"encoding":797},"\\theta_{\\text{rad}} = \\theta_{\\text{deg}} \\times \\frac{\\pi}{180}",[772,10827,10829,10890,10951],{"className":10828,"ariaHidden":803},[802],[772,10830,10832,10835,10881,10884,10887],{"className":10831},[807],[772,10833],{"className":10834,"style":5959},[811],[772,10836,10838,10841],{"className":10837},[816],[772,10839,793],{"className":10840,"style":818},[816,817],[772,10842,10844],{"className":10843},[1752],[772,10845,10847,10873],{"className":10846},[1756,3556],[772,10848,10850,10870],{"className":10849},[1760],[772,10851,10853],{"className":10852,"style":5978},[1764],[772,10854,10855,10858],{"style":5981},[772,10856],{"className":10857,"style":1773},[1772],[772,10859,10861],{"className":10860},[1777,1778,1779,1780],[772,10862,10864],{"className":10863},[816,1780],[772,10865,10867],{"className":10866},[816,1709,1780],[772,10868,5927],{"className":10869},[816,1780],[772,10871,3580],{"className":10872},[3579],[772,10874,10876],{"className":10875},[1760],[772,10877,10879],{"className":10878,"style":3587},[1764],[772,10880],{},[772,10882],{"className":10883,"style":1004},[1003],[772,10885,982],{"className":10886},[1008],[772,10888],{"className":10889,"style":1004},[1003],[772,10891,10893,10896,10942,10945,10948],{"className":10892},[807],[772,10894],{"className":10895,"style":6256},[811],[772,10897,10899,10902],{"className":10898},[816],[772,10900,793],{"className":10901,"style":818},[816,817],[772,10903,10905],{"className":10904},[1752],[772,10906,10908,10934],{"className":10907},[1756,3556],[772,10909,10911,10931],{"className":10910},[1760],[772,10912,10914],{"className":10913,"style":5978},[1764],[772,10915,10916,10919],{"style":5981},[772,10917],{"className":10918,"style":1773},[1772],[772,10920,10922],{"className":10921},[1777,1778,1779,1780],[772,10923,10925],{"className":10924},[816,1780],[772,10926,10928],{"className":10927},[816,1709,1780],[772,10929,6189],{"className":10930},[816,1780],[772,10932,3580],{"className":10933},[3579],[772,10935,10937],{"className":10936},[1760],[772,10938,10940],{"className":10939,"style":6301},[1764],[772,10941],{},[772,10943],{"className":10944,"style":1121},[1003],[772,10946,6382],{"className":10947},[1125],[772,10949],{"className":10950,"style":1121},[1003],[772,10952,10954,10958],{"className":10953},[807],[772,10955],{"className":10956,"style":10957},[811],"height:1.0404em;vertical-align:-0.345em;",[772,10959,10961,10964,11023],{"className":10960},[816],[772,10962],{"className":10963},[1113,5681],[772,10965,10967],{"className":10966},[5599],[772,10968,10970,11015],{"className":10969},[1756,3556],[772,10971,10973,11012],{"className":10972},[1760],[772,10974,10976,10990,10998],{"className":10975,"style":7589},[1764],[772,10977,10978,10981],{"style":5697},[772,10979],{"className":10980,"style":5422},[1772],[772,10982,10984],{"className":10983},[1777,1778,1779,1780],[772,10985,10987],{"className":10986},[816,1780],[772,10988,1086],{"className":10989},[816,1780],[772,10991,10992,10995],{"style":5719},[772,10993],{"className":10994,"style":5422},[1772],[772,10996],{"className":10997,"style":5727},[5726],[772,10999,11000,11003],{"style":6583},[772,11001],{"className":11002,"style":5422},[1772],[772,11004,11006],{"className":11005},[1777,1778,1779,1780],[772,11007,11009],{"className":11008},[816,1780],[772,11010,6105],{"className":11011,"style":6125},[816,817,1780],[772,11013,3580],{"className":11014},[3579],[772,11016,11018],{"className":11017},[1760],[772,11019,11021],{"className":11020,"style":6605},[1764],[772,11022],{},[772,11024],{"className":11025},[1141,5681],") yields incorrect coordinate deltas.",[706,11028,11030],{"id":11029},"parameter-order-in-dual-argument-arctangent-functions","Parameter order in dual-argument arctangent functions",[11,11032,11033,11034,11036],{},"The ",[2863,11035,5886],{}," function argument order varies across environments. For grid azimuth from North (clockwise), Surveying Core uses the same convention as JavaScript and Python:",[851,11038,11039,11052],{},[854,11040,11041],{},[857,11042,11043,11046,11049],{},[860,11044,11045],{},"Environment",[860,11047,11048],{},"Function syntax",[860,11050,11051],{},"Surveying mapping",[876,11053,11054,11069,11145],{},[857,11055,11056,11059,11064],{},[881,11057,11058],{},"JavaScript, Python, Surveying Core",[881,11060,11061],{},[2863,11062,11063],{},"atan2(y, x)",[881,11065,11066],{},[2863,11067,11068],{},"atan2(ΔE, ΔN)",[857,11070,11071,11074,11079],{},[881,11072,11073],{},"Spreadsheets (Excel, Google Sheets)",[881,11075,11076],{},[2863,11077,11078],{},"ATAN2(x, y)",[881,11080,11081,11084,11085,11114,11115],{},[2863,11082,11083],{},"ATAN2(ΔE, ΔN)"," when Easting = ",[772,11086,11088,11102],{"className":11087},[775],[772,11089,11091],{"className":11090},[779],[781,11092,11093],{"xmlns":783},[785,11094,11095,11100],{},[788,11096,11097],{},[791,11098,11099],{},"x",[795,11101,11099],{"encoding":797},[772,11103,11105],{"className":11104,"ariaHidden":803},[802],[772,11106,11108,11111],{"className":11107},[807],[772,11109],{"className":11110,"style":845},[811],[772,11112,11099],{"className":11113},[816,817]," and Northing = ",[772,11116,11118,11132],{"className":11117},[775],[772,11119,11121],{"className":11120},[779],[781,11122,11123],{"xmlns":783},[785,11124,11125,11130],{},[788,11126,11127],{},[791,11128,11129],{},"y",[795,11131,11129],{"encoding":797},[772,11133,11135],{"className":11134,"ariaHidden":803},[802],[772,11136,11138,11142],{"className":11137},[807],[772,11139],{"className":11140,"style":11141},[811],"height:0.625em;vertical-align:-0.1944em;",[772,11143,11129],{"className":11144,"style":6125},[816,817],[857,11146,11147,11150,11155],{},[881,11148,11149],{},"Standard calculators",[881,11151,11152],{},[2863,11153,11154],{},"atan(y\u002Fx)",[881,11156,11157],{},"`atan(",[11,11159,11160],{},"Always confirm argument order in your spreadsheet or language documentation before inverting coordinates.",[323,11162,11164],{"id":11163},"field-practices-datums-and-coordinate-reference-systems","Field Practices, Datums, and Coordinate Reference Systems",[11,11166,11167],{},"Applying coordinate geometry in the field requires translating theoretical formulas onto the physical, curved surface of the Earth.",[706,11169,11171],{"id":11170},"grid-azimuth-vs-magnetic-azimuth","Grid Azimuth vs. Magnetic Azimuth",[11,11173,11174],{},"A critical field distinction is the difference between grid azimuths and magnetic azimuths:",[335,11176,11177,11183],{},[338,11178,11179,11182],{},[318,11180,11181],{},"Magnetic azimuth",": Referenced to the Earth's local magnetic north pole, which shifts continuously over time and space. Magnetic azimuths are measured directly using hand compasses or older transit instruments.",[338,11184,11185,11188],{},[318,11186,11187],{},"Grid azimuth",": Tied to a flat map projection (such as State Plane or UTM) where Grid North is parallel to the central meridian.",[11,11190,11191],{},"Surveyors must convert magnetic azimuths to grid azimuths by applying the local magnetic declination. In modern engineering design and construction layout, grid or local project azimuths are used exclusively to maintain spatial consistency across the site.",[706,11193,11195],{"id":11194},"coordinate-reference-systems-crs-and-scale-factors","Coordinate Reference Systems (CRS) and Scale Factors",[11,11197,11198],{},"Plane surveying formulas assume a flat surface, which is appropriate for localized construction projects spanning less than 15 kilometers. On larger infrastructure projects, the Earth's curvature must be accounted for.",[11,11200,11201],{},"When coordinates are projected onto a national grid system (like UTM), the measured ground distances must be reduced to sea level and multiplied by a localized scale factor to compute the corresponding grid distance before calculating coordinates. On smaller projects, surveyors often establish a local coordinate system with a scale factor of 1.0 (known as a \"ground system\") to allow direct layout of design distances without projection adjustments.",[323,11203,11205],{"id":11204},"next-steps","Next Steps",[11,11207,11208],{},"A strong grasp of coordinate geometry is essential for civil engineering and surveying professionals. Transitioning between polar vectors and rectangular grid coordinates is a daily requirement for setting out building foundations, calculating boundary locations, and adjusting traverse loops.",[11,11210,11211,11212,11214],{},"To quickly verify field measurements, compute coordinates, or check manual traverse calculations, use the ",[693,11213,696],{"href":695},". This dedicated tool automates forward and inverse coordinate calculations, handles degree-to-radian conversions, and ensures millimeter-level accuracy for project quality control.",{"title":15,"searchDepth":16,"depth":16,"links":11216},[11217,11218,11223,11228,11229,11230,11231,11235,11240,11244],{"id":684,"depth":16,"text":685},{"id":700,"depth":16,"text":701,"children":11219},[11220,11221,11222],{"id":708,"depth":73,"text":709},{"id":715,"depth":73,"text":716},{"id":722,"depth":73,"text":723},{"id":729,"depth":16,"text":730,"children":11224},[11225,11226,11227],{"id":736,"depth":73,"text":737},{"id":763,"depth":73,"text":764},{"id":1596,"depth":73,"text":1597},{"id":2724,"depth":16,"text":2725},{"id":3094,"depth":16,"text":3095},{"id":4631,"depth":16,"text":4632},{"id":7018,"depth":16,"text":7019,"children":11232},[11233,11234],{"id":7022,"depth":73,"text":7023},{"id":8696,"depth":73,"text":8697},{"id":10517,"depth":16,"text":10518,"children":11236},[11237,11238,11239],{"id":10524,"depth":73,"text":10525},{"id":10784,"depth":73,"text":10785},{"id":11029,"depth":73,"text":11030},{"id":11163,"depth":16,"text":11164,"children":11241},[11242,11243],{"id":11170,"depth":73,"text":11171},{"id":11194,"depth":73,"text":11195},{"id":11204,"depth":16,"text":11205},"tools","Coordinate geometry for plane surveying — azimuth and quadrant bearings, polar ↔ rectangular conversion, worked examples, and common errors.",{},"\u002Flearn\u002Ftools\u002Fbearing-distance",{"title":679,"description":11246},{"loc":11248},"learn\u002Ftools\u002Fbearing-distance","ox0hasMcQmIJtaGlJFXk361ZthJlirgnSUGw1AAdmBE",{"id":11254,"title":11255,"body":11256,"category":11245,"description":13223,"extension":20,"meta":13224,"navigation":22,"order":40,"path":13225,"seo":13226,"sitemap":13227,"stem":13228,"__hash__":13229},"learn\u002Flearn\u002Ftools\u002Ffile-conversion-overview.md","File Conversion Overview",{"type":8,"value":11257,"toc":13186},[11258,11262,11265,11268,11289,11305,11309,11312,11315,11318,11321,11324,11328,11448,11454,11457,11477,11480,11576,11579,11583,11586,11590,11636,11640,11643,11672,11676,11683,11690,11694,11700,11704,11707,11721,11727,11731,11734,11769,11772,11889,11900,11904,11907,11914,11920,11960,11974,11978,11984,12022,12026,12032,12062,12066,12072,12101,12105,12111,12135,12139,12145,12176,12180,12186,12209,12213,12220,12224,12227,12238,12244,12247,12251,12262,12266,12269,12273,12279,12283,12286,12291,12295,12298,12301,12606,12813,12816,12910,12993,12997,13000,13011,13015,13018,13066,13070,13159,13161,13164],[323,11259,11261],{"id":11260},"high-precision-spatial-data-exchange-a-technical-overview-of-survey-file-conversion","High-Precision Spatial Data Exchange: A Technical Overview of Survey File Conversion",[11,11263,11264],{},"In geomatics and land surveying, migrating coordinate and observation datasets across field instruments, office spreadsheet applications, and computer-aided design (CAD) environments is critical to maintaining data integrity. For decades, surveyors and technicians have wrestled with discrepancies between proprietary instrument files, raw text documents, and complex vector drawings.",[11,11266,11267],{},"The Surveying Core platform provides browser-native utilities designed to resolve these data translation friction points:",[335,11269,11270,11277,11283],{},[338,11271,11033,11272,11276],{},[693,11273,11275],{"href":11274},"\u002Ftools\u002Fformat-converter","GSI → CSV converter"," bridges the gap between raw field observations and tabular office formats.",[338,11278,11033,11279,11282],{},[693,11280,11281],{"href":11274},"CSV → GSI converter"," translates adjusted coordinate tables back into instrument-ready files for layout operations.",[338,11284,11033,11285,11288],{},[693,11286,11287],{"href":11274},"CSV → DXF converter"," builds structured 2D CAD point layers optimized for drafting and layout reference.",[11,11290,11291,11292,11295,11296,11300,11301,5868],{},"Converter pages are public for discovery, but ",[318,11293,11294],{},"file conversion requires sign-in",". Calculators remain free; conversion requires an account — see ",[693,11297,11299],{"href":11298},"\u002Fpricing","Pricing",". All parsing and export run in the browser — coordinate files are not uploaded to external servers for conversion. For field reference tables, see the ",[693,11302,11304],{"href":11303},"\u002Flibrary?tab=field","Library",[323,11306,11308],{"id":11307},"formats-in-the-survey-workflow","Formats in the Survey Workflow",[11,11310,11311],{},"The lifecycle of survey data is cyclical, beginning with field acquisition, moving to office analysis and drafting, and returning to the field for stakeout and construction layout. Each stage relies on distinct file structures optimized for hardware constraints, software flexibility, or drawing manipulation.",[11,11313,11314],{},"During field acquisition, electronic total stations capture spatial information using optical encoders and electronic distance meters (EDMs), recording raw measurements such as horizontal circle readings, vertical angles, and slope distances. Proprietary file formats, such as Leica’s Geo Serial Interface (GSI), write these observations along with computed Cartesian coordinates directly to on-board storage.",[11,11316,11317],{},"Once field operations conclude, office technicians must process and adjust these measurements. For least-squares network adjustments, coordinate comparisons, or manual attribute editing, raw database structures are highly impractical. Technicians require flat, human-readable formats like Comma-Separated Values (CSV) that open in Microsoft Excel or database management systems. Historically, this step relied on legacy MS-DOS utilities or custom Excel VBA macros — solutions that are increasingly difficult to maintain on modern operating systems and lack cross-platform compatibility.",[11,11319,11320],{},"After data cleaning, adjusted coordinates are brought into CAD programs like AutoCAD or Civil 3D to draft boundary lines, design alignments, and model physical structures. For this drafting stage, coordinate lists are converted into vector formats, with the Drawing Exchange Format (DXF) acting as the primary open standard for coordinate point imports.",[11,11322,11323],{},"Finally, design coordinates created in CAD must return to the field. Laying out proposed foundation piles, utility lines, or roadway corridors requires exporting design coordinates from CAD, converting them back into GSI format, and uploading them to a field total station. Modern geomatics students must master these format transitions to understand how coordinate geometry behaves across different operating systems, instrument firmwares, and design software.",[323,11325,11327],{"id":11326},"the-conversion-pipeline-and-the-universal-survey-data-model-usdm-v1","The Conversion Pipeline and the Universal Survey Data Model (USDM v1)",[11,11329,11330,11331,11359,11360,11444,11445,5868],{},"Converting files between multiple formats can lead to complex development challenges as the number of formats grows. If every file format required a custom converter for every other format, supporting ",[772,11332,11334,11347],{"className":11333},[775],[772,11335,11337],{"className":11336},[779],[781,11338,11339],{"xmlns":783},[785,11340,11341,11345],{},[788,11342,11343],{},[791,11344,2839],{},[795,11346,2839],{"encoding":797},[772,11348,11350],{"className":11349,"ariaHidden":803},[802],[772,11351,11353,11356],{"className":11352},[807],[772,11354],{"className":11355,"style":2758},[811],[772,11357,2839],{"className":11358,"style":2854},[816,817]," files would require ",[772,11361,11363,11389],{"className":11362},[775],[772,11364,11366],{"className":11365},[779],[781,11367,11368],{"xmlns":783},[785,11369,11370,11386],{},[788,11371,11372,11374,11376,11378,11380,11382,11384],{},[791,11373,2839],{},[980,11375,6382],{},[980,11377,1082],{"stretchy":1081},[791,11379,2839],{},[980,11381,1091],{},[1084,11383,2170],{},[980,11385,1096],{"stretchy":1081},[795,11387,11388],{"encoding":797},"N \\times (N-1)",[772,11390,11392,11411,11432],{"className":11391,"ariaHidden":803},[802],[772,11393,11395,11399,11402,11405,11408],{"className":11394},[807],[772,11396],{"className":11397,"style":11398},[811],"height:0.7667em;vertical-align:-0.0833em;",[772,11400,2839],{"className":11401,"style":2854},[816,817],[772,11403],{"className":11404,"style":1121},[1003],[772,11406,6382],{"className":11407},[1125],[772,11409],{"className":11410,"style":1121},[1003],[772,11412,11414,11417,11420,11423,11426,11429],{"className":11413},[807],[772,11415],{"className":11416,"style":1109},[811],[772,11418,1082],{"className":11419},[1113],[772,11421,2839],{"className":11422,"style":2854},[816,817],[772,11424],{"className":11425,"style":1121},[1003],[772,11427,1091],{"className":11428},[1125],[772,11430],{"className":11431,"style":1121},[1003],[772,11433,11435,11438,11441],{"className":11434},[807],[772,11436],{"className":11437,"style":1109},[811],[772,11439,2170],{"className":11440},[816],[772,11442,1096],{"className":11443},[1141]," distinct conversion paths. To prevent this complexity, Surveying Core uses a centralized translation architecture based on the ",[318,11446,11447],{},"Universal Survey Data Model version 1 (USDM v1)",[2857,11449,11452],{"className":11450,"code":11451,"language":1709},[2860],"Incoming File ──► [Parser] ──► USDM v1 (in-memory) ──► [Exporter] ──► Target File\n",[2863,11453,11451],{"__ignoreMap":15},[11,11455,11456],{},"The conversion pipeline consists of three distinct phases:",[1602,11458,11459,11465,11471],{},[338,11460,11461,11464],{},[318,11462,11463],{},"Parsing:"," The parser reads the source format (such as GSI or CSV), validates the file structure, and extracts core spatial parameters into local browser memory.",[338,11466,11467,11470],{},[318,11468,11469],{},"Normalization:"," Parsed data is structured into a unified in-memory representation. USDM v1 uses a clean point schema that strips instrument-specific headers while preserving coordinate precision.",[338,11472,11473,11476],{},[318,11474,11475],{},"Exporting:"," The exporter reads the USDM v1 structure and writes coordinate points into the target file format (such as GSI, CSV, or DXF).",[11,11478,11479],{},"The USDM v1 point schema is detailed in the table below:",[851,11481,11482,11498],{},[854,11483,11484],{},[857,11485,11486,11489,11492,11495],{},[860,11487,11488],{},"Property",[860,11490,11491],{},"Type",[860,11493,11494],{},"Constraints",[860,11496,11497],{},"Purpose",[876,11499,11500,11516,11532,11546,11562],{},[857,11501,11502,11507,11510,11513],{},[881,11503,11504],{},[2863,11505,11506],{},"id",[881,11508,11509],{},"string",[881,11511,11512],{},"Required, trimmed",[881,11514,11515],{},"Point identifier or label",[857,11517,11518,11523,11526,11529],{},[881,11519,11520],{},[2863,11521,11522],{},"easting",[881,11524,11525],{},"number",[881,11527,11528],{},"Required",[881,11530,11531],{},"East–west Cartesian position",[857,11533,11534,11539,11541,11543],{},[881,11535,11536],{},[2863,11537,11538],{},"northing",[881,11540,11525],{},[881,11542,11528],{},[881,11544,11545],{},"North–south Cartesian position",[857,11547,11548,11553,11556,11559],{},[881,11549,11550],{},[2863,11551,11552],{},"elevation",[881,11554,11555],{},"number | null",[881,11557,11558],{},"Optional",[881,11560,11561],{},"Vertical height",[857,11563,11564,11568,11570,11573],{},[881,11565,11566],{},[2863,11567,2863],{},[881,11569,11509],{},[881,11571,11572],{},"Optional, max 32 chars",[881,11574,11575],{},"Feature or attribute code",[11,11577,11578],{},"By converting all files to USDM v1 first, the platform decouples input logic from output logic. A parser only needs to translate its specific format into USDM v1; the system can then export that data to any supported target format without custom file-to-file translation code.",[323,11580,11582],{"id":11581},"csv-conventions-and-spatial-office-pitfalls","CSV Conventions and Spatial Office Pitfalls",[11,11584,11585],{},"The Comma-Separated Values (CSV) format is widely used for tabular data exchange, but it lacks a single formal specification. In surveying offices, this flexibility can lead to coordinate mapping and translation errors.",[706,11587,11589],{"id":11588},"delimiters-and-encodings","Delimiters and encodings",[11,11591,11592,11593,11635],{},"CSV files use delimiters to separate data columns — typically commas, semicolons, or tabs. In regions where a comma is used as a decimal separator (such as ",[772,11594,11596,11611],{"className":11595},[775],[772,11597,11599],{"className":11598},[779],[781,11600,11601],{"xmlns":783},[785,11602,11603,11608],{},[788,11604,11605],{},[1084,11606,11607],{},"15204,321",[795,11609,11610],{"encoding":797},"15204{,}321",[772,11612,11614],{"className":11613,"ariaHidden":803},[802],[772,11615,11617,11621,11625,11631],{"className":11616},[807],[772,11618],{"className":11619,"style":11620},[811],"height:0.8389em;vertical-align:-0.1944em;",[772,11622,11624],{"className":11623},[816],"15204",[772,11626,11628],{"className":11627},[816],[772,11629,5940],{"className":11630},[6041],[772,11632,11634],{"className":11633},[816],"321","), semicolons are often the column delimiter. In Anglo-American regions, commas separate columns and periods separate decimals. Text encodings can also cause issues; legacy total stations require standard ASCII or UTF-8 without a Byte Order Mark (BOM) to prevent corruption on upload.",[706,11637,11639],{"id":11638},"coordinate-order-layouts","Coordinate order layouts",[11,11641,11642],{},"Geomatics professionals use several coordinate ordering layouts. If a system interprets coordinates using the wrong layout, the entire dataset can be rotated or flipped, leading to severe construction errors. Common layouts include:",[335,11644,11645,11651,11660,11666],{},[338,11646,11647,11650],{},[318,11648,11649],{},"PNEZD:"," Point ID, Northing, Easting, Elevation, Description — common in regional grid workflows.",[338,11652,11653,11656,11657,2925],{},[318,11654,11655],{},"PENZD:"," Point ID, Easting, Northing, Elevation, Description — the default column order for Surveying Core CSV export (",[2863,11658,11659],{},"PointNumber,Easting,Northing,Elevation,Description",[338,11661,11662,11665],{},[318,11663,11664],{},"ENZ:"," Easting, Northing, Elevation — typical for surface modeling imports.",[338,11667,11668,11671],{},[318,11669,11670],{},"NEZ:"," Northing, Easting, Elevation.",[706,11673,11675],{"id":11674},"header-row-pitfalls","Header row pitfalls",[11,11677,11678,11679,11682],{},"Field instrument firmware cannot parse string headers. If a coordinate file contains a header row like ",[2863,11680,11681],{},"Point_ID,Northing,Easting,Elevation,Description",", the instrument may reject the upload. Office software and web parsers, however, need headers to map columns automatically.",[11,11684,11685,11686,11689],{},"Surveying Core uses an intelligent CSV import path. On upload, the parser reads the initial lines. If it detects standard geomatics header strings, it maps columns automatically. If headers are missing or ambiguous, a ",[318,11687,11688],{},"column mapping wizard"," lets you verify Easting, Northing, Elevation, ID, and Code assignments before conversion.",[323,11691,11693],{"id":11692},"gsi-overview","GSI Overview",[11,11695,11696,11697,5868],{},"The Geo Serial Interface (GSI) is a fixed-width, block-structured format used by Leica Geosystems total stations and digital levels. For bit-level structures and word-index detail, see the ",[693,11698,11699],{"href":11303},"GSI format guide",[706,11701,11703],{"id":11702},"gsi-8-versus-gsi-16-structure","GSI-8 versus GSI-16 structure",[11,11705,11706],{},"Two primary variants exist, distinguished by word length:",[335,11708,11709,11715],{},[338,11710,11711,11714],{},[318,11712,11713],{},"GSI-8:"," Each data block uses 8-character data payloads. Common on legacy total stations (TPS300, TPS400, TPS700 series).",[338,11716,11717,11720],{},[318,11718,11719],{},"GSI-16:"," Uses 16-character payloads for higher precision and longer point IDs. GSI-16 blocks are identified by an asterisk at the start of the line.",[2857,11722,11725],{"className":11723,"code":11724,"language":1709},[2860],"GSI-8 coordinate block example:\n110001+00000101 81..00+01234567 82..00+02345678 83..00+00100000\n└─ Pt ID       └─ Easting      └─ Northing     └─ Elevation\n\nGSI-16 block example:\n*110001+000000000PNC0055 81.002+0000000012345678 82.002+0000000056789012\n└─ Pt ID                 └─ Easting              └─ Northing\n",[2863,11726,11724],{"__ignoreMap":15},[706,11728,11730],{"id":11729},"word-indexing-wi-and-block-architecture","Word indexing (WI) and block architecture",[11,11732,11733],{},"GSI records are written in lines called blocks, with words separated by spaces. Each word contains:",[1602,11735,11736,11750,11756],{},[338,11737,11738,11741,11742,11745,11746,11749],{},[318,11739,11740],{},"Word Index (WI):"," Positions 1–2 define the parameter (",[2863,11743,11744],{},"11"," = Point ID, ",[2863,11747,11748],{},"81"," = Easting, etc.).",[338,11751,11752,11755],{},[318,11753,11754],{},"Information fields (AI):"," Positions 3–6 describe sequence, units, or input mode.",[338,11757,11758,11761,11762,11764,11765,11768],{},[318,11759,11760],{},"Data payload (DA):"," Remaining characters hold the value, prefixed with ",[2863,11763,1353],{}," or ",[2863,11766,11767],{},"-"," and zero-padded to fixed width.",[11,11770,11771],{},"Critical word indices for coordinate exchange:",[851,11773,11774,11790],{},[854,11775,11776],{},[857,11777,11778,11781,11784,11787],{},[860,11779,11780],{},"WI",[860,11782,11783],{},"Parameter",[860,11785,11786],{},"Typical representation",[860,11788,11789],{},"USDM v1 mapping",[876,11791,11792,11808,11824,11840,11856,11873],{},[857,11793,11794,11798,11801,11804],{},[881,11795,11796],{},[318,11797,11744],{},[881,11799,11800],{},"Point ID",[881,11802,11803],{},"Text (padded)",[881,11805,11806],{},[2863,11807,11506],{},[857,11809,11810,11814,11817,11820],{},[881,11811,11812],{},[318,11813,11748],{},[881,11815,11816],{},"Target easting",[881,11818,11819],{},"Meters or feet (per AI)",[881,11821,11822],{},[2863,11823,11522],{},[857,11825,11826,11831,11834,11836],{},[881,11827,11828],{},[318,11829,11830],{},"82",[881,11832,11833],{},"Target northing",[881,11835,11819],{},[881,11837,11838],{},[2863,11839,11538],{},[857,11841,11842,11847,11850,11852],{},[881,11843,11844],{},[318,11845,11846],{},"83",[881,11848,11849],{},"Target elevation",[881,11851,11819],{},[881,11853,11854],{},[2863,11855,11552],{},[857,11857,11858,11863,11866,11869],{},[881,11859,11860],{},[318,11861,11862],{},"71",[881,11864,11865],{},"Point \u002F feature code",[881,11867,11868],{},"Alphanumeric",[881,11870,11871],{},[2863,11872,2863],{},[857,11874,11875,11880,11883,11886],{},[881,11876,11877],{},[318,11878,11879],{},"84–86",[881,11881,11882],{},"Station E\u002FN\u002FH",[881,11884,11885],{},"Instrument setup",[881,11887,11888],{},"Metadata (not stakeout points)",[11,11890,11891,11892,2896,11894,11896,11897,11899],{},"When parsing GSI, the importer scans each line for these word indices. When WIs ",[2863,11893,11748],{},[2863,11895,11830],{},", and ",[2863,11898,11846],{}," are present, payloads are extracted, decimal scaling is applied from the unit flag, and coordinates are written to USDM v1 in memory.",[323,11901,11903],{"id":11902},"dxf-stakeout-export-mechanics","DXF Stakeout Export Mechanics",[11,11905,11906],{},"The Drawing Exchange Format (DXF) is a tagged vector format developed by Autodesk. Unlike proprietary DWG files, ASCII DXF is plain text and widely compatible across CAD and GIS software.",[11,11908,11909,11910,11913],{},"For stakeout, surveyors typically need a clean set of point nodes — not complex linework, hatching, or block libraries. The Surveying Core converter exports minimal ",[318,11911,11912],{},"POINT"," entities suitable for layout reference.",[11,11915,11916,11917,11919],{},"The MVP exporter writes ASCII DXF with a lightweight header and one ",[2863,11918,11912],{}," entity per USDM point. Each point includes:",[335,11921,11922,11930,11939,11945,11951],{},[338,11923,11924,11927,11928,1096],{},[318,11925,11926],{},"Group code 0:"," Entity type (",[2863,11929,11912],{},[338,11931,11932,11935,11936,11938],{},[318,11933,11934],{},"Group code 8:"," Layer name (default layer ",[2863,11937,1748],{}," in the current MVP)",[338,11940,11941,11944],{},[318,11942,11943],{},"Group code 10:"," X-coordinate (easting)",[338,11946,11947,11950],{},[318,11948,11949],{},"Group code 20:"," Y-coordinate (northing)",[338,11952,11953,11956,11957,11959],{},[318,11954,11955],{},"Group code 30:"," Z-coordinate (elevation; ",[2863,11958,1748],{}," when elevation is missing)",[11,11961,11962,11963,11965,11966,11969,11970,11973],{},"CAD programs may display bare ",[2863,11964,11912],{}," entities as small markers. If points are hard to see, adjust point display style in your CAD software (for example AutoCAD ",[2863,11967,11968],{},"PDMODE"," \u002F ",[2863,11971,11972],{},"PDSIZE",") after import.",[323,11975,11977],{"id":11976},"tool-guide-gsi-csv","Tool Guide: GSI → CSV",[11,11979,11033,11980,11983],{},[318,11981,11982],{},"GSI → CSV"," tool supports field-to-office workflows — converting Leica memory-card coordinates into spreadsheets for Excel, Google Sheets, or QGIS.",[335,11985,11986,11992,12002,12014],{},[338,11987,11988,11991],{},[318,11989,11990],{},"Primary use case:"," Transforming total-station GSI jobs into tabular CSV for office processing.",[338,11993,11994,11997,11998,12001],{},[318,11995,11996],{},"Input:"," A valid Leica ",[2863,11999,12000],{},".gsi"," file (GSI-8 or GSI-16).",[338,12003,12004,12007,12008,12011,12012,2925],{},[318,12005,12006],{},"Output:"," Comma-delimited ",[2863,12009,12010],{},".csv"," with header row (",[2863,12013,11659],{},[338,12015,12016,12019,12020],{},[318,12017,12018],{},"Converter:"," ",[693,12021,11982],{"href":11274},[706,12023,12025],{"id":12024},"step-by-step-field-to-office-pipeline","Step-by-step field-to-office pipeline",[2857,12027,12030],{"className":12028,"code":12029,"language":1709},[2860],"Leica instrument ──► SD card ──► GSI file ──► CSV ──► spreadsheets \u002F QGIS\n",[2863,12031,12029],{"__ignoreMap":15},[1602,12033,12034,12037,12043,12049,12055],{},[338,12035,12036],{},"On the Leica total station, export the desired job in standard GSI format.",[338,12038,12039,12040,12042],{},"Transfer the ",[2863,12041,12000],{}," file to your computer or mobile device.",[338,12044,12045,12046,12048],{},"Open the ",[693,12047,11275],{"href":11274}," and sign in.",[338,12050,12051,12052,12054],{},"Select the ",[2863,12053,12000],{}," file. The browser parses it locally through USDM v1.",[338,12056,12057,12058,12061],{},"Click ",[318,12059,12060],{},"Convert to CSV"," and download the result.",[323,12063,12065],{"id":12064},"tool-guide-csv-gsi","Tool Guide: CSV → GSI",[11,12067,11033,12068,12071],{},[318,12069,12070],{},"CSV → GSI"," tool reverses the office workflow, preparing coordinate lists for field upload.",[335,12073,12074,12079,12087,12095],{},[338,12075,12076,12078],{},[318,12077,11990],{}," Preparing stakeout point lists from design spreadsheets for Leica total stations.",[338,12080,12081,12083,12084,12086],{},[318,12082,11996],{}," Delimited ",[2863,12085,12010],{}," with numeric coordinates and point identifiers.",[338,12088,12089,12091,12092,12094],{},[318,12090,12006],{}," Formatted Leica ",[2863,12093,12000],{}," text (GSI-8 in the current MVP UI).",[338,12096,12097,12019,12099],{},[318,12098,12018],{},[693,12100,12070],{"href":11274},[706,12102,12104],{"id":12103},"step-by-step-prep-to-instrument-pipeline","Step-by-step prep-to-instrument pipeline",[2857,12106,12109],{"className":12107,"code":12108,"language":1709},[2860],"Design coordinates (CSV) ──► GSI file ──► SD card \u002F USB ──► total station layout\n",[2863,12110,12108],{"__ignoreMap":15},[1602,12112,12113,12116,12120,12123,12132],{},[338,12114,12115],{},"Export layout points from design software as CSV.",[338,12117,12045,12118,12048],{},[693,12119,11281],{"href":11274},[338,12121,12122],{},"Upload the CSV. If columns are not auto-detected, map them to ID, Easting, Northing, Elevation, and Code in the mapping wizard.",[338,12124,12057,12125,12128,12129,12131],{},[318,12126,12127],{},"Convert to GSI"," and download the ",[2863,12130,12000],{}," file.",[338,12133,12134],{},"Copy the file to your Leica SD card or USB drive and load it into the instrument job.",[323,12136,12138],{"id":12137},"tool-guide-csv-dxf","Tool Guide: CSV → DXF",[11,12140,11033,12141,12144],{},[318,12142,12143],{},"CSV → DXF"," tool generates lightweight point drawings from coordinate lists.",[335,12146,12147,12152,12158,12170],{},[338,12148,12149,12151],{},[318,12150,11990],{}," Creating CAD point layers with elevation for reference in AutoCAD, Civil 3D, or BricsCAD.",[338,12153,12154,12083,12156,12131],{},[318,12155,11996],{},[2863,12157,12010],{},[338,12159,12160,12162,12163,12166,12167,12169],{},[318,12161,12006],{}," ASCII ",[2863,12164,12165],{},".dxf"," with one ",[2863,12168,11912],{}," entity per coordinate.",[338,12171,12172,12019,12174],{},[318,12173,12018],{},[693,12175,12143],{"href":11274},[706,12177,12179],{"id":12178},"step-by-step-cad-import-pipeline","Step-by-step CAD import pipeline",[2857,12181,12184],{"className":12182,"code":12183,"language":1709},[2860],"Survey points (CSV) ──► DXF file ──► CAD drawing area\n",[2863,12185,12183],{"__ignoreMap":15},[1602,12187,12188,12191,12195,12198,12204],{},[338,12189,12190],{},"Prepare your coordinate list as CSV.",[338,12192,12045,12193,12048],{},[693,12194,11287],{"href":11274},[338,12196,12197],{},"Upload the file and confirm column mapping (Easting → X, Northing → Y, Elevation → Z).",[338,12199,12057,12200,12203],{},[318,12201,12202],{},"Convert to DXF"," and download the drawing.",[338,12205,12045,12206,12208],{},[2863,12207,12165],{}," in your CAD program or attach it as an external reference (XREF).",[323,12210,12212],{"id":12211},"client-side-processing-and-privacy","Client-Side Processing and Privacy",[11,12214,12215,12216,12219],{},"A core design feature is that file conversion executes ",[318,12217,12218],{},"entirely in the user’s browser",". No coordinate data, raw measurement files, or project point lists are uploaded to external servers for parsing or export.",[706,12221,12223],{"id":12222},"the-privacy-advantage","The privacy advantage",[11,12225,12226],{},"Land survey coordinate databases represent sensitive intellectual property, proprietary infrastructure layouts, and boundary control. Uploading these files to third-party servers creates security and compliance risk.",[11,12228,12229,12230,12233,12234,12237],{},"Surveying Core uses the HTML5 ",[318,12231,12232],{},"FileReader API"," to read the source file from local storage. The conversion engine (",[2863,12235,12236],{},"lib\u002Fformats\u002F*",") runs in JavaScript, normalizes data to USDM v1 in RAM, and generates a download via a local blob URL. Spatial records never leave your machine during conversion.",[2857,12239,12242],{"className":12240,"code":12241,"language":1709},[2860],"┌─────────────────────────────────────┐\n│         Your local browser          │\n│                                     │\n│  User file ──► FileReader           │\n│            ──► Parse → USDM v1      │──► Download\n│            ──► Export target format │\n└─────────────────────────────────────┘\n",[2863,12243,12241],{"__ignoreMap":15},[11,12245,12246],{},"Credits are checked and consumed via API calls (metadata only — format, file size, point count), but the coordinate file itself stays client-side.",[706,12248,12250],{"id":12249},"re-export-without-extra-credits","Re-export without extra credits",[11,12252,12253,12254,12257,12258,12261],{},"If you convert the same USDM dataset to a different export format within ",[318,12255,12256],{},"five minutes",", a re-export session applies and ",[318,12259,12260],{},"no additional credit is charged",". This supports exploring CSV, GSI, and DXF outputs from one import without penalizing format trials.",[323,12263,12265],{"id":12264},"mathematical-and-spatial-validation-protocols","Mathematical and Spatial Validation Protocols",[11,12267,12268],{},"Conversion must not be treated as simple text translation. Spatial data integrity demands structured quality-assurance steps after every job.",[706,12270,12272],{"id":12271},"_1-point-count-verification","1. Point count verification",[11,12274,12275,12276,12278],{},"Verify that the record count in your source file matches the output. If your CSV has 1,250 data rows, your GSI export should contain 1,250 coordinate blocks and your DXF should contain 1,250 ",[2863,12277,11912],{}," entities. A discrepancy usually indicates blank rows, malformed headers, or parsing errors.",[706,12280,12282],{"id":12281},"_2-spatial-extents-sanity-check","2. Spatial extents sanity check",[11,12284,12285],{},"After import to CAD or GIS, zoom to extents and confirm coordinates fall in the expected range for your project. If eastings should be near 400,000 m and northings near 5,000,000 m but the drawing shows swapped axes, re-run the conversion with corrected column mapping.",[11,12287,12288,12289,5868],{},"You can also verify bearings and distances between control points with the ",[693,12290,696],{"href":695},[706,12292,12294],{"id":12293},"_3-spot-checking-control-points","3. Spot-checking control points",[11,12296,12297],{},"Identify at least one known monument or control point in the source file and compare coordinates character-for-character in the output — horizontal and vertical.",[11,12299,12300],{},"Verify horizontal distance between two points:",[11,12302,12303],{},[772,12304,12306,12361],{"className":12305},[775],[772,12307,12309],{"className":12308},[779],[781,12310,12311],{"xmlns":783},[785,12312,12313,12358],{},[788,12314,12315,12317,12319,12321,12327,12329,12335,12338,12340,12342,12344,12350,12352],{},[791,12316,2963],{"mathvariant":906},[791,12318,2808],{},[980,12320,982],{},[3525,12322,12323,12325],{},[791,12324,2808],{},[1084,12326,1872],{},[980,12328,1091],{},[3525,12330,12331,12333],{},[791,12332,2808],{},[1084,12334,2170],{},[1003,12336],{"width":12337},"2em",[791,12339,2963],{"mathvariant":906},[791,12341,2839],{},[980,12343,982],{},[3525,12345,12346,12348],{},[791,12347,2839],{},[1084,12349,1872],{},[980,12351,1091],{},[3525,12353,12354,12356],{},[791,12355,2839],{},[1084,12357,2170],{},[795,12359,12360],{"encoding":797},"\\Delta E = E_2 - E_1 \\qquad \\Delta N = N_2 - N_1",[772,12362,12364,12385,12440,12505,12560],{"className":12363,"ariaHidden":803},[802],[772,12365,12367,12370,12373,12376,12379,12382],{"className":12366},[807],[772,12368],{"className":12369,"style":2758},[811],[772,12371,2963],{"className":12372},[816],[772,12374,2808],{"className":12375,"style":2823},[816,817],[772,12377],{"className":12378,"style":1004},[1003],[772,12380,982],{"className":12381},[1008],[772,12383],{"className":12384,"style":1004},[1003],[772,12386,12388,12391,12431,12434,12437],{"className":12387},[807],[772,12389],{"className":12390,"style":3543},[811],[772,12392,12394,12397],{"className":12393},[816],[772,12395,2808],{"className":12396,"style":2823},[816,817],[772,12398,12400],{"className":12399},[1752],[772,12401,12403,12423],{"className":12402},[1756,3556],[772,12404,12406,12420],{"className":12405},[1760],[772,12407,12409],{"className":12408,"style":3563},[1764],[772,12410,12411,12414],{"style":3566},[772,12412],{"className":12413,"style":1773},[1772],[772,12415,12417],{"className":12416},[1777,1778,1779,1780],[772,12418,1872],{"className":12419},[816,1780],[772,12421,3580],{"className":12422},[3579],[772,12424,12426],{"className":12425},[1760],[772,12427,12429],{"className":12428,"style":3587},[1764],[772,12430],{},[772,12432],{"className":12433,"style":1121},[1003],[772,12435,1091],{"className":12436},[1125],[772,12438],{"className":12439,"style":1121},[1003],[772,12441,12443,12446,12486,12490,12493,12496,12499,12502],{"className":12442},[807],[772,12444],{"className":12445,"style":3543},[811],[772,12447,12449,12452],{"className":12448},[816],[772,12450,2808],{"className":12451,"style":2823},[816,817],[772,12453,12455],{"className":12454},[1752],[772,12456,12458,12478],{"className":12457},[1756,3556],[772,12459,12461,12475],{"className":12460},[1760],[772,12462,12464],{"className":12463,"style":3563},[1764],[772,12465,12466,12469],{"style":3566},[772,12467],{"className":12468,"style":1773},[1772],[772,12470,12472],{"className":12471},[1777,1778,1779,1780],[772,12473,2170],{"className":12474},[816,1780],[772,12476,3580],{"className":12477},[3579],[772,12479,12481],{"className":12480},[1760],[772,12482,12484],{"className":12483,"style":3587},[1764],[772,12485],{},[772,12487],{"className":12488,"style":12489},[1003],"margin-right:2em;",[772,12491,2963],{"className":12492},[816],[772,12494,2839],{"className":12495,"style":2854},[816,817],[772,12497],{"className":12498,"style":1004},[1003],[772,12500,982],{"className":12501},[1008],[772,12503],{"className":12504,"style":1004},[1003],[772,12506,12508,12511,12551,12554,12557],{"className":12507},[807],[772,12509],{"className":12510,"style":3543},[811],[772,12512,12514,12517],{"className":12513},[816],[772,12515,2839],{"className":12516,"style":2854},[816,817],[772,12518,12520],{"className":12519},[1752],[772,12521,12523,12543],{"className":12522},[1756,3556],[772,12524,12526,12540],{"className":12525},[1760],[772,12527,12529],{"className":12528,"style":3563},[1764],[772,12530,12531,12534],{"style":3640},[772,12532],{"className":12533,"style":1773},[1772],[772,12535,12537],{"className":12536},[1777,1778,1779,1780],[772,12538,1872],{"className":12539},[816,1780],[772,12541,3580],{"className":12542},[3579],[772,12544,12546],{"className":12545},[1760],[772,12547,12549],{"className":12548,"style":3587},[1764],[772,12550],{},[772,12552],{"className":12553,"style":1121},[1003],[772,12555,1091],{"className":12556},[1125],[772,12558],{"className":12559,"style":1121},[1003],[772,12561,12563,12566],{"className":12562},[807],[772,12564],{"className":12565,"style":3543},[811],[772,12567,12569,12572],{"className":12568},[816],[772,12570,2839],{"className":12571,"style":2854},[816,817],[772,12573,12575],{"className":12574},[1752],[772,12576,12578,12598],{"className":12577},[1756,3556],[772,12579,12581,12595],{"className":12580},[1760],[772,12582,12584],{"className":12583,"style":3563},[1764],[772,12585,12586,12589],{"style":3640},[772,12587],{"className":12588,"style":1773},[1772],[772,12590,12592],{"className":12591},[1777,1778,1779,1780],[772,12593,2170],{"className":12594},[816,1780],[772,12596,3580],{"className":12597},[3579],[772,12599,12601],{"className":12600},[1760],[772,12602,12604],{"className":12603,"style":3587},[1764],[772,12605],{},[11,12607,12608],{},[772,12609,12611,12657],{"className":12610},[775],[772,12612,12614],{"className":12613},[779],[781,12615,12616],{"xmlns":783},[785,12617,12618,12654],{},[788,12619,12620,12622,12624],{},[791,12621,3180],{},[980,12623,982],{},[5347,12625,12626],{},[788,12627,12628,12630,12632,12634,12640,12642,12644,12646,12648],{},[980,12629,1082],{"stretchy":1081},[791,12631,2963],{"mathvariant":906},[791,12633,2808],{},[1636,12635,12636,12638],{},[980,12637,1096],{"stretchy":1081},[1084,12639,1872],{},[980,12641,1353],{},[980,12643,1082],{"stretchy":1081},[791,12645,2963],{"mathvariant":906},[791,12647,2839],{},[1636,12649,12650,12652],{},[980,12651,1096],{"stretchy":1081},[1084,12653,1872],{},[795,12655,12656],{"encoding":797},"D = \\sqrt{(\\Delta E)^2 + (\\Delta N)^2}",[772,12658,12660,12678],{"className":12659,"ariaHidden":803},[802],[772,12661,12663,12666,12669,12672,12675],{"className":12662},[807],[772,12664],{"className":12665,"style":2758},[811],[772,12667,3180],{"className":12668,"style":818},[816,817],[772,12670],{"className":12671,"style":1004},[1003],[772,12673,982],{"className":12674},[1008],[772,12676],{"className":12677,"style":1004},[1003],[772,12679,12681,12684],{"className":12680},[807],[772,12682],{"className":12683,"style":9438},[811],[772,12685,12687],{"className":12686},[816,5403],[772,12688,12690,12805],{"className":12689},[1756,3556],[772,12691,12693,12802],{"className":12692},[1760],[772,12694,12696,12790],{"className":12695,"style":9451},[1764],[772,12697,12699,12702],{"className":12698,"style":9455},[5417],[772,12700],{"className":12701,"style":9459},[1772],[772,12703,12705,12708,12711,12714,12743,12746,12749,12752,12755,12758,12761],{"className":12704,"style":9463},[816],[772,12706,1082],{"className":12707},[1113],[772,12709,2963],{"className":12710},[816],[772,12712,2808],{"className":12713,"style":2823},[816,817],[772,12715,12717,12720],{"className":12716},[1141],[772,12718,1096],{"className":12719},[1141],[772,12721,12723],{"className":12722},[1752],[772,12724,12726],{"className":12725},[1756],[772,12727,12729],{"className":12728},[1760],[772,12730,12732],{"className":12731,"style":5448},[1764],[772,12733,12734,12737],{"style":5451},[772,12735],{"className":12736,"style":1773},[1772],[772,12738,12740],{"className":12739},[1777,1778,1779,1780],[772,12741,1872],{"className":12742},[816,1780],[772,12744],{"className":12745,"style":1121},[1003],[772,12747,1353],{"className":12748},[1125],[772,12750],{"className":12751,"style":1121},[1003],[772,12753,1082],{"className":12754},[1113],[772,12756,2963],{"className":12757},[816],[772,12759,2839],{"className":12760,"style":2854},[816,817],[772,12762,12764,12767],{"className":12763},[1141],[772,12765,1096],{"className":12766},[1141],[772,12768,12770],{"className":12769},[1752],[772,12771,12773],{"className":12772},[1756],[772,12774,12776],{"className":12775},[1760],[772,12777,12779],{"className":12778,"style":5448},[1764],[772,12780,12781,12784],{"style":5451},[772,12782],{"className":12783,"style":1773},[1772],[772,12785,12787],{"className":12786},[1777,1778,1779,1780],[772,12788,1872],{"className":12789},[816,1780],[772,12791,12792,12795],{"style":9548},[772,12793],{"className":12794,"style":9459},[1772],[772,12796,12798],{"className":12797,"style":9555},[5511],[5514,12799,12800],{"xmlns":5516,"width":5517,"height":9558,"viewBox":9559,"preserveAspectRatio":5520},[5522,12801],{"d":9562},[772,12803,3580],{"className":12804},[3579],[772,12806,12808],{"className":12807},[1760],[772,12809,12811],{"className":12810,"style":9572},[1764],[772,12812],{},[11,12814,12815],{},"Grid bearing from north (clockwise):",[11,12817,12818],{},[772,12819,12821,12853],{"className":12820},[775],[772,12822,12824],{"className":12823},[779],[781,12825,12826],{"xmlns":783},[785,12827,12828,12850],{},[788,12829,12830,12832,12834,12836,12838,12840,12842,12844,12846,12848],{},[791,12831,793],{},[980,12833,982],{},[1625,12835,5886],{},[980,12837,1082],{"stretchy":1081},[791,12839,2963],{"mathvariant":906},[791,12841,2808],{},[980,12843,5940],{"separator":803},[791,12845,2963],{"mathvariant":906},[791,12847,2839],{},[980,12849,1096],{"stretchy":1081},[795,12851,12852],{"encoding":797},"\\theta = \\text{atan2}(\\Delta E, \\Delta N)",[772,12854,12856,12874],{"className":12855,"ariaHidden":803},[802],[772,12857,12859,12862,12865,12868,12871],{"className":12858},[807],[772,12860],{"className":12861,"style":812},[811],[772,12863,793],{"className":12864,"style":818},[816,817],[772,12866],{"className":12867,"style":1004},[1003],[772,12869,982],{"className":12870},[1008],[772,12872],{"className":12873,"style":1004},[1003],[772,12875,12877,12880,12886,12889,12892,12895,12898,12901,12904,12907],{"className":12876},[807],[772,12878],{"className":12879,"style":1109},[811],[772,12881,12883],{"className":12882},[816,1709],[772,12884,5886],{"className":12885},[816],[772,12887,1082],{"className":12888},[1113],[772,12890,2963],{"className":12891},[816],[772,12893,2808],{"className":12894,"style":2823},[816,817],[772,12896,5940],{"className":12897},[6041],[772,12899],{"className":12900,"style":3294},[1003],[772,12902,2963],{"className":12903},[816],[772,12905,2839],{"className":12906,"style":2854},[816,817],[772,12908,1096],{"className":12909},[1141],[11,12911,12912,12913,1096],{},"(normalize to ",[772,12914,12916,12942],{"className":12915},[775],[772,12917,12919],{"className":12918},[779],[781,12920,12921],{"xmlns":783},[785,12922,12923,12939],{},[788,12924,12925,12927,12929,12931,12933,12935,12937],{},[1084,12926,1748],{},[791,12928,907],{"mathvariant":906},[980,12930,6182],{},[791,12932,793],{},[980,12934,6192],{},[1084,12936,1468],{},[791,12938,907],{"mathvariant":906},[795,12940,12941],{"encoding":797},"0° \\le \\theta \u003C 360°",[772,12943,12945,12965,12984],{"className":12944,"ariaHidden":803},[802],[772,12946,12948,12952,12956,12959,12962],{"className":12947},[807],[772,12949],{"className":12950,"style":12951},[811],"height:0.8304em;vertical-align:-0.136em;",[772,12953,12955],{"className":12954},[816],"0°",[772,12957],{"className":12958,"style":1004},[1003],[772,12960,6182],{"className":12961},[1008],[772,12963],{"className":12964,"style":1004},[1003],[772,12966,12968,12972,12975,12978,12981],{"className":12967},[807],[772,12969],{"className":12970,"style":12971},[811],"height:0.7335em;vertical-align:-0.0391em;",[772,12973,793],{"className":12974,"style":818},[816,817],[772,12976],{"className":12977,"style":1004},[1003],[772,12979,6192],{"className":12980},[1008],[772,12982],{"className":12983,"style":1004},[1003],[772,12985,12987,12990],{"className":12986},[807],[772,12988],{"className":12989,"style":812},[811],[772,12991,1495],{"className":12992},[816],[706,12994,12996],{"id":12995},"_4-re-export-round-trip-testing","4. Re-export round-trip testing",[11,12998,12999],{},"The gold standard of coordinate validation is the round-trip test:",[1602,13001,13002,13005,13008],{},[338,13003,13004],{},"Convert raw GSI to CSV.",[338,13006,13007],{},"Convert that CSV back to GSI.",[338,13009,13010],{},"Compare initial and final GSI files. Trailing-zero formatting may differ, but coordinate values must match.",[323,13012,13014],{"id":13013},"mvp-limits-and-future-capabilities","MVP Limits and Future Capabilities",[11,13016,13017],{},"The current MVP focuses on direct, point-based spatial conversions. Be aware of these limits:",[335,13019,13020,13026,13035,13041,13054,13060],{},[338,13021,13022,13025],{},[318,13023,13024],{},"No bulk\u002Fbatch processing:"," One file per conversion job.",[338,13027,13028,13031,13032,13034],{},[318,13029,13030],{},"No direct DWG export:"," Output is ASCII DXF with ",[2863,13033,11912],{}," entities only.",[338,13036,13037,13040],{},[318,13038,13039],{},"No LandXML \u002F alignment geometry:"," Discrete USDM v1 points only — not road alignments, cross-sections, or surfaces.",[338,13042,13043,13050,13051,13053],{},[318,13044,13045,13046,13049],{},"No direct ",[2863,13047,13048],{},".xlsx"," parsing:"," Save spreadsheets as ",[2863,13052,12010],{}," before conversion.",[338,13055,13056,13059],{},[318,13057,13058],{},"No cloud project storage:"," Processed coordinates live in browser memory; refreshing clears unsaved work (by design for privacy).",[338,13061,13062,13065],{},[318,13063,13064],{},"GSI-8 default on CSV → GSI UI:"," The engine supports GSI-16 round-trips in tests; the live converter page currently exports GSI-8.",[323,13067,13069],{"id":13068},"troubleshooting-common-conversion-failures","Troubleshooting Common Conversion Failures",[851,13071,13072,13085],{},[854,13073,13074],{},[857,13075,13076,13079,13082],{},[860,13077,13078],{},"Symptom",[860,13080,13081],{},"Probable cause",[860,13083,13084],{},"Corrective action",[876,13086,13087,13101,13112,13126,13137,13148],{},[857,13088,13089,13092,13095],{},[881,13090,13091],{},"Parser rejects GSI input",[881,13093,13094],{},"Missing sign flags or inconsistent formatting",[881,13096,13097,13098],{},"Use instrument-native export; GSI-16 lines must start with ",[2863,13099,13100],{},"*",[857,13102,13103,13106,13109],{},[881,13104,13105],{},"Points rotated 90° in CAD",[881,13107,13108],{},"Easting and Northing columns swapped",[881,13110,13111],{},"Re-upload CSV and swap Easting\u002FNorthing in the mapping wizard",[857,13113,13114,13117,13120],{},[881,13115,13116],{},"Points at elevation 0.0",[881,13118,13119],{},"Elevation column unmapped or missing",[881,13121,13122,13123,13125],{},"Map the elevation column; missing values export as ",[2863,13124,1748],{}," in DXF",[857,13127,13128,13131,13134],{},[881,13129,13130],{},"Total station rejects GSI upload",[881,13132,13133],{},"GSI-8 vs GSI-16 mismatch",[881,13135,13136],{},"Match instrument settings to exported variant",[857,13138,13139,13142,13145],{},[881,13140,13141],{},"Special characters corrupted in point IDs",[881,13143,13144],{},"UTF-16 or UTF-8 BOM encoding",[881,13146,13147],{},"Re-save as UTF-8 (no BOM) or ANSI in a plain-text editor",[857,13149,13150,13153,13156],{},[881,13151,13152],{},"Conversion blocked",[881,13154,13155],{},"Not signed in or insufficient credits",[881,13157,13158],{},"Sign in; check remaining credits on your account",[323,13160,11205],{"id":11204},[11,13162,13163],{},"Maximize field efficiency and protect data privacy with Surveying Core’s browser-native tools:",[335,13165,13166,13171,13176,13181],{},[338,13167,13168,13170],{},[693,13169,11982],{"href":11274}," — Leica field files to office spreadsheets",[338,13172,13173,13175],{},[693,13174,12070],{"href":11274}," — design coordinates back to the instrument",[338,13177,13178,13180],{},[693,13179,12143],{"href":11274}," — coordinate lists to CAD point layers",[338,13182,13183,13185],{},[693,13184,11699],{"href":11303}," — deep reference for Leica GSI structure",{"title":15,"searchDepth":16,"depth":16,"links":13187},[13188,13189,13190,13191,13196,13200,13201,13204,13207,13210,13214,13220,13221,13222],{"id":11260,"depth":16,"text":11261},{"id":11307,"depth":16,"text":11308},{"id":11326,"depth":16,"text":11327},{"id":11581,"depth":16,"text":11582,"children":13192},[13193,13194,13195],{"id":11588,"depth":73,"text":11589},{"id":11638,"depth":73,"text":11639},{"id":11674,"depth":73,"text":11675},{"id":11692,"depth":16,"text":11693,"children":13197},[13198,13199],{"id":11702,"depth":73,"text":11703},{"id":11729,"depth":73,"text":11730},{"id":11902,"depth":16,"text":11903},{"id":11976,"depth":16,"text":11977,"children":13202},[13203],{"id":12024,"depth":73,"text":12025},{"id":12064,"depth":16,"text":12065,"children":13205},[13206],{"id":12103,"depth":73,"text":12104},{"id":12137,"depth":16,"text":12138,"children":13208},[13209],{"id":12178,"depth":73,"text":12179},{"id":12211,"depth":16,"text":12212,"children":13211},[13212,13213],{"id":12222,"depth":73,"text":12223},{"id":12249,"depth":73,"text":12250},{"id":12264,"depth":16,"text":12265,"children":13215},[13216,13217,13218,13219],{"id":12271,"depth":73,"text":12272},{"id":12281,"depth":73,"text":12282},{"id":12293,"depth":73,"text":12294},{"id":12995,"depth":73,"text":12996},{"id":13013,"depth":16,"text":13014},{"id":13068,"depth":16,"text":13069},{"id":11204,"depth":16,"text":11205},"Browser-native GSI, CSV, and DXF conversion — survey workflows, USDM v1, client-side privacy, and quality checks.",{},"\u002Flearn\u002Ftools\u002Ffile-conversion-overview",{"title":11255,"description":13223},{"loc":13225},"learn\u002Ftools\u002Ffile-conversion-overview","xpzNSmP9GdHjgSr6EvRqow9AukvYXRp0TYaAz0RBtkM",{"id":13231,"title":13232,"body":13233,"category":11245,"description":22285,"extension":20,"meta":22286,"navigation":22,"order":90,"path":22287,"seo":22288,"sitemap":22289,"stem":22290,"__hash__":22291},"learn\u002Flearn\u002Ftools\u002Fpolygon-area.md","Polygon Area",{"type":8,"value":13234,"toc":22267},[13235,13237,13245,13249,13252,13255,13259,13320,13323,13471,13598,13602,13609,13686,13809,13813,13816,13849,14314,14608,15086,15089,15498,15783,15787,16224,16690,16693,17196,17346,17594,17598,17601,18578,18985,19435,20370,20373,20909,21133,21136,21203,21206,21377,21383,21387,21390,21394,21534,21706,21710,21717,21721,21724,21728,21731,21979,21982,21986,22195,22199,22206,22208],[323,13236,685],{"id":684},[11,13238,13239,13240,13244],{},"In civil engineering, land surveying, and geomatics, calculating the planimetric area and perimeter of bounded land parcels is a core requirement. Whether verifying boundary dimensions for cadastral land transfers, determining footprints of structural foundations, or computing earthwork quantities for infrastructure developments, precise geometric calculations are paramount. Modern workflows rely on digital systems to process field observations, making coordinate geometry (COGO) the analytical standard over legacy manual drafting or planimeter tracing. The ",[693,13241,13243],{"href":13242},"\u002Ftools\u002Fpolygon-area","Polygon area calculator"," provides a fast, reliable interface for ordered easting\u002Fnorthing pairs, outputting plan area and perimeter with minimal manual overhead.",[323,13246,13248],{"id":13247},"plan-area-determination-in-engineering-surveying","Plan Area Determination in Engineering Surveying",[11,13250,13251],{},"Before computerized coordinate geometry, surveyors calculated parcel areas using physical measurements recorded in field books. Early boundaries were measured with Gunter's chains or steel tapes, with areas derived by partitioning irregular tracts into triangles or by taking perpendicular offsets from a baseline using the trapezoidal or Simpson's rules. These methods suffered from error propagation across the calculation and required heavy approximation for irregular curves.",[11,13253,13254],{},"Mandatory theodolite observations, electronic distance measurement (EDM), and high-precision GNSS catalyzed a shift to coordinate-based computation. By defining boundaries as ordered grid coordinate pairs on a projected Cartesian plane, surveyors replaced graphical approximations with exact algebraic formulations. This shift eliminated many drafting errors and established a framework that integrates with GIS and CAD. Professional bodies such as RICS emphasize coordinate competence for conveyancing and land planning.",[323,13256,13258],{"id":13257},"coordinate-systems-and-projected-grids","Coordinate Systems and Projected Grids",[11,13260,13261,13262,13290,13291,13319],{},"In plane surveying, calculations assume a flat two-dimensional Cartesian plane where horizontal locations are identified by easting (",[772,13263,13265,13278],{"className":13264},[775],[772,13266,13268],{"className":13267},[779],[781,13269,13270],{"xmlns":783},[785,13271,13272,13276],{},[788,13273,13274],{},[791,13275,2808],{},[795,13277,2808],{"encoding":797},[772,13279,13281],{"className":13280,"ariaHidden":803},[802],[772,13282,13284,13287],{"className":13283},[807],[772,13285],{"className":13286,"style":2758},[811],[772,13288,2808],{"className":13289,"style":2823},[816,817],") and northing (",[772,13292,13294,13307],{"className":13293},[775],[772,13295,13297],{"className":13296},[779],[781,13298,13299],{"xmlns":783},[785,13300,13301,13305],{},[788,13302,13303],{},[791,13304,2839],{},[795,13306,2839],{"encoding":797},[772,13308,13310],{"className":13309,"ariaHidden":803},[802],[772,13311,13313,13316],{"className":13312},[807],[772,13314],{"className":13315,"style":2758},[811],[772,13317,2839],{"className":13318,"style":2854},[816,817],") pairs. This model simplifies ellipsoidal geometry into Euclidean math, which is highly accurate for localized engineering projects. Grid systems — UTM, State Plane, and regional grids — project the curved Earth onto a 2D plane.",[11,13321,13322],{},"Grid distances differ from ground distances by projection and height factors. Surveyors must account for these when translating field measurements to the coordinate plane. When converting bearings and horizontal distances to grid coordinates:",[11,13324,13325,13398],{},[772,13326,13328,13353],{"className":13327},[775],[772,13329,13331],{"className":13330},[779],[781,13332,13333],{"xmlns":783},[785,13334,13335,13351],{},[788,13336,13337,13339,13341,13343,13345,13347,13349],{},[791,13338,2963],{"mathvariant":906},[791,13340,2808],{},[980,13342,982],{},[791,13344,3180],{},[791,13346,3249],{},[980,13348,3252],{},[791,13350,793],{},[795,13352,10661],{"encoding":797},[772,13354,13356,13377],{"className":13355,"ariaHidden":803},[802],[772,13357,13359,13362,13365,13368,13371,13374],{"className":13358},[807],[772,13360],{"className":13361,"style":2758},[811],[772,13363,2963],{"className":13364},[816],[772,13366,2808],{"className":13367,"style":2823},[816,817],[772,13369],{"className":13370,"style":1004},[1003],[772,13372,982],{"className":13373},[1008],[772,13375],{"className":13376,"style":1004},[1003],[772,13378,13380,13383,13386,13389,13392,13395],{"className":13379},[807],[772,13381],{"className":13382,"style":812},[811],[772,13384,3180],{"className":13385,"style":818},[816,817],[772,13387],{"className":13388,"style":3294},[1003],[772,13390,3249],{"className":13391},[3298],[772,13393],{"className":13394,"style":3294},[1003],[772,13396,793],{"className":13397,"style":818},[816,817],[772,13399,13401,13426],{"className":13400},[775],[772,13402,13404],{"className":13403},[779],[781,13405,13406],{"xmlns":783},[785,13407,13408,13424],{},[788,13409,13410,13412,13414,13416,13418,13420,13422],{},[791,13411,2963],{"mathvariant":906},[791,13413,2839],{},[980,13415,982],{},[791,13417,3180],{},[791,13419,3327],{},[980,13421,3252],{},[791,13423,793],{},[795,13425,10735],{"encoding":797},[772,13427,13429,13450],{"className":13428,"ariaHidden":803},[802],[772,13430,13432,13435,13438,13441,13444,13447],{"className":13431},[807],[772,13433],{"className":13434,"style":2758},[811],[772,13436,2963],{"className":13437},[816],[772,13439,2839],{"className":13440,"style":2854},[816,817],[772,13442],{"className":13443,"style":1004},[1003],[772,13445,982],{"className":13446},[1008],[772,13448],{"className":13449,"style":1004},[1003],[772,13451,13453,13456,13459,13462,13465,13468],{"className":13452},[807],[772,13454],{"className":13455,"style":812},[811],[772,13457,3180],{"className":13458,"style":818},[816,817],[772,13460],{"className":13461,"style":3294},[1003],[772,13463,3327],{"className":13464},[3298],[772,13466],{"className":13467,"style":3294},[1003],[772,13469,793],{"className":13470,"style":818},[816,817],[11,13472,13473,13474,13593,13594,13597],{},"Absolute position follows ",[772,13475,13477,13505],{"className":13476},[775],[772,13478,13480],{"className":13479},[779],[781,13481,13482],{"xmlns":783},[785,13483,13484,13502],{},[788,13485,13486,13488,13490,13496,13498,13500],{},[791,13487,2808],{},[980,13489,982],{},[3525,13491,13492,13494],{},[791,13493,2808],{},[1084,13495,1748],{},[980,13497,1353],{},[791,13499,2963],{"mathvariant":906},[791,13501,2808],{},[795,13503,13504],{"encoding":797},"E = E_0 + \\Delta E",[772,13506,13508,13526,13581],{"className":13507,"ariaHidden":803},[802],[772,13509,13511,13514,13517,13520,13523],{"className":13510},[807],[772,13512],{"className":13513,"style":2758},[811],[772,13515,2808],{"className":13516,"style":2823},[816,817],[772,13518],{"className":13519,"style":1004},[1003],[772,13521,982],{"className":13522},[1008],[772,13524],{"className":13525,"style":1004},[1003],[772,13527,13529,13532,13572,13575,13578],{"className":13528},[807],[772,13530],{"className":13531,"style":3543},[811],[772,13533,13535,13538],{"className":13534},[816],[772,13536,2808],{"className":13537,"style":2823},[816,817],[772,13539,13541],{"className":13540},[1752],[772,13542,13544,13564],{"className":13543},[1756,3556],[772,13545,13547,13561],{"className":13546},[1760],[772,13548,13550],{"className":13549,"style":3563},[1764],[772,13551,13552,13555],{"style":3566},[772,13553],{"className":13554,"style":1773},[1772],[772,13556,13558],{"className":13557},[1777,1778,1779,1780],[772,13559,1748],{"className":13560},[816,1780],[772,13562,3580],{"className":13563},[3579],[772,13565,13567],{"className":13566},[1760],[772,13568,13570],{"className":13569,"style":3587},[1764],[772,13571],{},[772,13573],{"className":13574,"style":1121},[1003],[772,13576,1353],{"className":13577},[1125],[772,13579],{"className":13580,"style":1121},[1003],[772,13582,13584,13587,13590],{"className":13583},[807],[772,13585],{"className":13586,"style":2758},[811],[772,13588,2963],{"className":13589},[816],[772,13591,2808],{"className":13592,"style":2823},[816,817]," (and similarly for northing). For forward\u002Finverse coordinate steps, use the ",[693,13595,696],{"href":13596},"\u002Ftools\u002Fcalculators\u002Fbearing-distance",". Once vertices lie on the grid, planimetric area follows from coordinate algorithms.",[323,13599,13601],{"id":13600},"vertex-winding-and-signed-area","Vertex Winding and Signed Area",[11,13603,13604,13605,13608],{},"When computing area from coordinates, vertex order matters. The Shoelace sum yields a ",[318,13606,13607],{},"signed area"," depending on whether the boundary is traversed clockwise or counter-clockwise (CCW).",[11,13610,13611,13612,13646,13647,13682,13683,13685],{},"In a standard mathematical frame with ",[772,13613,13615,13631],{"className":13614},[775],[772,13616,13618],{"className":13617},[779],[781,13619,13620],{"xmlns":783},[785,13621,13622,13628],{},[788,13623,13624,13626],{},[980,13625,1353],{},[791,13627,11099],{},[795,13629,13630],{"encoding":797},"+x",[772,13632,13634],{"className":13633,"ariaHidden":803},[802],[772,13635,13637,13640,13643],{"className":13636},[807],[772,13638],{"className":13639,"style":6118},[811],[772,13641,1353],{"className":13642},[816],[772,13644,11099],{"className":13645},[816,817]," east and ",[772,13648,13650,13666],{"className":13649},[775],[772,13651,13653],{"className":13652},[779],[781,13654,13655],{"xmlns":783},[785,13656,13657,13663],{},[788,13658,13659,13661],{},[980,13660,1353],{},[791,13662,11129],{},[795,13664,13665],{"encoding":797},"+y",[772,13667,13669],{"className":13668,"ariaHidden":803},[802],[772,13670,13672,13676,13679],{"className":13671},[807],[772,13673],{"className":13674,"style":13675},[811],"height:0.7778em;vertical-align:-0.1944em;",[772,13677,1353],{"className":13678},[816],[772,13680,11129],{"className":13681,"style":6125},[816,817]," north, CCW traversal typically yields positive signed area; CW yields negative signed area. Physical parcel area is always non-negative, so practitioners take the absolute value of the signed result — as the ",[693,13684,13243],{"href":13242}," does.",[851,13687,13688,13704],{},[854,13689,13690],{},[857,13691,13692,13695,13698,13701],{},[860,13693,13694],{},"Winding order",[860,13696,13697],{},"Traversal",[860,13699,13700],{},"Sign of result",[860,13702,13703],{},"Reported plan area",[876,13705,13706,13759],{},[857,13707,13708,13711,13714,13716],{},[881,13709,13710],{},"Anticlockwise",[881,13712,13713],{},"CCW",[881,13715,3038],{},[881,13717,13718],{},[772,13719,13721,13741],{"className":13720},[775],[772,13722,13724],{"className":13723},[779],[781,13725,13726],{"xmlns":783},[785,13727,13728,13738],{},[788,13729,13730,13733,13736],{},[791,13731,13732],{"mathvariant":906},"∥",[791,13734,13735],{},"A",[791,13737,13732],{"mathvariant":906},[795,13739,13740],{"encoding":797},"\\|A\\|",[772,13742,13744],{"className":13743,"ariaHidden":803},[802],[772,13745,13747,13750,13753,13756],{"className":13746},[807],[772,13748],{"className":13749,"style":1109},[811],[772,13751,13732],{"className":13752},[816],[772,13754,13735],{"className":13755},[816,817],[772,13757,13732],{"className":13758},[816],[857,13760,13761,13764,13767,13769],{},[881,13762,13763],{},"Clockwise",[881,13765,13766],{},"CW",[881,13768,3058],{},[881,13770,13771],{},[772,13772,13774,13791],{"className":13773},[775],[772,13775,13777],{"className":13776},[779],[781,13778,13779],{"xmlns":783},[785,13780,13781,13789],{},[788,13782,13783,13785,13787],{},[791,13784,13732],{"mathvariant":906},[791,13786,13735],{},[791,13788,13732],{"mathvariant":906},[795,13790,13740],{"encoding":797},[772,13792,13794],{"className":13793,"ariaHidden":803},[802],[772,13795,13797,13800,13803,13806],{"className":13796},[807],[772,13798],{"className":13799,"style":1109},[811],[772,13801,13732],{"className":13802},[816],[772,13804,13735],{"className":13805},[816,817],[772,13807,13732],{"className":13808},[816],[323,13810,13812],{"id":13811},"the-shoelace-formula-surveyors-formula","The Shoelace Formula (Surveyor's Formula)",[11,13814,13815],{},"The algorithm for a simple polygon from Cartesian coordinates is known as the Shoelace formula, Gauss's area formula, or the surveyor's formula. It is a discrete form of Green's theorem: a line integral around a closed boundary expressed as a sum of cross-products between consecutive vertices.",[11,13817,13818,13819,13848],{},"For ",[772,13820,13822,13836],{"className":13821},[775],[772,13823,13825],{"className":13824},[779],[781,13826,13827],{"xmlns":783},[785,13828,13829,13834],{},[788,13830,13831],{},[791,13832,13833],{},"n",[795,13835,13833],{"encoding":797},[772,13837,13839],{"className":13838,"ariaHidden":803},[802],[772,13840,13842,13845],{"className":13841},[807],[772,13843],{"className":13844,"style":845},[811],[772,13846,13833],{"className":13847},[816,817]," vertices ordered around the perimeter:",[11,13850,13851],{},[772,13852,13854,13941],{"className":13853},[775],[772,13855,13857],{"className":13856},[779],[781,13858,13859],{"xmlns":783},[785,13860,13861,13938],{},[788,13862,13863,13865,13867,13873],{},[791,13864,13735],{},[980,13866,982],{},[5599,13868,13869,13871],{},[1084,13870,2170],{},[1084,13872,1872],{},[788,13874,13875,13877,13894,13896,13902,13914,13916,13928,13934,13936],{},[980,13876,5605],{"fence":803},[13878,13879,13880,13883,13892],"msubsup",{},[980,13881,13882],{},"∑",[788,13884,13885,13888,13890],{},[791,13886,13887],{},"i",[980,13889,982],{},[1084,13891,2170],{},[791,13893,13833],{},[980,13895,1082],{"stretchy":1081},[3525,13897,13898,13900],{},[791,13899,2808],{},[791,13901,13887],{},[3525,13903,13904,13906],{},[791,13905,2839],{},[788,13907,13908,13910,13912],{},[791,13909,13887],{},[980,13911,1353],{},[1084,13913,2170],{},[980,13915,1091],{},[3525,13917,13918,13920],{},[791,13919,2808],{},[788,13921,13922,13924,13926],{},[791,13923,13887],{},[980,13925,1353],{},[1084,13927,2170],{},[3525,13929,13930,13932],{},[791,13931,2839],{},[791,13933,13887],{},[980,13935,1096],{"stretchy":1081},[980,13937,5605],{"fence":803},[795,13939,13940],{"encoding":797},"A = \\frac{1}{2} \\left| \\sum_{i=1}^{n} (E_i N_{i+1} - E_{i+1} N_i) \\right|",[772,13942,13944,13962],{"className":13943,"ariaHidden":803},[802],[772,13945,13947,13950,13953,13956,13959],{"className":13946},[807],[772,13948],{"className":13949,"style":2758},[811],[772,13951,13735],{"className":13952},[816,817],[772,13954],{"className":13955,"style":1004},[1003],[772,13957,982],{"className":13958},[1008],[772,13960],{"className":13961,"style":1004},[1003],[772,13963,13965,13969,14037,14040],{"className":13964},[807],[772,13966],{"className":13967,"style":13968},[811],"height:1.1901em;vertical-align:-0.345em;",[772,13970,13972,13975,14034],{"className":13971},[816],[772,13973],{"className":13974},[1113,5681],[772,13976,13978],{"className":13977},[5599],[772,13979,13981,14026],{"className":13980},[1756,3556],[772,13982,13984,14023],{"className":13983},[1760],[772,13985,13987,14001,14009],{"className":13986,"style":6558},[1764],[772,13988,13989,13992],{"style":5697},[772,13990],{"className":13991,"style":5422},[1772],[772,13993,13995],{"className":13994},[1777,1778,1779,1780],[772,13996,13998],{"className":13997},[816,1780],[772,13999,1872],{"className":14000},[816,1780],[772,14002,14003,14006],{"style":5719},[772,14004],{"className":14005,"style":5422},[1772],[772,14007],{"className":14008,"style":5727},[5726],[772,14010,14011,14014],{"style":6583},[772,14012],{"className":14013,"style":5422},[1772],[772,14015,14017],{"className":14016},[1777,1778,1779,1780],[772,14018,14020],{"className":14019},[816,1780],[772,14021,2170],{"className":14022},[816,1780],[772,14024,3580],{"className":14025},[3579],[772,14027,14029],{"className":14028},[1760],[772,14030,14032],{"className":14031,"style":6605},[1764],[772,14033],{},[772,14035],{"className":14036},[1141,5681],[772,14038],{"className":14039,"style":3294},[1003],[772,14041,14043,14046,14116,14119,14160,14210,14213,14216,14219,14268,14308,14311],{"className":14042},[5664],[772,14044,5605],{"className":14045,"style":5669},[1113,5668],[772,14047,14049,14055],{"className":14048},[3298],[772,14050,13882],{"className":14051,"style":14054},[3298,14052,14053],"op-symbol","small-op","position:relative;top:0em;",[772,14056,14058],{"className":14057},[1752],[772,14059,14061,14107],{"className":14060},[1756,3556],[772,14062,14064,14104],{"className":14063},[1760],[772,14065,14068,14089],{"className":14066,"style":14067},[1764],"height:0.8043em;",[772,14069,14071,14074],{"style":14070},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[772,14072],{"className":14073,"style":1773},[1772],[772,14075,14077],{"className":14076},[1777,1778,1779,1780],[772,14078,14080,14083,14086],{"className":14079},[816,1780],[772,14081,13887],{"className":14082},[816,817,1780],[772,14084,982],{"className":14085},[1008,1780],[772,14087,2170],{"className":14088},[816,1780],[772,14090,14092,14095],{"style":14091},"top:-3.2029em;margin-right:0.05em;",[772,14093],{"className":14094,"style":1773},[1772],[772,14096,14098],{"className":14097},[1777,1778,1779,1780],[772,14099,14101],{"className":14100},[816,1780],[772,14102,13833],{"className":14103},[816,817,1780],[772,14105,3580],{"className":14106},[3579],[772,14108,14110],{"className":14109},[1760],[772,14111,14114],{"className":14112,"style":14113},[1764],"height:0.2997em;",[772,14115],{},[772,14117,1082],{"className":14118},[1113],[772,14120,14122,14125],{"className":14121},[816],[772,14123,2808],{"className":14124,"style":2823},[816,817],[772,14126,14128],{"className":14127},[1752],[772,14129,14131,14152],{"className":14130},[1756,3556],[772,14132,14134,14149],{"className":14133},[1760],[772,14135,14138],{"className":14136,"style":14137},[1764],"height:0.3117em;",[772,14139,14140,14143],{"style":3566},[772,14141],{"className":14142,"style":1773},[1772],[772,14144,14146],{"className":14145},[1777,1778,1779,1780],[772,14147,13887],{"className":14148},[816,817,1780],[772,14150,3580],{"className":14151},[3579],[772,14153,14155],{"className":14154},[1760],[772,14156,14158],{"className":14157,"style":3587},[1764],[772,14159],{},[772,14161,14163,14166],{"className":14162},[816],[772,14164,2839],{"className":14165,"style":2854},[816,817],[772,14167,14169],{"className":14168},[1752],[772,14170,14172,14201],{"className":14171},[1756,3556],[772,14173,14175,14198],{"className":14174},[1760],[772,14176,14178],{"className":14177,"style":14137},[1764],[772,14179,14180,14183],{"style":3640},[772,14181],{"className":14182,"style":1773},[1772],[772,14184,14186],{"className":14185},[1777,1778,1779,1780],[772,14187,14189,14192,14195],{"className":14188},[816,1780],[772,14190,13887],{"className":14191},[816,817,1780],[772,14193,1353],{"className":14194},[1125,1780],[772,14196,2170],{"className":14197},[816,1780],[772,14199,3580],{"className":14200},[3579],[772,14202,14204],{"className":14203},[1760],[772,14205,14208],{"className":14206,"style":14207},[1764],"height:0.2083em;",[772,14209],{},[772,14211],{"className":14212,"style":1121},[1003],[772,14214,1091],{"className":14215},[1125],[772,14217],{"className":14218,"style":1121},[1003],[772,14220,14222,14225],{"className":14221},[816],[772,14223,2808],{"className":14224,"style":2823},[816,817],[772,14226,14228],{"className":14227},[1752],[772,14229,14231,14260],{"className":14230},[1756,3556],[772,14232,14234,14257],{"className":14233},[1760],[772,14235,14237],{"className":14236,"style":14137},[1764],[772,14238,14239,14242],{"style":3566},[772,14240],{"className":14241,"style":1773},[1772],[772,14243,14245],{"className":14244},[1777,1778,1779,1780],[772,14246,14248,14251,14254],{"className":14247},[816,1780],[772,14249,13887],{"className":14250},[816,817,1780],[772,14252,1353],{"className":14253},[1125,1780],[772,14255,2170],{"className":14256},[816,1780],[772,14258,3580],{"className":14259},[3579],[772,14261,14263],{"className":14262},[1760],[772,14264,14266],{"className":14265,"style":14207},[1764],[772,14267],{},[772,14269,14271,14274],{"className":14270},[816],[772,14272,2839],{"className":14273,"style":2854},[816,817],[772,14275,14277],{"className":14276},[1752],[772,14278,14280,14300],{"className":14279},[1756,3556],[772,14281,14283,14297],{"className":14282},[1760],[772,14284,14286],{"className":14285,"style":14137},[1764],[772,14287,14288,14291],{"style":3640},[772,14289],{"className":14290,"style":1773},[1772],[772,14292,14294],{"className":14293},[1777,1778,1779,1780],[772,14295,13887],{"className":14296},[816,817,1780],[772,14298,3580],{"className":14299},[3579],[772,14301,14303],{"className":14302},[1760],[772,14304,14306],{"className":14305,"style":3587},[1764],[772,14307],{},[772,14309,1096],{"className":14310},[1141],[772,14312,5605],{"className":14313,"style":5669},[1141,5668],[11,14315,14316,14317,5868],{},"Indices wrap so ",[772,14318,14320,14382],{"className":14319},[775],[772,14321,14323],{"className":14322},[779],[781,14324,14325],{"xmlns":783},[785,14326,14327,14379],{},[788,14328,14329,14331,14343,14345,14357,14359,14361,14363,14369,14371,14377],{},[980,14330,1082],{"stretchy":1081},[3525,14332,14333,14335],{},[791,14334,2808],{},[788,14336,14337,14339,14341],{},[791,14338,13833],{},[980,14340,1353],{},[1084,14342,2170],{},[980,14344,5940],{"separator":803},[3525,14346,14347,14349],{},[791,14348,2839],{},[788,14350,14351,14353,14355],{},[791,14352,13833],{},[980,14354,1353],{},[1084,14356,2170],{},[980,14358,1096],{"stretchy":1081},[980,14360,982],{},[980,14362,1082],{"stretchy":1081},[3525,14364,14365,14367],{},[791,14366,2808],{},[1084,14368,2170],{},[980,14370,5940],{"separator":803},[3525,14372,14373,14375],{},[791,14374,2839],{},[1084,14376,2170],{},[980,14378,1096],{"stretchy":1081},[795,14380,14381],{"encoding":797},"(E_{n+1}, N_{n+1}) = (E_1, N_1)",[772,14383,14385,14510],{"className":14384,"ariaHidden":803},[802],[772,14386,14388,14391,14394,14443,14446,14449,14498,14501,14504,14507],{"className":14387},[807],[772,14389],{"className":14390,"style":1109},[811],[772,14392,1082],{"className":14393},[1113],[772,14395,14397,14400],{"className":14396},[816],[772,14398,2808],{"className":14399,"style":2823},[816,817],[772,14401,14403],{"className":14402},[1752],[772,14404,14406,14435],{"className":14405},[1756,3556],[772,14407,14409,14432],{"className":14408},[1760],[772,14410,14412],{"className":14411,"style":3563},[1764],[772,14413,14414,14417],{"style":3566},[772,14415],{"className":14416,"style":1773},[1772],[772,14418,14420],{"className":14419},[1777,1778,1779,1780],[772,14421,14423,14426,14429],{"className":14422},[816,1780],[772,14424,13833],{"className":14425},[816,817,1780],[772,14427,1353],{"className":14428},[1125,1780],[772,14430,2170],{"className":14431},[816,1780],[772,14433,3580],{"className":14434},[3579],[772,14436,14438],{"className":14437},[1760],[772,14439,14441],{"className":14440,"style":14207},[1764],[772,14442],{},[772,14444,5940],{"className":14445},[6041],[772,14447],{"className":14448,"style":3294},[1003],[772,14450,14452,14455],{"className":14451},[816],[772,14453,2839],{"className":14454,"style":2854},[816,817],[772,14456,14458],{"className":14457},[1752],[772,14459,14461,14490],{"className":14460},[1756,3556],[772,14462,14464,14487],{"className":14463},[1760],[772,14465,14467],{"className":14466,"style":3563},[1764],[772,14468,14469,14472],{"style":3640},[772,14470],{"className":14471,"style":1773},[1772],[772,14473,14475],{"className":14474},[1777,1778,1779,1780],[772,14476,14478,14481,14484],{"className":14477},[816,1780],[772,14479,13833],{"className":14480},[816,817,1780],[772,14482,1353],{"className":14483},[1125,1780],[772,14485,2170],{"className":14486},[816,1780],[772,14488,3580],{"className":14489},[3579],[772,14491,14493],{"className":14492},[1760],[772,14494,14496],{"className":14495,"style":14207},[1764],[772,14497],{},[772,14499,1096],{"className":14500},[1141],[772,14502],{"className":14503,"style":1004},[1003],[772,14505,982],{"className":14506},[1008],[772,14508],{"className":14509,"style":1004},[1003],[772,14511,14513,14516,14519,14559,14562,14565,14605],{"className":14512},[807],[772,14514],{"className":14515,"style":1109},[811],[772,14517,1082],{"className":14518},[1113],[772,14520,14522,14525],{"className":14521},[816],[772,14523,2808],{"className":14524,"style":2823},[816,817],[772,14526,14528],{"className":14527},[1752],[772,14529,14531,14551],{"className":14530},[1756,3556],[772,14532,14534,14548],{"className":14533},[1760],[772,14535,14537],{"className":14536,"style":3563},[1764],[772,14538,14539,14542],{"style":3566},[772,14540],{"className":14541,"style":1773},[1772],[772,14543,14545],{"className":14544},[1777,1778,1779,1780],[772,14546,2170],{"className":14547},[816,1780],[772,14549,3580],{"className":14550},[3579],[772,14552,14554],{"className":14553},[1760],[772,14555,14557],{"className":14556,"style":3587},[1764],[772,14558],{},[772,14560,5940],{"className":14561},[6041],[772,14563],{"className":14564,"style":3294},[1003],[772,14566,14568,14571],{"className":14567},[816],[772,14569,2839],{"className":14570,"style":2854},[816,817],[772,14572,14574],{"className":14573},[1752],[772,14575,14577,14597],{"className":14576},[1756,3556],[772,14578,14580,14594],{"className":14579},[1760],[772,14581,14583],{"className":14582,"style":3563},[1764],[772,14584,14585,14588],{"style":3640},[772,14586],{"className":14587,"style":1773},[1772],[772,14589,14591],{"className":14590},[1777,1778,1779,1780],[772,14592,2170],{"className":14593},[816,1780],[772,14595,3580],{"className":14596},[3579],[772,14598,14600],{"className":14599},[1760],[772,14601,14603],{"className":14602,"style":3587},[1764],[772,14604],{},[772,14606,1096],{"className":14607},[1141],[11,14609,14610,14611,14867,14868,14926,14927,2896,15000,15085],{},"Each term ",[772,14612,14614,14664],{"className":14613},[775],[772,14615,14617],{"className":14616},[779],[781,14618,14619],{"xmlns":783},[785,14620,14621,14661],{},[788,14622,14623,14629,14641,14643,14655],{},[3525,14624,14625,14627],{},[791,14626,2808],{},[791,14628,13887],{},[3525,14630,14631,14633],{},[791,14632,2839],{},[788,14634,14635,14637,14639],{},[791,14636,13887],{},[980,14638,1353],{},[1084,14640,2170],{},[980,14642,1091],{},[3525,14644,14645,14647],{},[791,14646,2808],{},[788,14648,14649,14651,14653],{},[791,14650,13887],{},[980,14652,1353],{},[1084,14654,2170],{},[3525,14656,14657,14659],{},[791,14658,2839],{},[791,14660,13887],{},[795,14662,14663],{"encoding":797},"E_i N_{i+1} - E_{i+1} N_i",[772,14665,14667,14772],{"className":14666,"ariaHidden":803},[802],[772,14668,14670,14674,14714,14763,14766,14769],{"className":14669},[807],[772,14671],{"className":14672,"style":14673},[811],"height:0.8917em;vertical-align:-0.2083em;",[772,14675,14677,14680],{"className":14676},[816],[772,14678,2808],{"className":14679,"style":2823},[816,817],[772,14681,14683],{"className":14682},[1752],[772,14684,14686,14706],{"className":14685},[1756,3556],[772,14687,14689,14703],{"className":14688},[1760],[772,14690,14692],{"className":14691,"style":14137},[1764],[772,14693,14694,14697],{"style":3566},[772,14695],{"className":14696,"style":1773},[1772],[772,14698,14700],{"className":14699},[1777,1778,1779,1780],[772,14701,13887],{"className":14702},[816,817,1780],[772,14704,3580],{"className":14705},[3579],[772,14707,14709],{"className":14708},[1760],[772,14710,14712],{"className":14711,"style":3587},[1764],[772,14713],{},[772,14715,14717,14720],{"className":14716},[816],[772,14718,2839],{"className":14719,"style":2854},[816,817],[772,14721,14723],{"className":14722},[1752],[772,14724,14726,14755],{"className":14725},[1756,3556],[772,14727,14729,14752],{"className":14728},[1760],[772,14730,14732],{"className":14731,"style":14137},[1764],[772,14733,14734,14737],{"style":3640},[772,14735],{"className":14736,"style":1773},[1772],[772,14738,14740],{"className":14739},[1777,1778,1779,1780],[772,14741,14743,14746,14749],{"className":14742},[816,1780],[772,14744,13887],{"className":14745},[816,817,1780],[772,14747,1353],{"className":14748},[1125,1780],[772,14750,2170],{"className":14751},[816,1780],[772,14753,3580],{"className":14754},[3579],[772,14756,14758],{"className":14757},[1760],[772,14759,14761],{"className":14760,"style":14207},[1764],[772,14762],{},[772,14764],{"className":14765,"style":1121},[1003],[772,14767,1091],{"className":14768},[1125],[772,14770],{"className":14771,"style":1121},[1003],[772,14773,14775,14778,14827],{"className":14774},[807],[772,14776],{"className":14777,"style":14673},[811],[772,14779,14781,14784],{"className":14780},[816],[772,14782,2808],{"className":14783,"style":2823},[816,817],[772,14785,14787],{"className":14786},[1752],[772,14788,14790,14819],{"className":14789},[1756,3556],[772,14791,14793,14816],{"className":14792},[1760],[772,14794,14796],{"className":14795,"style":14137},[1764],[772,14797,14798,14801],{"style":3566},[772,14799],{"className":14800,"style":1773},[1772],[772,14802,14804],{"className":14803},[1777,1778,1779,1780],[772,14805,14807,14810,14813],{"className":14806},[816,1780],[772,14808,13887],{"className":14809},[816,817,1780],[772,14811,1353],{"className":14812},[1125,1780],[772,14814,2170],{"className":14815},[816,1780],[772,14817,3580],{"className":14818},[3579],[772,14820,14822],{"className":14821},[1760],[772,14823,14825],{"className":14824,"style":14207},[1764],[772,14826],{},[772,14828,14830,14833],{"className":14829},[816],[772,14831,2839],{"className":14832,"style":2854},[816,817],[772,14834,14836],{"className":14835},[1752],[772,14837,14839,14859],{"className":14838},[1756,3556],[772,14840,14842,14856],{"className":14841},[1760],[772,14843,14845],{"className":14844,"style":14137},[1764],[772,14846,14847,14850],{"style":3640},[772,14848],{"className":14849,"style":1773},[1772],[772,14851,14853],{"className":14852},[1777,1778,1779,1780],[772,14854,13887],{"className":14855},[816,817,1780],[772,14857,3580],{"className":14858},[3579],[772,14860,14862],{"className":14861},[1760],[772,14863,14865],{"className":14864,"style":3587},[1764],[772,14866],{}," is twice the signed area of the triangle formed by the origin ",[772,14869,14871,14896],{"className":14870},[775],[772,14872,14874],{"className":14873},[779],[781,14875,14876],{"xmlns":783},[785,14877,14878,14893],{},[788,14879,14880,14883,14885,14887,14889,14891],{},[791,14881,14882],{},"O",[980,14884,1082],{"stretchy":1081},[1084,14886,1748],{},[980,14888,5940],{"separator":803},[1084,14890,1748],{},[980,14892,1096],{"stretchy":1081},[795,14894,14895],{"encoding":797},"O(0,0)",[772,14897,14899],{"className":14898,"ariaHidden":803},[802],[772,14900,14902,14905,14908,14911,14914,14917,14920,14923],{"className":14901},[807],[772,14903],{"className":14904,"style":1109},[811],[772,14906,14882],{"className":14907,"style":818},[816,817],[772,14909,1082],{"className":14910},[1113],[772,14912,1748],{"className":14913},[816],[772,14915,5940],{"className":14916},[6041],[772,14918],{"className":14919,"style":3294},[1003],[772,14921,1748],{"className":14922},[816],[772,14924,1096],{"className":14925},[1141]," and vertices ",[772,14928,14930,14949],{"className":14929},[775],[772,14931,14933],{"className":14932},[779],[781,14934,14935],{"xmlns":783},[785,14936,14937,14946],{},[788,14938,14939],{},[3525,14940,14941,14944],{},[791,14942,14943],{},"P",[791,14945,13887],{},[795,14947,14948],{"encoding":797},"P_i",[772,14950,14952],{"className":14951,"ariaHidden":803},[802],[772,14953,14955,14958],{"className":14954},[807],[772,14956],{"className":14957,"style":3543},[811],[772,14959,14961,14965],{"className":14960},[816],[772,14962,14943],{"className":14963,"style":14964},[816,817],"margin-right:0.1389em;",[772,14966,14968],{"className":14967},[1752],[772,14969,14971,14992],{"className":14970},[1756,3556],[772,14972,14974,14989],{"className":14973},[1760],[772,14975,14977],{"className":14976,"style":14137},[1764],[772,14978,14980,14983],{"style":14979},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[772,14981],{"className":14982,"style":1773},[1772],[772,14984,14986],{"className":14985},[1777,1778,1779,1780],[772,14987,13887],{"className":14988},[816,817,1780],[772,14990,3580],{"className":14991},[3579],[772,14993,14995],{"className":14994},[1760],[772,14996,14998],{"className":14997,"style":3587},[1764],[772,14999],{},[772,15001,15003,15027],{"className":15002},[775],[772,15004,15006],{"className":15005},[779],[781,15007,15008],{"xmlns":783},[785,15009,15010,15024],{},[788,15011,15012],{},[3525,15013,15014,15016],{},[791,15015,14943],{},[788,15017,15018,15020,15022],{},[791,15019,13887],{},[980,15021,1353],{},[1084,15023,2170],{},[795,15025,15026],{"encoding":797},"P_{i+1}",[772,15028,15030],{"className":15029,"ariaHidden":803},[802],[772,15031,15033,15036],{"className":15032},[807],[772,15034],{"className":15035,"style":14673},[811],[772,15037,15039,15042],{"className":15038},[816],[772,15040,14943],{"className":15041,"style":14964},[816,817],[772,15043,15045],{"className":15044},[1752],[772,15046,15048,15077],{"className":15047},[1756,3556],[772,15049,15051,15074],{"className":15050},[1760],[772,15052,15054],{"className":15053,"style":14137},[1764],[772,15055,15056,15059],{"style":14979},[772,15057],{"className":15058,"style":1773},[1772],[772,15060,15062],{"className":15061},[1777,1778,1779,1780],[772,15063,15065,15068,15071],{"className":15064},[816,1780],[772,15066,13887],{"className":15067},[816,817,1780],[772,15069,1353],{"className":15070},[1125,1780],[772,15072,2170],{"className":15073},[816,1780],[772,15075,3580],{"className":15076},[3579],[772,15078,15080],{"className":15079},[1760],[772,15081,15083],{"className":15082,"style":14207},[1764],[772,15084],{},". Summing these terms cancels exterior regions, leaving the enclosed area.",[11,15087,15088],{},"An equivalent form using coordinate differences reduces multiplications:",[11,15090,15091],{},[772,15092,15094,15172],{"className":15093},[775],[772,15095,15097],{"className":15096},[779],[781,15098,15099],{"xmlns":783},[785,15100,15101,15169],{},[788,15102,15103,15105,15107,15113],{},[791,15104,13735],{},[980,15106,982],{},[5599,15108,15109,15111],{},[1084,15110,2170],{},[1084,15112,1872],{},[788,15114,15115,15117,15131,15137,15139,15151,15153,15165,15167],{},[980,15116,5605],{"fence":803},[13878,15118,15119,15121,15129],{},[980,15120,13882],{},[788,15122,15123,15125,15127],{},[791,15124,13887],{},[980,15126,982],{},[1084,15128,2170],{},[791,15130,13833],{},[3525,15132,15133,15135],{},[791,15134,2808],{},[791,15136,13887],{},[980,15138,1082],{"stretchy":1081},[3525,15140,15141,15143],{},[791,15142,2839],{},[788,15144,15145,15147,15149],{},[791,15146,13887],{},[980,15148,1353],{},[1084,15150,2170],{},[980,15152,1091],{},[3525,15154,15155,15157],{},[791,15156,2839],{},[788,15158,15159,15161,15163],{},[791,15160,13887],{},[980,15162,1091],{},[1084,15164,2170],{},[980,15166,1096],{"stretchy":1081},[980,15168,5605],{"fence":803},[795,15170,15171],{"encoding":797},"A = \\frac{1}{2} \\left| \\sum_{i=1}^{n} E_i(N_{i+1} - N_{i-1}) \\right|",[772,15173,15175,15193],{"className":15174,"ariaHidden":803},[802],[772,15176,15178,15181,15184,15187,15190],{"className":15177},[807],[772,15179],{"className":15180,"style":2758},[811],[772,15182,13735],{"className":15183},[816,817],[772,15185],{"className":15186,"style":1004},[1003],[772,15188,982],{"className":15189},[1008],[772,15191],{"className":15192,"style":1004},[1003],[772,15194,15196,15199,15267,15270],{"className":15195},[807],[772,15197],{"className":15198,"style":13968},[811],[772,15200,15202,15205,15264],{"className":15201},[816],[772,15203],{"className":15204},[1113,5681],[772,15206,15208],{"className":15207},[5599],[772,15209,15211,15256],{"className":15210},[1756,3556],[772,15212,15214,15253],{"className":15213},[1760],[772,15215,15217,15231,15239],{"className":15216,"style":6558},[1764],[772,15218,15219,15222],{"style":5697},[772,15220],{"className":15221,"style":5422},[1772],[772,15223,15225],{"className":15224},[1777,1778,1779,1780],[772,15226,15228],{"className":15227},[816,1780],[772,15229,1872],{"className":15230},[816,1780],[772,15232,15233,15236],{"style":5719},[772,15234],{"className":15235,"style":5422},[1772],[772,15237],{"className":15238,"style":5727},[5726],[772,15240,15241,15244],{"style":6583},[772,15242],{"className":15243,"style":5422},[1772],[772,15245,15247],{"className":15246},[1777,1778,1779,1780],[772,15248,15250],{"className":15249},[816,1780],[772,15251,2170],{"className":15252},[816,1780],[772,15254,3580],{"className":15255},[3579],[772,15257,15259],{"className":15258},[1760],[772,15260,15262],{"className":15261,"style":6605},[1764],[772,15263],{},[772,15265],{"className":15266},[1141,5681],[772,15268],{"className":15269,"style":3294},[1003],[772,15271,15273,15276,15339,15342,15382,15385,15434,15437,15440,15443,15492,15495],{"className":15272},[5664],[772,15274,5605],{"className":15275,"style":5669},[1113,5668],[772,15277,15279,15282],{"className":15278},[3298],[772,15280,13882],{"className":15281,"style":14054},[3298,14052,14053],[772,15283,15285],{"className":15284},[1752],[772,15286,15288,15331],{"className":15287},[1756,3556],[772,15289,15291,15328],{"className":15290},[1760],[772,15292,15294,15314],{"className":15293,"style":14067},[1764],[772,15295,15296,15299],{"style":14070},[772,15297],{"className":15298,"style":1773},[1772],[772,15300,15302],{"className":15301},[1777,1778,1779,1780],[772,15303,15305,15308,15311],{"className":15304},[816,1780],[772,15306,13887],{"className":15307},[816,817,1780],[772,15309,982],{"className":15310},[1008,1780],[772,15312,2170],{"className":15313},[816,1780],[772,15315,15316,15319],{"style":14091},[772,15317],{"className":15318,"style":1773},[1772],[772,15320,15322],{"className":15321},[1777,1778,1779,1780],[772,15323,15325],{"className":15324},[816,1780],[772,15326,13833],{"className":15327},[816,817,1780],[772,15329,3580],{"className":15330},[3579],[772,15332,15334],{"className":15333},[1760],[772,15335,15337],{"className":15336,"style":14113},[1764],[772,15338],{},[772,15340],{"className":15341,"style":3294},[1003],[772,15343,15345,15348],{"className":15344},[816],[772,15346,2808],{"className":15347,"style":2823},[816,817],[772,15349,15351],{"className":15350},[1752],[772,15352,15354,15374],{"className":15353},[1756,3556],[772,15355,15357,15371],{"className":15356},[1760],[772,15358,15360],{"className":15359,"style":14137},[1764],[772,15361,15362,15365],{"style":3566},[772,15363],{"className":15364,"style":1773},[1772],[772,15366,15368],{"className":15367},[1777,1778,1779,1780],[772,15369,13887],{"className":15370},[816,817,1780],[772,15372,3580],{"className":15373},[3579],[772,15375,15377],{"className":15376},[1760],[772,15378,15380],{"className":15379,"style":3587},[1764],[772,15381],{},[772,15383,1082],{"className":15384},[1113],[772,15386,15388,15391],{"className":15387},[816],[772,15389,2839],{"className":15390,"style":2854},[816,817],[772,15392,15394],{"className":15393},[1752],[772,15395,15397,15426],{"className":15396},[1756,3556],[772,15398,15400,15423],{"className":15399},[1760],[772,15401,15403],{"className":15402,"style":14137},[1764],[772,15404,15405,15408],{"style":3640},[772,15406],{"className":15407,"style":1773},[1772],[772,15409,15411],{"className":15410},[1777,1778,1779,1780],[772,15412,15414,15417,15420],{"className":15413},[816,1780],[772,15415,13887],{"className":15416},[816,817,1780],[772,15418,1353],{"className":15419},[1125,1780],[772,15421,2170],{"className":15422},[816,1780],[772,15424,3580],{"className":15425},[3579],[772,15427,15429],{"className":15428},[1760],[772,15430,15432],{"className":15431,"style":14207},[1764],[772,15433],{},[772,15435],{"className":15436,"style":1121},[1003],[772,15438,1091],{"className":15439},[1125],[772,15441],{"className":15442,"style":1121},[1003],[772,15444,15446,15449],{"className":15445},[816],[772,15447,2839],{"className":15448,"style":2854},[816,817],[772,15450,15452],{"className":15451},[1752],[772,15453,15455,15484],{"className":15454},[1756,3556],[772,15456,15458,15481],{"className":15457},[1760],[772,15459,15461],{"className":15460,"style":14137},[1764],[772,15462,15463,15466],{"style":3640},[772,15464],{"className":15465,"style":1773},[1772],[772,15467,15469],{"className":15468},[1777,1778,1779,1780],[772,15470,15472,15475,15478],{"className":15471},[816,1780],[772,15473,13887],{"className":15474},[816,817,1780],[772,15476,1091],{"className":15477},[1125,1780],[772,15479,2170],{"className":15480},[816,1780],[772,15482,3580],{"className":15483},[3579],[772,15485,15487],{"className":15486},[1760],[772,15488,15490],{"className":15489,"style":14207},[1764],[772,15491],{},[772,15493,1096],{"className":15494},[1141],[772,15496,5605],{"className":15497,"style":5669},[1141,5668],[11,15499,15500,15501,3474,15635,5868],{},"with ",[772,15502,15504,15530],{"className":15503},[775],[772,15505,15507],{"className":15506},[779],[781,15508,15509],{"xmlns":783},[785,15510,15511,15527],{},[788,15512,15513,15519,15521],{},[3525,15514,15515,15517],{},[791,15516,2839],{},[1084,15518,1748],{},[980,15520,982],{},[3525,15522,15523,15525],{},[791,15524,2839],{},[791,15526,13833],{},[795,15528,15529],{"encoding":797},"N_0 = N_n",[772,15531,15533,15588],{"className":15532,"ariaHidden":803},[802],[772,15534,15536,15539,15579,15582,15585],{"className":15535},[807],[772,15537],{"className":15538,"style":3543},[811],[772,15540,15542,15545],{"className":15541},[816],[772,15543,2839],{"className":15544,"style":2854},[816,817],[772,15546,15548],{"className":15547},[1752],[772,15549,15551,15571],{"className":15550},[1756,3556],[772,15552,15554,15568],{"className":15553},[1760],[772,15555,15557],{"className":15556,"style":3563},[1764],[772,15558,15559,15562],{"style":3640},[772,15560],{"className":15561,"style":1773},[1772],[772,15563,15565],{"className":15564},[1777,1778,1779,1780],[772,15566,1748],{"className":15567},[816,1780],[772,15569,3580],{"className":15570},[3579],[772,15572,15574],{"className":15573},[1760],[772,15575,15577],{"className":15576,"style":3587},[1764],[772,15578],{},[772,15580],{"className":15581,"style":1004},[1003],[772,15583,982],{"className":15584},[1008],[772,15586],{"className":15587,"style":1004},[1003],[772,15589,15591,15594],{"className":15590},[807],[772,15592],{"className":15593,"style":3543},[811],[772,15595,15597,15600],{"className":15596},[816],[772,15598,2839],{"className":15599,"style":2854},[816,817],[772,15601,15603],{"className":15602},[1752],[772,15604,15606,15627],{"className":15605},[1756,3556],[772,15607,15609,15624],{"className":15608},[1760],[772,15610,15613],{"className":15611,"style":15612},[1764],"height:0.1514em;",[772,15614,15615,15618],{"style":3640},[772,15616],{"className":15617,"style":1773},[1772],[772,15619,15621],{"className":15620},[1777,1778,1779,1780],[772,15622,13833],{"className":15623},[816,817,1780],[772,15625,3580],{"className":15626},[3579],[772,15628,15630],{"className":15629},[1760],[772,15631,15633],{"className":15632,"style":3587},[1764],[772,15634],{},[772,15636,15638,15670],{"className":15637},[775],[772,15639,15641],{"className":15640},[779],[781,15642,15643],{"xmlns":783},[785,15644,15645,15667],{},[788,15646,15647,15659,15661],{},[3525,15648,15649,15651],{},[791,15650,2839],{},[788,15652,15653,15655,15657],{},[791,15654,13833],{},[980,15656,1353],{},[1084,15658,2170],{},[980,15660,982],{},[3525,15662,15663,15665],{},[791,15664,2839],{},[1084,15666,2170],{},[795,15668,15669],{"encoding":797},"N_{n+1} = N_1",[772,15671,15673,15737],{"className":15672,"ariaHidden":803},[802],[772,15674,15676,15679,15728,15731,15734],{"className":15675},[807],[772,15677],{"className":15678,"style":14673},[811],[772,15680,15682,15685],{"className":15681},[816],[772,15683,2839],{"className":15684,"style":2854},[816,817],[772,15686,15688],{"className":15687},[1752],[772,15689,15691,15720],{"className":15690},[1756,3556],[772,15692,15694,15717],{"className":15693},[1760],[772,15695,15697],{"className":15696,"style":3563},[1764],[772,15698,15699,15702],{"style":3640},[772,15700],{"className":15701,"style":1773},[1772],[772,15703,15705],{"className":15704},[1777,1778,1779,1780],[772,15706,15708,15711,15714],{"className":15707},[816,1780],[772,15709,13833],{"className":15710},[816,817,1780],[772,15712,1353],{"className":15713},[1125,1780],[772,15715,2170],{"className":15716},[816,1780],[772,15718,3580],{"className":15719},[3579],[772,15721,15723],{"className":15722},[1760],[772,15724,15726],{"className":15725,"style":14207},[1764],[772,15727],{},[772,15729],{"className":15730,"style":1004},[1003],[772,15732,982],{"className":15733},[1008],[772,15735],{"className":15736,"style":1004},[1003],[772,15738,15740,15743],{"className":15739},[807],[772,15741],{"className":15742,"style":3543},[811],[772,15744,15746,15749],{"className":15745},[816],[772,15747,2839],{"className":15748,"style":2854},[816,817],[772,15750,15752],{"className":15751},[1752],[772,15753,15755,15775],{"className":15754},[1756,3556],[772,15756,15758,15772],{"className":15757},[1760],[772,15759,15761],{"className":15760,"style":3563},[1764],[772,15762,15763,15766],{"style":3640},[772,15764],{"className":15765,"style":1773},[1772],[772,15767,15769],{"className":15768},[1777,1778,1779,1780],[772,15770,2170],{"className":15771},[816,1780],[772,15773,3580],{"className":15774},[3579],[772,15776,15778],{"className":15777},[1760],[772,15779,15781],{"className":15780,"style":3587},[1764],[772,15782],{},[323,15784,15786],{"id":15785},"planimetric-perimeter-and-polyline-closure","Planimetric Perimeter and Polyline Closure",[11,15788,15789,15790,15818,15819,3474,15999,4135],{},"Perimeter (",[772,15791,15793,15806],{"className":15792},[775],[772,15794,15796],{"className":15795},[779],[781,15797,15798],{"xmlns":783},[785,15799,15800,15804],{},[788,15801,15802],{},[791,15803,14943],{},[795,15805,14943],{"encoding":797},[772,15807,15809],{"className":15808,"ariaHidden":803},[802],[772,15810,15812,15815],{"className":15811},[807],[772,15813],{"className":15814,"style":2758},[811],[772,15816,14943],{"className":15817,"style":14964},[816,817],") is the sum of horizontal Euclidean distances between consecutive vertices, including the closing leg from the last vertex back to the first. Segment length between ",[772,15820,15822,15858],{"className":15821},[775],[772,15823,15825],{"className":15824},[779],[781,15826,15827],{"xmlns":783},[785,15828,15829,15855],{},[788,15830,15831,15837,15839,15845,15847,15853],{},[3525,15832,15833,15835],{},[791,15834,14943],{},[791,15836,13887],{},[980,15838,1082],{"stretchy":1081},[3525,15840,15841,15843],{},[791,15842,2808],{},[791,15844,13887],{},[980,15846,5940],{"separator":803},[3525,15848,15849,15851],{},[791,15850,2839],{},[791,15852,13887],{},[980,15854,1096],{"stretchy":1081},[795,15856,15857],{"encoding":797},"P_i(E_i, N_i)",[772,15859,15861],{"className":15860,"ariaHidden":803},[802],[772,15862,15864,15867,15907,15910,15950,15953,15956,15996],{"className":15863},[807],[772,15865],{"className":15866,"style":1109},[811],[772,15868,15870,15873],{"className":15869},[816],[772,15871,14943],{"className":15872,"style":14964},[816,817],[772,15874,15876],{"className":15875},[1752],[772,15877,15879,15899],{"className":15878},[1756,3556],[772,15880,15882,15896],{"className":15881},[1760],[772,15883,15885],{"className":15884,"style":14137},[1764],[772,15886,15887,15890],{"style":14979},[772,15888],{"className":15889,"style":1773},[1772],[772,15891,15893],{"className":15892},[1777,1778,1779,1780],[772,15894,13887],{"className":15895},[816,817,1780],[772,15897,3580],{"className":15898},[3579],[772,15900,15902],{"className":15901},[1760],[772,15903,15905],{"className":15904,"style":3587},[1764],[772,15906],{},[772,15908,1082],{"className":15909},[1113],[772,15911,15913,15916],{"className":15912},[816],[772,15914,2808],{"className":15915,"style":2823},[816,817],[772,15917,15919],{"className":15918},[1752],[772,15920,15922,15942],{"className":15921},[1756,3556],[772,15923,15925,15939],{"className":15924},[1760],[772,15926,15928],{"className":15927,"style":14137},[1764],[772,15929,15930,15933],{"style":3566},[772,15931],{"className":15932,"style":1773},[1772],[772,15934,15936],{"className":15935},[1777,1778,1779,1780],[772,15937,13887],{"className":15938},[816,817,1780],[772,15940,3580],{"className":15941},[3579],[772,15943,15945],{"className":15944},[1760],[772,15946,15948],{"className":15947,"style":3587},[1764],[772,15949],{},[772,15951,5940],{"className":15952},[6041],[772,15954],{"className":15955,"style":3294},[1003],[772,15957,15959,15962],{"className":15958},[816],[772,15960,2839],{"className":15961,"style":2854},[816,817],[772,15963,15965],{"className":15964},[1752],[772,15966,15968,15988],{"className":15967},[1756,3556],[772,15969,15971,15985],{"className":15970},[1760],[772,15972,15974],{"className":15973,"style":14137},[1764],[772,15975,15976,15979],{"style":3640},[772,15977],{"className":15978,"style":1773},[1772],[772,15980,15982],{"className":15981},[1777,1778,1779,1780],[772,15983,13887],{"className":15984},[816,817,1780],[772,15986,3580],{"className":15987},[3579],[772,15989,15991],{"className":15990},[1760],[772,15992,15994],{"className":15993,"style":3587},[1764],[772,15995],{},[772,15997,1096],{"className":15998},[1141],[772,16000,16002,16056],{"className":16001},[775],[772,16003,16005],{"className":16004},[779],[781,16006,16007],{"xmlns":783},[785,16008,16009,16053],{},[788,16010,16011,16023,16025,16037,16039,16051],{},[3525,16012,16013,16015],{},[791,16014,14943],{},[788,16016,16017,16019,16021],{},[791,16018,13887],{},[980,16020,1353],{},[1084,16022,2170],{},[980,16024,1082],{"stretchy":1081},[3525,16026,16027,16029],{},[791,16028,2808],{},[788,16030,16031,16033,16035],{},[791,16032,13887],{},[980,16034,1353],{},[1084,16036,2170],{},[980,16038,5940],{"separator":803},[3525,16040,16041,16043],{},[791,16042,2839],{},[788,16044,16045,16047,16049],{},[791,16046,13887],{},[980,16048,1353],{},[1084,16050,2170],{},[980,16052,1096],{"stretchy":1081},[795,16054,16055],{"encoding":797},"P_{i+1}(E_{i+1}, N_{i+1})",[772,16057,16059],{"className":16058,"ariaHidden":803},[802],[772,16060,16062,16065,16114,16117,16166,16169,16172,16221],{"className":16061},[807],[772,16063],{"className":16064,"style":1109},[811],[772,16066,16068,16071],{"className":16067},[816],[772,16069,14943],{"className":16070,"style":14964},[816,817],[772,16072,16074],{"className":16073},[1752],[772,16075,16077,16106],{"className":16076},[1756,3556],[772,16078,16080,16103],{"className":16079},[1760],[772,16081,16083],{"className":16082,"style":14137},[1764],[772,16084,16085,16088],{"style":14979},[772,16086],{"className":16087,"style":1773},[1772],[772,16089,16091],{"className":16090},[1777,1778,1779,1780],[772,16092,16094,16097,16100],{"className":16093},[816,1780],[772,16095,13887],{"className":16096},[816,817,1780],[772,16098,1353],{"className":16099},[1125,1780],[772,16101,2170],{"className":16102},[816,1780],[772,16104,3580],{"className":16105},[3579],[772,16107,16109],{"className":16108},[1760],[772,16110,16112],{"className":16111,"style":14207},[1764],[772,16113],{},[772,16115,1082],{"className":16116},[1113],[772,16118,16120,16123],{"className":16119},[816],[772,16121,2808],{"className":16122,"style":2823},[816,817],[772,16124,16126],{"className":16125},[1752],[772,16127,16129,16158],{"className":16128},[1756,3556],[772,16130,16132,16155],{"className":16131},[1760],[772,16133,16135],{"className":16134,"style":14137},[1764],[772,16136,16137,16140],{"style":3566},[772,16138],{"className":16139,"style":1773},[1772],[772,16141,16143],{"className":16142},[1777,1778,1779,1780],[772,16144,16146,16149,16152],{"className":16145},[816,1780],[772,16147,13887],{"className":16148},[816,817,1780],[772,16150,1353],{"className":16151},[1125,1780],[772,16153,2170],{"className":16154},[816,1780],[772,16156,3580],{"className":16157},[3579],[772,16159,16161],{"className":16160},[1760],[772,16162,16164],{"className":16163,"style":14207},[1764],[772,16165],{},[772,16167,5940],{"className":16168},[6041],[772,16170],{"className":16171,"style":3294},[1003],[772,16173,16175,16178],{"className":16174},[816],[772,16176,2839],{"className":16177,"style":2854},[816,817],[772,16179,16181],{"className":16180},[1752],[772,16182,16184,16213],{"className":16183},[1756,3556],[772,16185,16187,16210],{"className":16186},[1760],[772,16188,16190],{"className":16189,"style":14137},[1764],[772,16191,16192,16195],{"style":3640},[772,16193],{"className":16194,"style":1773},[1772],[772,16196,16198],{"className":16197},[1777,1778,1779,1780],[772,16199,16201,16204,16207],{"className":16200},[816,1780],[772,16202,13887],{"className":16203},[816,817,1780],[772,16205,1353],{"className":16206},[1125,1780],[772,16208,2170],{"className":16209},[816,1780],[772,16211,3580],{"className":16212},[3579],[772,16214,16216],{"className":16215},[1760],[772,16217,16219],{"className":16218,"style":14207},[1764],[772,16220],{},[772,16222,1096],{"className":16223},[1141],[11,16225,16226],{},[772,16227,16229,16312],{"className":16228},[775],[772,16230,16232],{"className":16231},[779],[781,16233,16234],{"xmlns":783},[785,16235,16236,16309],{},[788,16237,16238,16245,16247],{},[3525,16239,16240,16243],{},[791,16241,16242],{},"d",[791,16244,13887],{},[980,16246,982],{},[5347,16248,16249],{},[788,16250,16251,16253,16265,16267,16273,16279,16281,16283,16295,16297,16303],{},[980,16252,1082],{"stretchy":1081},[3525,16254,16255,16257],{},[791,16256,2808],{},[788,16258,16259,16261,16263],{},[791,16260,13887],{},[980,16262,1353],{},[1084,16264,2170],{},[980,16266,1091],{},[3525,16268,16269,16271],{},[791,16270,2808],{},[791,16272,13887],{},[1636,16274,16275,16277],{},[980,16276,1096],{"stretchy":1081},[1084,16278,1872],{},[980,16280,1353],{},[980,16282,1082],{"stretchy":1081},[3525,16284,16285,16287],{},[791,16286,2839],{},[788,16288,16289,16291,16293],{},[791,16290,13887],{},[980,16292,1353],{},[1084,16294,2170],{},[980,16296,1091],{},[3525,16298,16299,16301],{},[791,16300,2839],{},[791,16302,13887],{},[1636,16304,16305,16307],{},[980,16306,1096],{"stretchy":1081},[1084,16308,1872],{},[795,16310,16311],{"encoding":797},"d_i = \\sqrt{(E_{i+1} - E_i)^2 + (N_{i+1} - N_i)^2}",[772,16313,16315,16371],{"className":16314,"ariaHidden":803},[802],[772,16316,16318,16321,16362,16365,16368],{"className":16317},[807],[772,16319],{"className":16320,"style":5959},[811],[772,16322,16324,16327],{"className":16323},[816],[772,16325,16242],{"className":16326},[816,817],[772,16328,16330],{"className":16329},[1752],[772,16331,16333,16354],{"className":16332},[1756,3556],[772,16334,16336,16351],{"className":16335},[1760],[772,16337,16339],{"className":16338,"style":14137},[1764],[772,16340,16342,16345],{"style":16341},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[772,16343],{"className":16344,"style":1773},[1772],[772,16346,16348],{"className":16347},[1777,1778,1779,1780],[772,16349,13887],{"className":16350},[816,817,1780],[772,16352,3580],{"className":16353},[3579],[772,16355,16357],{"className":16356},[1760],[772,16358,16360],{"className":16359,"style":3587},[1764],[772,16361],{},[772,16363],{"className":16364,"style":1004},[1003],[772,16366,982],{"className":16367},[1008],[772,16369],{"className":16370,"style":1004},[1003],[772,16372,16374,16377],{"className":16373},[807],[772,16375],{"className":16376,"style":9438},[811],[772,16378,16380],{"className":16379},[816,5403],[772,16381,16383,16682],{"className":16382},[1756,3556],[772,16384,16386,16679],{"className":16385},[1760],[772,16387,16389,16667],{"className":16388,"style":9451},[1764],[772,16390,16392,16395],{"className":16391,"style":9455},[5417],[772,16393],{"className":16394,"style":9459},[1772],[772,16396,16398,16401,16450,16453,16456,16459,16499,16528,16531,16534,16537,16540,16589,16592,16595,16598,16638],{"className":16397,"style":9463},[816],[772,16399,1082],{"className":16400},[1113],[772,16402,16404,16407],{"className":16403},[816],[772,16405,2808],{"className":16406,"style":2823},[816,817],[772,16408,16410],{"className":16409},[1752],[772,16411,16413,16442],{"className":16412},[1756,3556],[772,16414,16416,16439],{"className":16415},[1760],[772,16417,16419],{"className":16418,"style":14137},[1764],[772,16420,16421,16424],{"style":3566},[772,16422],{"className":16423,"style":1773},[1772],[772,16425,16427],{"className":16426},[1777,1778,1779,1780],[772,16428,16430,16433,16436],{"className":16429},[816,1780],[772,16431,13887],{"className":16432},[816,817,1780],[772,16434,1353],{"className":16435},[1125,1780],[772,16437,2170],{"className":16438},[816,1780],[772,16440,3580],{"className":16441},[3579],[772,16443,16445],{"className":16444},[1760],[772,16446,16448],{"className":16447,"style":14207},[1764],[772,16449],{},[772,16451],{"className":16452,"style":1121},[1003],[772,16454,1091],{"className":16455},[1125],[772,16457],{"className":16458,"style":1121},[1003],[772,16460,16462,16465],{"className":16461},[816],[772,16463,2808],{"className":16464,"style":2823},[816,817],[772,16466,16468],{"className":16467},[1752],[772,16469,16471,16491],{"className":16470},[1756,3556],[772,16472,16474,16488],{"className":16473},[1760],[772,16475,16477],{"className":16476,"style":14137},[1764],[772,16478,16479,16482],{"style":3566},[772,16480],{"className":16481,"style":1773},[1772],[772,16483,16485],{"className":16484},[1777,1778,1779,1780],[772,16486,13887],{"className":16487},[816,817,1780],[772,16489,3580],{"className":16490},[3579],[772,16492,16494],{"className":16493},[1760],[772,16495,16497],{"className":16496,"style":3587},[1764],[772,16498],{},[772,16500,16502,16505],{"className":16501},[1141],[772,16503,1096],{"className":16504},[1141],[772,16506,16508],{"className":16507},[1752],[772,16509,16511],{"className":16510},[1756],[772,16512,16514],{"className":16513},[1760],[772,16515,16517],{"className":16516,"style":5448},[1764],[772,16518,16519,16522],{"style":5451},[772,16520],{"className":16521,"style":1773},[1772],[772,16523,16525],{"className":16524},[1777,1778,1779,1780],[772,16526,1872],{"className":16527},[816,1780],[772,16529],{"className":16530,"style":1121},[1003],[772,16532,1353],{"className":16533},[1125],[772,16535],{"className":16536,"style":1121},[1003],[772,16538,1082],{"className":16539},[1113],[772,16541,16543,16546],{"className":16542},[816],[772,16544,2839],{"className":16545,"style":2854},[816,817],[772,16547,16549],{"className":16548},[1752],[772,16550,16552,16581],{"className":16551},[1756,3556],[772,16553,16555,16578],{"className":16554},[1760],[772,16556,16558],{"className":16557,"style":14137},[1764],[772,16559,16560,16563],{"style":3640},[772,16561],{"className":16562,"style":1773},[1772],[772,16564,16566],{"className":16565},[1777,1778,1779,1780],[772,16567,16569,16572,16575],{"className":16568},[816,1780],[772,16570,13887],{"className":16571},[816,817,1780],[772,16573,1353],{"className":16574},[1125,1780],[772,16576,2170],{"className":16577},[816,1780],[772,16579,3580],{"className":16580},[3579],[772,16582,16584],{"className":16583},[1760],[772,16585,16587],{"className":16586,"style":14207},[1764],[772,16588],{},[772,16590],{"className":16591,"style":1121},[1003],[772,16593,1091],{"className":16594},[1125],[772,16596],{"className":16597,"style":1121},[1003],[772,16599,16601,16604],{"className":16600},[816],[772,16602,2839],{"className":16603,"style":2854},[816,817],[772,16605,16607],{"className":16606},[1752],[772,16608,16610,16630],{"className":16609},[1756,3556],[772,16611,16613,16627],{"className":16612},[1760],[772,16614,16616],{"className":16615,"style":14137},[1764],[772,16617,16618,16621],{"style":3640},[772,16619],{"className":16620,"style":1773},[1772],[772,16622,16624],{"className":16623},[1777,1778,1779,1780],[772,16625,13887],{"className":16626},[816,817,1780],[772,16628,3580],{"className":16629},[3579],[772,16631,16633],{"className":16632},[1760],[772,16634,16636],{"className":16635,"style":3587},[1764],[772,16637],{},[772,16639,16641,16644],{"className":16640},[1141],[772,16642,1096],{"className":16643},[1141],[772,16645,16647],{"className":16646},[1752],[772,16648,16650],{"className":16649},[1756],[772,16651,16653],{"className":16652},[1760],[772,16654,16656],{"className":16655,"style":5448},[1764],[772,16657,16658,16661],{"style":5451},[772,16659],{"className":16660,"style":1773},[1772],[772,16662,16664],{"className":16663},[1777,1778,1779,1780],[772,16665,1872],{"className":16666},[816,1780],[772,16668,16669,16672],{"style":9548},[772,16670],{"className":16671,"style":9459},[1772],[772,16673,16675],{"className":16674,"style":9555},[5511],[5514,16676,16677],{"xmlns":5516,"width":5517,"height":9558,"viewBox":9559,"preserveAspectRatio":5520},[5522,16678],{"d":9562},[772,16680,3580],{"className":16681},[3579],[772,16683,16685],{"className":16684},[1760],[772,16686,16688],{"className":16687,"style":9572},[1764],[772,16689],{},[11,16691,16692],{},"Total perimeter:",[11,16694,16695],{},[772,16696,16698,16790],{"className":16697},[775],[772,16699,16701],{"className":16700},[779],[781,16702,16703],{"xmlns":783},[785,16704,16705,16787],{},[788,16706,16707,16709,16711,16725],{},[791,16708,14943],{},[980,16710,982],{},[13878,16712,16713,16715,16723],{},[980,16714,13882],{},[788,16716,16717,16719,16721],{},[791,16718,13887],{},[980,16720,982],{},[1084,16722,2170],{},[791,16724,13833],{},[5347,16726,16727],{},[788,16728,16729,16731,16743,16745,16751,16757,16759,16761,16773,16775,16781],{},[980,16730,1082],{"stretchy":1081},[3525,16732,16733,16735],{},[791,16734,2808],{},[788,16736,16737,16739,16741],{},[791,16738,13887],{},[980,16740,1353],{},[1084,16742,2170],{},[980,16744,1091],{},[3525,16746,16747,16749],{},[791,16748,2808],{},[791,16750,13887],{},[1636,16752,16753,16755],{},[980,16754,1096],{"stretchy":1081},[1084,16756,1872],{},[980,16758,1353],{},[980,16760,1082],{"stretchy":1081},[3525,16762,16763,16765],{},[791,16764,2839],{},[788,16766,16767,16769,16771],{},[791,16768,13887],{},[980,16770,1353],{},[1084,16772,2170],{},[980,16774,1091],{},[3525,16776,16777,16779],{},[791,16778,2839],{},[791,16780,13887],{},[1636,16782,16783,16785],{},[980,16784,1096],{"stretchy":1081},[1084,16786,1872],{},[795,16788,16789],{"encoding":797},"P = \\sum_{i=1}^{n} \\sqrt{(E_{i+1} - E_i)^2 + (N_{i+1} - N_i)^2}",[772,16791,16793,16811],{"className":16792,"ariaHidden":803},[802],[772,16794,16796,16799,16802,16805,16808],{"className":16795},[807],[772,16797],{"className":16798,"style":2758},[811],[772,16800,14943],{"className":16801,"style":14964},[816,817],[772,16803],{"className":16804,"style":1004},[1003],[772,16806,982],{"className":16807},[1008],[772,16809],{"className":16810,"style":1004},[1003],[772,16812,16814,16817,16880,16883],{"className":16813},[807],[772,16815],{"className":16816,"style":9438},[811],[772,16818,16820,16823],{"className":16819},[3298],[772,16821,13882],{"className":16822,"style":14054},[3298,14052,14053],[772,16824,16826],{"className":16825},[1752],[772,16827,16829,16872],{"className":16828},[1756,3556],[772,16830,16832,16869],{"className":16831},[1760],[772,16833,16835,16855],{"className":16834,"style":14067},[1764],[772,16836,16837,16840],{"style":14070},[772,16838],{"className":16839,"style":1773},[1772],[772,16841,16843],{"className":16842},[1777,1778,1779,1780],[772,16844,16846,16849,16852],{"className":16845},[816,1780],[772,16847,13887],{"className":16848},[816,817,1780],[772,16850,982],{"className":16851},[1008,1780],[772,16853,2170],{"className":16854},[816,1780],[772,16856,16857,16860],{"style":14091},[772,16858],{"className":16859,"style":1773},[1772],[772,16861,16863],{"className":16862},[1777,1778,1779,1780],[772,16864,16866],{"className":16865},[816,1780],[772,16867,13833],{"className":16868},[816,817,1780],[772,16870,3580],{"className":16871},[3579],[772,16873,16875],{"className":16874},[1760],[772,16876,16878],{"className":16877,"style":14113},[1764],[772,16879],{},[772,16881],{"className":16882,"style":3294},[1003],[772,16884,16886],{"className":16885},[816,5403],[772,16887,16889,17188],{"className":16888},[1756,3556],[772,16890,16892,17185],{"className":16891},[1760],[772,16893,16895,17173],{"className":16894,"style":9451},[1764],[772,16896,16898,16901],{"className":16897,"style":9455},[5417],[772,16899],{"className":16900,"style":9459},[1772],[772,16902,16904,16907,16956,16959,16962,16965,17005,17034,17037,17040,17043,17046,17095,17098,17101,17104,17144],{"className":16903,"style":9463},[816],[772,16905,1082],{"className":16906},[1113],[772,16908,16910,16913],{"className":16909},[816],[772,16911,2808],{"className":16912,"style":2823},[816,817],[772,16914,16916],{"className":16915},[1752],[772,16917,16919,16948],{"className":16918},[1756,3556],[772,16920,16922,16945],{"className":16921},[1760],[772,16923,16925],{"className":16924,"style":14137},[1764],[772,16926,16927,16930],{"style":3566},[772,16928],{"className":16929,"style":1773},[1772],[772,16931,16933],{"className":16932},[1777,1778,1779,1780],[772,16934,16936,16939,16942],{"className":16935},[816,1780],[772,16937,13887],{"className":16938},[816,817,1780],[772,16940,1353],{"className":16941},[1125,1780],[772,16943,2170],{"className":16944},[816,1780],[772,16946,3580],{"className":16947},[3579],[772,16949,16951],{"className":16950},[1760],[772,16952,16954],{"className":16953,"style":14207},[1764],[772,16955],{},[772,16957],{"className":16958,"style":1121},[1003],[772,16960,1091],{"className":16961},[1125],[772,16963],{"className":16964,"style":1121},[1003],[772,16966,16968,16971],{"className":16967},[816],[772,16969,2808],{"className":16970,"style":2823},[816,817],[772,16972,16974],{"className":16973},[1752],[772,16975,16977,16997],{"className":16976},[1756,3556],[772,16978,16980,16994],{"className":16979},[1760],[772,16981,16983],{"className":16982,"style":14137},[1764],[772,16984,16985,16988],{"style":3566},[772,16986],{"className":16987,"style":1773},[1772],[772,16989,16991],{"className":16990},[1777,1778,1779,1780],[772,16992,13887],{"className":16993},[816,817,1780],[772,16995,3580],{"className":16996},[3579],[772,16998,17000],{"className":16999},[1760],[772,17001,17003],{"className":17002,"style":3587},[1764],[772,17004],{},[772,17006,17008,17011],{"className":17007},[1141],[772,17009,1096],{"className":17010},[1141],[772,17012,17014],{"className":17013},[1752],[772,17015,17017],{"className":17016},[1756],[772,17018,17020],{"className":17019},[1760],[772,17021,17023],{"className":17022,"style":5448},[1764],[772,17024,17025,17028],{"style":5451},[772,17026],{"className":17027,"style":1773},[1772],[772,17029,17031],{"className":17030},[1777,1778,1779,1780],[772,17032,1872],{"className":17033},[816,1780],[772,17035],{"className":17036,"style":1121},[1003],[772,17038,1353],{"className":17039},[1125],[772,17041],{"className":17042,"style":1121},[1003],[772,17044,1082],{"className":17045},[1113],[772,17047,17049,17052],{"className":17048},[816],[772,17050,2839],{"className":17051,"style":2854},[816,817],[772,17053,17055],{"className":17054},[1752],[772,17056,17058,17087],{"className":17057},[1756,3556],[772,17059,17061,17084],{"className":17060},[1760],[772,17062,17064],{"className":17063,"style":14137},[1764],[772,17065,17066,17069],{"style":3640},[772,17067],{"className":17068,"style":1773},[1772],[772,17070,17072],{"className":17071},[1777,1778,1779,1780],[772,17073,17075,17078,17081],{"className":17074},[816,1780],[772,17076,13887],{"className":17077},[816,817,1780],[772,17079,1353],{"className":17080},[1125,1780],[772,17082,2170],{"className":17083},[816,1780],[772,17085,3580],{"className":17086},[3579],[772,17088,17090],{"className":17089},[1760],[772,17091,17093],{"className":17092,"style":14207},[1764],[772,17094],{},[772,17096],{"className":17097,"style":1121},[1003],[772,17099,1091],{"className":17100},[1125],[772,17102],{"className":17103,"style":1121},[1003],[772,17105,17107,17110],{"className":17106},[816],[772,17108,2839],{"className":17109,"style":2854},[816,817],[772,17111,17113],{"className":17112},[1752],[772,17114,17116,17136],{"className":17115},[1756,3556],[772,17117,17119,17133],{"className":17118},[1760],[772,17120,17122],{"className":17121,"style":14137},[1764],[772,17123,17124,17127],{"style":3640},[772,17125],{"className":17126,"style":1773},[1772],[772,17128,17130],{"className":17129},[1777,1778,1779,1780],[772,17131,13887],{"className":17132},[816,817,1780],[772,17134,3580],{"className":17135},[3579],[772,17137,17139],{"className":17138},[1760],[772,17140,17142],{"className":17141,"style":3587},[1764],[772,17143],{},[772,17145,17147,17150],{"className":17146},[1141],[772,17148,1096],{"className":17149},[1141],[772,17151,17153],{"className":17152},[1752],[772,17154,17156],{"className":17155},[1756],[772,17157,17159],{"className":17158},[1760],[772,17160,17162],{"className":17161,"style":5448},[1764],[772,17163,17164,17167],{"style":5451},[772,17165],{"className":17166,"style":1773},[1772],[772,17168,17170],{"className":17169},[1777,1778,1779,1780],[772,17171,1872],{"className":17172},[816,1780],[772,17174,17175,17178],{"style":9548},[772,17176],{"className":17177,"style":9459},[1772],[772,17179,17181],{"className":17180,"style":9555},[5511],[5514,17182,17183],{"xmlns":5516,"width":5517,"height":9558,"viewBox":9559,"preserveAspectRatio":5520},[5522,17184],{"d":9562},[772,17186,3580],{"className":17187},[3579],[772,17189,17191],{"className":17190},[1760],[772,17192,17194],{"className":17193,"style":9572},[1764],[772,17195],{},[11,17197,15500,17198,5868],{},[772,17199,17201,17233],{"className":17200},[775],[772,17202,17204],{"className":17203},[779],[781,17205,17206],{"xmlns":783},[785,17207,17208,17230],{},[788,17209,17210,17222,17224],{},[3525,17211,17212,17214],{},[791,17213,14943],{},[788,17215,17216,17218,17220],{},[791,17217,13833],{},[980,17219,1353],{},[1084,17221,2170],{},[980,17223,982],{},[3525,17225,17226,17228],{},[791,17227,14943],{},[1084,17229,2170],{},[795,17231,17232],{"encoding":797},"P_{n+1} = P_1",[772,17234,17236,17300],{"className":17235,"ariaHidden":803},[802],[772,17237,17239,17242,17291,17294,17297],{"className":17238},[807],[772,17240],{"className":17241,"style":14673},[811],[772,17243,17245,17248],{"className":17244},[816],[772,17246,14943],{"className":17247,"style":14964},[816,817],[772,17249,17251],{"className":17250},[1752],[772,17252,17254,17283],{"className":17253},[1756,3556],[772,17255,17257,17280],{"className":17256},[1760],[772,17258,17260],{"className":17259,"style":3563},[1764],[772,17261,17262,17265],{"style":14979},[772,17263],{"className":17264,"style":1773},[1772],[772,17266,17268],{"className":17267},[1777,1778,1779,1780],[772,17269,17271,17274,17277],{"className":17270},[816,1780],[772,17272,13833],{"className":17273},[816,817,1780],[772,17275,1353],{"className":17276},[1125,1780],[772,17278,2170],{"className":17279},[816,1780],[772,17281,3580],{"className":17282},[3579],[772,17284,17286],{"className":17285},[1760],[772,17287,17289],{"className":17288,"style":14207},[1764],[772,17290],{},[772,17292],{"className":17293,"style":1004},[1003],[772,17295,982],{"className":17296},[1008],[772,17298],{"className":17299,"style":1004},[1003],[772,17301,17303,17306],{"className":17302},[807],[772,17304],{"className":17305,"style":3543},[811],[772,17307,17309,17312],{"className":17308},[816],[772,17310,14943],{"className":17311,"style":14964},[816,817],[772,17313,17315],{"className":17314},[1752],[772,17316,17318,17338],{"className":17317},[1756,3556],[772,17319,17321,17335],{"className":17320},[1760],[772,17322,17324],{"className":17323,"style":3563},[1764],[772,17325,17326,17329],{"style":14979},[772,17327],{"className":17328,"style":1773},[1772],[772,17330,17332],{"className":17331},[1777,1778,1779,1780],[772,17333,2170],{"className":17334},[816,1780],[772,17336,3580],{"className":17337},[3579],[772,17339,17341],{"className":17340},[1760],[772,17342,17344],{"className":17343,"style":3587},[1764],[772,17345],{},[11,17347,17348,17349,17377,17378,17512,17513,17564,17565,17593],{},"A polygon must be topologically closed for a valid area. Some datasets store ",[772,17350,17352,17365],{"className":17351},[775],[772,17353,17355],{"className":17354},[779],[781,17356,17357],{"xmlns":783},[785,17358,17359,17363],{},[788,17360,17361],{},[791,17362,13833],{},[795,17364,13833],{"encoding":797},[772,17366,17368],{"className":17367,"ariaHidden":803},[802],[772,17369,17371,17374],{"className":17370},[807],[772,17372],{"className":17373,"style":845},[811],[772,17375,13833],{"className":17376},[816,817]," unique vertices and close implicitly (",[772,17379,17381,17408],{"className":17380},[775],[772,17382,17384],{"className":17383},[779],[781,17385,17386],{"xmlns":783},[785,17387,17388,17405],{},[788,17389,17390,17396,17399],{},[3525,17391,17392,17394],{},[791,17393,14943],{},[791,17395,13833],{},[980,17397,17398],{},"→",[3525,17400,17401,17403],{},[791,17402,14943],{},[1084,17404,2170],{},[795,17406,17407],{"encoding":797},"P_n \\to P_1",[772,17409,17411,17466],{"className":17410,"ariaHidden":803},[802],[772,17412,17414,17417,17457,17460,17463],{"className":17413},[807],[772,17415],{"className":17416,"style":3543},[811],[772,17418,17420,17423],{"className":17419},[816],[772,17421,14943],{"className":17422,"style":14964},[816,817],[772,17424,17426],{"className":17425},[1752],[772,17427,17429,17449],{"className":17428},[1756,3556],[772,17430,17432,17446],{"className":17431},[1760],[772,17433,17435],{"className":17434,"style":15612},[1764],[772,17436,17437,17440],{"style":14979},[772,17438],{"className":17439,"style":1773},[1772],[772,17441,17443],{"className":17442},[1777,1778,1779,1780],[772,17444,13833],{"className":17445},[816,817,1780],[772,17447,3580],{"className":17448},[3579],[772,17450,17452],{"className":17451},[1760],[772,17453,17455],{"className":17454,"style":3587},[1764],[772,17456],{},[772,17458],{"className":17459,"style":1004},[1003],[772,17461,17398],{"className":17462},[1008],[772,17464],{"className":17465,"style":1004},[1003],[772,17467,17469,17472],{"className":17468},[807],[772,17470],{"className":17471,"style":3543},[811],[772,17473,17475,17478],{"className":17474},[816],[772,17476,14943],{"className":17477,"style":14964},[816,817],[772,17479,17481],{"className":17480},[1752],[772,17482,17484,17504],{"className":17483},[1756,3556],[772,17485,17487,17501],{"className":17486},[1760],[772,17488,17490],{"className":17489,"style":3563},[1764],[772,17491,17492,17495],{"style":14979},[772,17493],{"className":17494,"style":1773},[1772],[772,17496,17498],{"className":17497},[1777,1778,1779,1780],[772,17499,2170],{"className":17500},[816,1780],[772,17502,3580],{"className":17503},[3579],[772,17505,17507],{"className":17506},[1760],[772,17508,17510],{"className":17509,"style":3587},[1764],[772,17511],{},"); others list ",[772,17514,17516,17534],{"className":17515},[775],[772,17517,17519],{"className":17518},[779],[781,17520,17521],{"xmlns":783},[785,17522,17523,17531],{},[788,17524,17525,17527,17529],{},[791,17526,13833],{},[980,17528,1353],{},[1084,17530,2170],{},[795,17532,17533],{"encoding":797},"n+1",[772,17535,17537,17555],{"className":17536,"ariaHidden":803},[802],[772,17538,17540,17543,17546,17549,17552],{"className":17539},[807],[772,17541],{"className":17542,"style":6118},[811],[772,17544,13833],{"className":17545},[816,817],[772,17547],{"className":17548,"style":1121},[1003],[772,17550,1353],{"className":17551},[1125],[772,17553],{"className":17554,"style":1121},[1003],[772,17556,17558,17561],{"className":17557},[807],[772,17559],{"className":17560,"style":5899},[811],[772,17562,2170],{"className":17563},[816]," points with the first and last identical. Surveying Core closes implicitly: enter ",[772,17566,17568,17581],{"className":17567},[775],[772,17569,17571],{"className":17570},[779],[781,17572,17573],{"xmlns":783},[785,17574,17575,17579],{},[788,17576,17577],{},[791,17578,13833],{},[795,17580,13833],{"encoding":797},[772,17582,17584],{"className":17583,"ariaHidden":803},[802],[772,17585,17587,17590],{"className":17586},[807],[772,17588],{"className":17589,"style":845},[811],[772,17591,13833],{"className":17592},[816,817]," vertices in boundary order and the calculator adds the final edge automatically.",[323,17595,17597],{"id":17596},"tabular-worked-computational-example","Tabular Worked Computational Example",[11,17599,17600],{},"Consider a pentagonal parcel on a regional grid near 10,000 m:",[335,17602,17603,17801,17997,18191,18387],{},[338,17604,17605,17675,17676,17739,17740,17800],{},[772,17606,17608,17626],{"className":17607},[775],[772,17609,17611],{"className":17610},[779],[781,17612,17613],{"xmlns":783},[785,17614,17615,17623],{},[788,17616,17617],{},[3525,17618,17619,17621],{},[791,17620,14943],{},[1084,17622,2170],{},[795,17624,17625],{"encoding":797},"P_1",[772,17627,17629],{"className":17628,"ariaHidden":803},[802],[772,17630,17632,17635],{"className":17631},[807],[772,17633],{"className":17634,"style":3543},[811],[772,17636,17638,17641],{"className":17637},[816],[772,17639,14943],{"className":17640,"style":14964},[816,817],[772,17642,17644],{"className":17643},[1752],[772,17645,17647,17667],{"className":17646},[1756,3556],[772,17648,17650,17664],{"className":17649},[1760],[772,17651,17653],{"className":17652,"style":3563},[1764],[772,17654,17655,17658],{"style":14979},[772,17656],{"className":17657,"style":1773},[1772],[772,17659,17661],{"className":17660},[1777,1778,1779,1780],[772,17662,2170],{"className":17663},[816,1780],[772,17665,3580],{"className":17666},[3579],[772,17668,17670],{"className":17669},[1760],[772,17671,17673],{"className":17672,"style":3587},[1764],[772,17674],{},": (",[772,17677,17679,17698],{"className":17678},[775],[772,17680,17682],{"className":17681},[779],[781,17683,17684],{"xmlns":783},[785,17685,17686,17695],{},[788,17687,17688,17690,17692],{},[791,17689,2808],{},[980,17691,982],{},[1084,17693,17694],{},"10,000.000",[795,17696,17697],{"encoding":797},"E = 10{,}000.000",[772,17699,17701,17719],{"className":17700,"ariaHidden":803},[802],[772,17702,17704,17707,17710,17713,17716],{"className":17703},[807],[772,17705],{"className":17706,"style":2758},[811],[772,17708,2808],{"className":17709,"style":2823},[816,817],[772,17711],{"className":17712,"style":1004},[1003],[772,17714,982],{"className":17715},[1008],[772,17717],{"className":17718,"style":1004},[1003],[772,17720,17722,17725,17729,17735],{"className":17721},[807],[772,17723],{"className":17724,"style":11620},[811],[772,17726,17728],{"className":17727},[816],"10",[772,17730,17732],{"className":17731},[816],[772,17733,5940],{"className":17734},[6041],[772,17736,17738],{"className":17737},[816],"000.000"," m, ",[772,17741,17743,17761],{"className":17742},[775],[772,17744,17746],{"className":17745},[779],[781,17747,17748],{"xmlns":783},[785,17749,17750,17758],{},[788,17751,17752,17754,17756],{},[791,17753,2839],{},[980,17755,982],{},[1084,17757,17694],{},[795,17759,17760],{"encoding":797},"N = 10{,}000.000",[772,17762,17764,17782],{"className":17763,"ariaHidden":803},[802],[772,17765,17767,17770,17773,17776,17779],{"className":17766},[807],[772,17768],{"className":17769,"style":2758},[811],[772,17771,2839],{"className":17772,"style":2854},[816,817],[772,17774],{"className":17775,"style":1004},[1003],[772,17777,982],{"className":17778},[1008],[772,17780],{"className":17781,"style":1004},[1003],[772,17783,17785,17788,17791,17797],{"className":17784},[807],[772,17786],{"className":17787,"style":11620},[811],[772,17789,17728],{"className":17790},[816],[772,17792,17794],{"className":17793},[816],[772,17795,5940],{"className":17796},[6041],[772,17798,17738],{"className":17799},[816]," m)",[338,17802,17803,17675,17873,17739,17935,17800],{},[772,17804,17806,17824],{"className":17805},[775],[772,17807,17809],{"className":17808},[779],[781,17810,17811],{"xmlns":783},[785,17812,17813,17821],{},[788,17814,17815],{},[3525,17816,17817,17819],{},[791,17818,14943],{},[1084,17820,1872],{},[795,17822,17823],{"encoding":797},"P_2",[772,17825,17827],{"className":17826,"ariaHidden":803},[802],[772,17828,17830,17833],{"className":17829},[807],[772,17831],{"className":17832,"style":3543},[811],[772,17834,17836,17839],{"className":17835},[816],[772,17837,14943],{"className":17838,"style":14964},[816,817],[772,17840,17842],{"className":17841},[1752],[772,17843,17845,17865],{"className":17844},[1756,3556],[772,17846,17848,17862],{"className":17847},[1760],[772,17849,17851],{"className":17850,"style":3563},[1764],[772,17852,17853,17856],{"style":14979},[772,17854],{"className":17855,"style":1773},[1772],[772,17857,17859],{"className":17858},[1777,1778,1779,1780],[772,17860,1872],{"className":17861},[816,1780],[772,17863,3580],{"className":17864},[3579],[772,17866,17868],{"className":17867},[1760],[772,17869,17871],{"className":17870,"style":3587},[1764],[772,17872],{},[772,17874,17876,17895],{"className":17875},[775],[772,17877,17879],{"className":17878},[779],[781,17880,17881],{"xmlns":783},[785,17882,17883,17892],{},[788,17884,17885,17887,17889],{},[791,17886,2808],{},[980,17888,982],{},[1084,17890,17891],{},"10,030.000",[795,17893,17894],{"encoding":797},"E = 10{,}030.000",[772,17896,17898,17916],{"className":17897,"ariaHidden":803},[802],[772,17899,17901,17904,17907,17910,17913],{"className":17900},[807],[772,17902],{"className":17903,"style":2758},[811],[772,17905,2808],{"className":17906,"style":2823},[816,817],[772,17908],{"className":17909,"style":1004},[1003],[772,17911,982],{"className":17912},[1008],[772,17914],{"className":17915,"style":1004},[1003],[772,17917,17919,17922,17925,17931],{"className":17918},[807],[772,17920],{"className":17921,"style":11620},[811],[772,17923,17728],{"className":17924},[816],[772,17926,17928],{"className":17927},[816],[772,17929,5940],{"className":17930},[6041],[772,17932,17934],{"className":17933},[816],"030.000",[772,17936,17938,17957],{"className":17937},[775],[772,17939,17941],{"className":17940},[779],[781,17942,17943],{"xmlns":783},[785,17944,17945,17954],{},[788,17946,17947,17949,17951],{},[791,17948,2839],{},[980,17950,982],{},[1084,17952,17953],{},"10,010.000",[795,17955,17956],{"encoding":797},"N = 10{,}010.000",[772,17958,17960,17978],{"className":17959,"ariaHidden":803},[802],[772,17961,17963,17966,17969,17972,17975],{"className":17962},[807],[772,17964],{"className":17965,"style":2758},[811],[772,17967,2839],{"className":17968,"style":2854},[816,817],[772,17970],{"className":17971,"style":1004},[1003],[772,17973,982],{"className":17974},[1008],[772,17976],{"className":17977,"style":1004},[1003],[772,17979,17981,17984,17987,17993],{"className":17980},[807],[772,17982],{"className":17983,"style":11620},[811],[772,17985,17728],{"className":17986},[816],[772,17988,17990],{"className":17989},[816],[772,17991,5940],{"className":17992},[6041],[772,17994,17996],{"className":17995},[816],"010.000",[338,17998,17999,17675,18069,17739,18131,17800],{},[772,18000,18002,18020],{"className":18001},[775],[772,18003,18005],{"className":18004},[779],[781,18006,18007],{"xmlns":783},[785,18008,18009,18017],{},[788,18010,18011],{},[3525,18012,18013,18015],{},[791,18014,14943],{},[1084,18016,1845],{},[795,18018,18019],{"encoding":797},"P_3",[772,18021,18023],{"className":18022,"ariaHidden":803},[802],[772,18024,18026,18029],{"className":18025},[807],[772,18027],{"className":18028,"style":3543},[811],[772,18030,18032,18035],{"className":18031},[816],[772,18033,14943],{"className":18034,"style":14964},[816,817],[772,18036,18038],{"className":18037},[1752],[772,18039,18041,18061],{"className":18040},[1756,3556],[772,18042,18044,18058],{"className":18043},[1760],[772,18045,18047],{"className":18046,"style":3563},[1764],[772,18048,18049,18052],{"style":14979},[772,18050],{"className":18051,"style":1773},[1772],[772,18053,18055],{"className":18054},[1777,1778,1779,1780],[772,18056,1845],{"className":18057},[816,1780],[772,18059,3580],{"className":18060},[3579],[772,18062,18064],{"className":18063},[1760],[772,18065,18067],{"className":18066,"style":3587},[1764],[772,18068],{},[772,18070,18072,18091],{"className":18071},[775],[772,18073,18075],{"className":18074},[779],[781,18076,18077],{"xmlns":783},[785,18078,18079,18088],{},[788,18080,18081,18083,18085],{},[791,18082,2808],{},[980,18084,982],{},[1084,18086,18087],{},"10,040.000",[795,18089,18090],{"encoding":797},"E = 10{,}040.000",[772,18092,18094,18112],{"className":18093,"ariaHidden":803},[802],[772,18095,18097,18100,18103,18106,18109],{"className":18096},[807],[772,18098],{"className":18099,"style":2758},[811],[772,18101,2808],{"className":18102,"style":2823},[816,817],[772,18104],{"className":18105,"style":1004},[1003],[772,18107,982],{"className":18108},[1008],[772,18110],{"className":18111,"style":1004},[1003],[772,18113,18115,18118,18121,18127],{"className":18114},[807],[772,18116],{"className":18117,"style":11620},[811],[772,18119,17728],{"className":18120},[816],[772,18122,18124],{"className":18123},[816],[772,18125,5940],{"className":18126},[6041],[772,18128,18130],{"className":18129},[816],"040.000",[772,18132,18134,18152],{"className":18133},[775],[772,18135,18137],{"className":18136},[779],[781,18138,18139],{"xmlns":783},[785,18140,18141,18149],{},[788,18142,18143,18145,18147],{},[791,18144,2839],{},[980,18146,982],{},[1084,18148,18087],{},[795,18150,18151],{"encoding":797},"N = 10{,}040.000",[772,18153,18155,18173],{"className":18154,"ariaHidden":803},[802],[772,18156,18158,18161,18164,18167,18170],{"className":18157},[807],[772,18159],{"className":18160,"style":2758},[811],[772,18162,2839],{"className":18163,"style":2854},[816,817],[772,18165],{"className":18166,"style":1004},[1003],[772,18168,982],{"className":18169},[1008],[772,18171],{"className":18172,"style":1004},[1003],[772,18174,18176,18179,18182,18188],{"className":18175},[807],[772,18177],{"className":18178,"style":11620},[811],[772,18180,17728],{"className":18181},[816],[772,18183,18185],{"className":18184},[816],[772,18186,5940],{"className":18187},[6041],[772,18189,18130],{"className":18190},[816],[338,18192,18193,17675,18263,17739,18325,17800],{},[772,18194,18196,18214],{"className":18195},[775],[772,18197,18199],{"className":18198},[779],[781,18200,18201],{"xmlns":783},[785,18202,18203,18211],{},[788,18204,18205],{},[3525,18206,18207,18209],{},[791,18208,14943],{},[1084,18210,1977],{},[795,18212,18213],{"encoding":797},"P_4",[772,18215,18217],{"className":18216,"ariaHidden":803},[802],[772,18218,18220,18223],{"className":18219},[807],[772,18221],{"className":18222,"style":3543},[811],[772,18224,18226,18229],{"className":18225},[816],[772,18227,14943],{"className":18228,"style":14964},[816,817],[772,18230,18232],{"className":18231},[1752],[772,18233,18235,18255],{"className":18234},[1756,3556],[772,18236,18238,18252],{"className":18237},[1760],[772,18239,18241],{"className":18240,"style":3563},[1764],[772,18242,18243,18246],{"style":14979},[772,18244],{"className":18245,"style":1773},[1772],[772,18247,18249],{"className":18248},[1777,1778,1779,1780],[772,18250,1977],{"className":18251},[816,1780],[772,18253,3580],{"className":18254},[3579],[772,18256,18258],{"className":18257},[1760],[772,18259,18261],{"className":18260,"style":3587},[1764],[772,18262],{},[772,18264,18266,18285],{"className":18265},[775],[772,18267,18269],{"className":18268},[779],[781,18270,18271],{"xmlns":783},[785,18272,18273,18282],{},[788,18274,18275,18277,18279],{},[791,18276,2808],{},[980,18278,982],{},[1084,18280,18281],{},"10,015.000",[795,18283,18284],{"encoding":797},"E = 10{,}015.000",[772,18286,18288,18306],{"className":18287,"ariaHidden":803},[802],[772,18289,18291,18294,18297,18300,18303],{"className":18290},[807],[772,18292],{"className":18293,"style":2758},[811],[772,18295,2808],{"className":18296,"style":2823},[816,817],[772,18298],{"className":18299,"style":1004},[1003],[772,18301,982],{"className":18302},[1008],[772,18304],{"className":18305,"style":1004},[1003],[772,18307,18309,18312,18315,18321],{"className":18308},[807],[772,18310],{"className":18311,"style":11620},[811],[772,18313,17728],{"className":18314},[816],[772,18316,18318],{"className":18317},[816],[772,18319,5940],{"className":18320},[6041],[772,18322,18324],{"className":18323},[816],"015.000",[772,18326,18328,18347],{"className":18327},[775],[772,18329,18331],{"className":18330},[779],[781,18332,18333],{"xmlns":783},[785,18334,18335,18344],{},[788,18336,18337,18339,18341],{},[791,18338,2839],{},[980,18340,982],{},[1084,18342,18343],{},"10,050.000",[795,18345,18346],{"encoding":797},"N = 10{,}050.000",[772,18348,18350,18368],{"className":18349,"ariaHidden":803},[802],[772,18351,18353,18356,18359,18362,18365],{"className":18352},[807],[772,18354],{"className":18355,"style":2758},[811],[772,18357,2839],{"className":18358,"style":2854},[816,817],[772,18360],{"className":18361,"style":1004},[1003],[772,18363,982],{"className":18364},[1008],[772,18366],{"className":18367,"style":1004},[1003],[772,18369,18371,18374,18377,18383],{"className":18370},[807],[772,18372],{"className":18373,"style":11620},[811],[772,18375,17728],{"className":18376},[816],[772,18378,18380],{"className":18379},[816],[772,18381,5940],{"className":18382},[6041],[772,18384,18386],{"className":18385},[816],"050.000",[338,18388,18389,17675,18459,17739,18518,17800],{},[772,18390,18392,18410],{"className":18391},[775],[772,18393,18395],{"className":18394},[779],[781,18396,18397],{"xmlns":783},[785,18398,18399,18407],{},[788,18400,18401],{},[3525,18402,18403,18405],{},[791,18404,14943],{},[1084,18406,2177],{},[795,18408,18409],{"encoding":797},"P_5",[772,18411,18413],{"className":18412,"ariaHidden":803},[802],[772,18414,18416,18419],{"className":18415},[807],[772,18417],{"className":18418,"style":3543},[811],[772,18420,18422,18425],{"className":18421},[816],[772,18423,14943],{"className":18424,"style":14964},[816,817],[772,18426,18428],{"className":18427},[1752],[772,18429,18431,18451],{"className":18430},[1756,3556],[772,18432,18434,18448],{"className":18433},[1760],[772,18435,18437],{"className":18436,"style":3563},[1764],[772,18438,18439,18442],{"style":14979},[772,18440],{"className":18441,"style":1773},[1772],[772,18443,18445],{"className":18444},[1777,1778,1779,1780],[772,18446,2177],{"className":18447},[816,1780],[772,18449,3580],{"className":18450},[3579],[772,18452,18454],{"className":18453},[1760],[772,18455,18457],{"className":18456,"style":3587},[1764],[772,18458],{},[772,18460,18462,18479],{"className":18461},[775],[772,18463,18465],{"className":18464},[779],[781,18466,18467],{"xmlns":783},[785,18468,18469,18477],{},[788,18470,18471,18473,18475],{},[791,18472,2808],{},[980,18474,982],{},[1084,18476,17694],{},[795,18478,17697],{"encoding":797},[772,18480,18482,18500],{"className":18481,"ariaHidden":803},[802],[772,18483,18485,18488,18491,18494,18497],{"className":18484},[807],[772,18486],{"className":18487,"style":2758},[811],[772,18489,2808],{"className":18490,"style":2823},[816,817],[772,18492],{"className":18493,"style":1004},[1003],[772,18495,982],{"className":18496},[1008],[772,18498],{"className":18499,"style":1004},[1003],[772,18501,18503,18506,18509,18515],{"className":18502},[807],[772,18504],{"className":18505,"style":11620},[811],[772,18507,17728],{"className":18508},[816],[772,18510,18512],{"className":18511},[816],[772,18513,5940],{"className":18514},[6041],[772,18516,17738],{"className":18517},[816],[772,18519,18521,18539],{"className":18520},[775],[772,18522,18524],{"className":18523},[779],[781,18525,18526],{"xmlns":783},[785,18527,18528,18536],{},[788,18529,18530,18532,18534],{},[791,18531,2839],{},[980,18533,982],{},[1084,18535,17891],{},[795,18537,18538],{"encoding":797},"N = 10{,}030.000",[772,18540,18542,18560],{"className":18541,"ariaHidden":803},[802],[772,18543,18545,18548,18551,18554,18557],{"className":18544},[807],[772,18546],{"className":18547,"style":2758},[811],[772,18549,2839],{"className":18550,"style":2854},[816,817],[772,18552],{"className":18553,"style":1004},[1003],[772,18555,982],{"className":18556},[1008],[772,18558],{"className":18559,"style":1004},[1003],[772,18561,18563,18566,18569,18575],{"className":18562},[807],[772,18564],{"className":18565,"style":11620},[811],[772,18567,17728],{"className":18568},[816],[772,18570,18572],{"className":18571},[816],[772,18573,5940],{"className":18574},[6041],[772,18576,17934],{"className":18577},[816],[11,18579,18580,18581,3474,18699,18814,18815,4135],{},"Subtract ",[772,18582,18584,18609],{"className":18583},[775],[772,18585,18587],{"className":18586},[779],[781,18588,18589],{"xmlns":783},[785,18590,18591,18606],{},[788,18592,18593,18602,18604],{},[3525,18594,18595,18597,18600],{},[791,18596,2808],{},[791,18598,18599],{},"min",[980,18601,3252],{},[980,18603,982],{},[1084,18605,17694],{},[795,18607,18608],{"encoding":797},"E_{\\min} = 10{,}000.000",[772,18610,18612,18681],{"className":18611,"ariaHidden":803},[802],[772,18613,18615,18618,18672,18675,18678],{"className":18614},[807],[772,18616],{"className":18617,"style":3543},[811],[772,18619,18621,18624],{"className":18620},[816],[772,18622,2808],{"className":18623,"style":2823},[816,817],[772,18625,18627],{"className":18626},[1752],[772,18628,18630,18664],{"className":18629},[1756,3556],[772,18631,18633,18661],{"className":18632},[1760],[772,18634,18637],{"className":18635,"style":18636},[1764],"height:0.3175em;",[772,18638,18639,18642],{"style":3566},[772,18640],{"className":18641,"style":1773},[1772],[772,18643,18645],{"className":18644},[1777,1778,1779,1780],[772,18646,18648],{"className":18647},[816,1780],[772,18649,18651,18655,18658],{"className":18650},[3298,1780],[772,18652,18654],{"className":18653},[1780],"m",[772,18656,13887],{"className":18657},[1780],[772,18659,13833],{"className":18660},[1780],[772,18662,3580],{"className":18663},[3579],[772,18665,18667],{"className":18666},[1760],[772,18668,18670],{"className":18669,"style":3587},[1764],[772,18671],{},[772,18673],{"className":18674,"style":1004},[1003],[772,18676,982],{"className":18677},[1008],[772,18679],{"className":18680,"style":1004},[1003],[772,18682,18684,18687,18690,18696],{"className":18683},[807],[772,18685],{"className":18686,"style":11620},[811],[772,18688,17728],{"className":18689},[816],[772,18691,18693],{"className":18692},[816],[772,18694,5940],{"className":18695},[6041],[772,18697,17738],{"className":18698},[816],[772,18700,18702,18726],{"className":18701},[775],[772,18703,18705],{"className":18704},[779],[781,18706,18707],{"xmlns":783},[785,18708,18709,18723],{},[788,18710,18711,18719,18721],{},[3525,18712,18713,18715,18717],{},[791,18714,2839],{},[791,18716,18599],{},[980,18718,3252],{},[980,18720,982],{},[1084,18722,17694],{},[795,18724,18725],{"encoding":797},"N_{\\min} = 10{,}000.000",[772,18727,18729,18796],{"className":18728,"ariaHidden":803},[802],[772,18730,18732,18735,18787,18790,18793],{"className":18731},[807],[772,18733],{"className":18734,"style":3543},[811],[772,18736,18738,18741],{"className":18737},[816],[772,18739,2839],{"className":18740,"style":2854},[816,817],[772,18742,18744],{"className":18743},[1752],[772,18745,18747,18779],{"className":18746},[1756,3556],[772,18748,18750,18776],{"className":18749},[1760],[772,18751,18753],{"className":18752,"style":18636},[1764],[772,18754,18755,18758],{"style":3640},[772,18756],{"className":18757,"style":1773},[1772],[772,18759,18761],{"className":18760},[1777,1778,1779,1780],[772,18762,18764],{"className":18763},[816,1780],[772,18765,18767,18770,18773],{"className":18766},[3298,1780],[772,18768,18654],{"className":18769},[1780],[772,18771,13887],{"className":18772},[1780],[772,18774,13833],{"className":18775},[1780],[772,18777,3580],{"className":18778},[3579],[772,18780,18782],{"className":18781},[1760],[772,18783,18785],{"className":18784,"style":3587},[1764],[772,18786],{},[772,18788],{"className":18789,"style":1004},[1003],[772,18791,982],{"className":18792},[1008],[772,18794],{"className":18795,"style":1004},[1003],[772,18797,18799,18802,18805,18811],{"className":18798},[807],[772,18800],{"className":18801,"style":11620},[811],[772,18803,17728],{"className":18804},[816],[772,18806,18808],{"className":18807},[816],[772,18809,5940],{"className":18810},[6041],[772,18812,17738],{"className":18813},[816]," for local coordinates ",[772,18816,18818,18852],{"className":18817},[775],[772,18819,18821],{"className":18820},[779],[781,18822,18823],{"xmlns":783},[785,18824,18825,18849],{},[788,18826,18827,18829,18837,18839,18847],{},[980,18828,1082],{"stretchy":1081},[13878,18830,18831,18833,18835],{},[791,18832,2808],{},[791,18834,13887],{},[980,18836,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,18838,5940],{"separator":803},[13878,18840,18841,18843,18845],{},[791,18842,2839],{},[791,18844,13887],{},[980,18846,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,18848,1096],{"stretchy":1081},[795,18850,18851],{"encoding":797},"(E'_i, N'_i)",[772,18853,18855],{"className":18854,"ariaHidden":803},[802],[772,18856,18858,18862,18865,18921,18924,18927,18982],{"className":18857},[807],[772,18859],{"className":18860,"style":18861},[811],"height:1.0106em;vertical-align:-0.2587em;",[772,18863,1082],{"className":18864},[1113],[772,18866,18868,18871],{"className":18867},[816],[772,18869,2808],{"className":18870,"style":2823},[816,817],[772,18872,18874],{"className":18873},[1752],[772,18875,18877,18912],{"className":18876},[1756,3556],[772,18878,18880,18909],{"className":18879},[1760],[772,18881,18883,18895],{"className":18882,"style":1805},[1764],[772,18884,18886,18889],{"style":18885},"top:-2.4413em;margin-left:-0.0576em;margin-right:0.05em;",[772,18887],{"className":18888,"style":1773},[1772],[772,18890,18892],{"className":18891},[1777,1778,1779,1780],[772,18893,13887],{"className":18894},[816,817,1780],[772,18896,18897,18900],{"style":1768},[772,18898],{"className":18899,"style":1773},[1772],[772,18901,18903],{"className":18902},[1777,1778,1779,1780],[772,18904,18906],{"className":18905},[816,1780],[772,18907,1651],{"className":18908},[816,1780],[772,18910,3580],{"className":18911},[3579],[772,18913,18915],{"className":18914},[1760],[772,18916,18919],{"className":18917,"style":18918},[1764],"height:0.2587em;",[772,18920],{},[772,18922,5940],{"className":18923},[6041],[772,18925],{"className":18926,"style":3294},[1003],[772,18928,18930,18933],{"className":18929},[816],[772,18931,2839],{"className":18932,"style":2854},[816,817],[772,18934,18936],{"className":18935},[1752],[772,18937,18939,18974],{"className":18938},[1756,3556],[772,18940,18942,18971],{"className":18941},[1760],[772,18943,18945,18957],{"className":18944,"style":1805},[1764],[772,18946,18948,18951],{"style":18947},"top:-2.4413em;margin-left:-0.109em;margin-right:0.05em;",[772,18949],{"className":18950,"style":1773},[1772],[772,18952,18954],{"className":18953},[1777,1778,1779,1780],[772,18955,13887],{"className":18956},[816,817,1780],[772,18958,18959,18962],{"style":1768},[772,18960],{"className":18961,"style":1773},[1772],[772,18963,18965],{"className":18964},[1777,1778,1779,1780],[772,18966,18968],{"className":18967},[816,1780],[772,18969,1651],{"className":18970},[816,1780],[772,18972,3580],{"className":18973},[3579],[772,18975,18977],{"className":18976},[1760],[772,18978,18980],{"className":18979,"style":18918},[1764],[772,18981],{},[772,18983,1096],{"className":18984},[1141],[335,18986,18987,19079,19168,19257,19346],{},[338,18988,18989,19078],{},[772,18990,18992,19012],{"className":18991},[775],[772,18993,18995],{"className":18994},[779],[781,18996,18997],{"xmlns":783},[785,18998,18999,19009],{},[788,19000,19001],{},[13878,19002,19003,19005,19007],{},[791,19004,14943],{},[1084,19006,2170],{},[980,19008,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19010,19011],{"encoding":797},"P'_1",[772,19013,19015],{"className":19014,"ariaHidden":803},[802],[772,19016,19018,19022],{"className":19017},[807],[772,19019],{"className":19020,"style":19021},[811],"height:1em;vertical-align:-0.2481em;",[772,19023,19025,19028],{"className":19024},[816],[772,19026,14943],{"className":19027,"style":14964},[816,817],[772,19029,19031],{"className":19030},[1752],[772,19032,19034,19069],{"className":19033},[1756,3556],[772,19035,19037,19066],{"className":19036},[1760],[772,19038,19040,19052],{"className":19039,"style":1805},[1764],[772,19041,19043,19046],{"style":19042},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[772,19044],{"className":19045,"style":1773},[1772],[772,19047,19049],{"className":19048},[1777,1778,1779,1780],[772,19050,2170],{"className":19051},[816,1780],[772,19053,19054,19057],{"style":1768},[772,19055],{"className":19056,"style":1773},[1772],[772,19058,19060],{"className":19059},[1777,1778,1779,1780],[772,19061,19063],{"className":19062},[816,1780],[772,19064,1651],{"className":19065},[816,1780],[772,19067,3580],{"className":19068},[3579],[772,19070,19072],{"className":19071},[1760],[772,19073,19076],{"className":19074,"style":19075},[1764],"height:0.2481em;",[772,19077],{},": (0.000, 0.000)",[338,19080,19081,19167],{},[772,19082,19084,19104],{"className":19083},[775],[772,19085,19087],{"className":19086},[779],[781,19088,19089],{"xmlns":783},[785,19090,19091,19101],{},[788,19092,19093],{},[13878,19094,19095,19097,19099],{},[791,19096,14943],{},[1084,19098,1872],{},[980,19100,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19102,19103],{"encoding":797},"P'_2",[772,19105,19107],{"className":19106,"ariaHidden":803},[802],[772,19108,19110,19113],{"className":19109},[807],[772,19111],{"className":19112,"style":19021},[811],[772,19114,19116,19119],{"className":19115},[816],[772,19117,14943],{"className":19118,"style":14964},[816,817],[772,19120,19122],{"className":19121},[1752],[772,19123,19125,19159],{"className":19124},[1756,3556],[772,19126,19128,19156],{"className":19127},[1760],[772,19129,19131,19142],{"className":19130,"style":1805},[1764],[772,19132,19133,19136],{"style":19042},[772,19134],{"className":19135,"style":1773},[1772],[772,19137,19139],{"className":19138},[1777,1778,1779,1780],[772,19140,1872],{"className":19141},[816,1780],[772,19143,19144,19147],{"style":1768},[772,19145],{"className":19146,"style":1773},[1772],[772,19148,19150],{"className":19149},[1777,1778,1779,1780],[772,19151,19153],{"className":19152},[816,1780],[772,19154,1651],{"className":19155},[816,1780],[772,19157,3580],{"className":19158},[3579],[772,19160,19162],{"className":19161},[1760],[772,19163,19165],{"className":19164,"style":19075},[1764],[772,19166],{},": (30.000, 10.000)",[338,19169,19170,19256],{},[772,19171,19173,19193],{"className":19172},[775],[772,19174,19176],{"className":19175},[779],[781,19177,19178],{"xmlns":783},[785,19179,19180,19190],{},[788,19181,19182],{},[13878,19183,19184,19186,19188],{},[791,19185,14943],{},[1084,19187,1845],{},[980,19189,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19191,19192],{"encoding":797},"P'_3",[772,19194,19196],{"className":19195,"ariaHidden":803},[802],[772,19197,19199,19202],{"className":19198},[807],[772,19200],{"className":19201,"style":19021},[811],[772,19203,19205,19208],{"className":19204},[816],[772,19206,14943],{"className":19207,"style":14964},[816,817],[772,19209,19211],{"className":19210},[1752],[772,19212,19214,19248],{"className":19213},[1756,3556],[772,19215,19217,19245],{"className":19216},[1760],[772,19218,19220,19231],{"className":19219,"style":1805},[1764],[772,19221,19222,19225],{"style":19042},[772,19223],{"className":19224,"style":1773},[1772],[772,19226,19228],{"className":19227},[1777,1778,1779,1780],[772,19229,1845],{"className":19230},[816,1780],[772,19232,19233,19236],{"style":1768},[772,19234],{"className":19235,"style":1773},[1772],[772,19237,19239],{"className":19238},[1777,1778,1779,1780],[772,19240,19242],{"className":19241},[816,1780],[772,19243,1651],{"className":19244},[816,1780],[772,19246,3580],{"className":19247},[3579],[772,19249,19251],{"className":19250},[1760],[772,19252,19254],{"className":19253,"style":19075},[1764],[772,19255],{},": (40.000, 40.000)",[338,19258,19259,19345],{},[772,19260,19262,19282],{"className":19261},[775],[772,19263,19265],{"className":19264},[779],[781,19266,19267],{"xmlns":783},[785,19268,19269,19279],{},[788,19270,19271],{},[13878,19272,19273,19275,19277],{},[791,19274,14943],{},[1084,19276,1977],{},[980,19278,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19280,19281],{"encoding":797},"P'_4",[772,19283,19285],{"className":19284,"ariaHidden":803},[802],[772,19286,19288,19291],{"className":19287},[807],[772,19289],{"className":19290,"style":19021},[811],[772,19292,19294,19297],{"className":19293},[816],[772,19295,14943],{"className":19296,"style":14964},[816,817],[772,19298,19300],{"className":19299},[1752],[772,19301,19303,19337],{"className":19302},[1756,3556],[772,19304,19306,19334],{"className":19305},[1760],[772,19307,19309,19320],{"className":19308,"style":1805},[1764],[772,19310,19311,19314],{"style":19042},[772,19312],{"className":19313,"style":1773},[1772],[772,19315,19317],{"className":19316},[1777,1778,1779,1780],[772,19318,1977],{"className":19319},[816,1780],[772,19321,19322,19325],{"style":1768},[772,19323],{"className":19324,"style":1773},[1772],[772,19326,19328],{"className":19327},[1777,1778,1779,1780],[772,19329,19331],{"className":19330},[816,1780],[772,19332,1651],{"className":19333},[816,1780],[772,19335,3580],{"className":19336},[3579],[772,19338,19340],{"className":19339},[1760],[772,19341,19343],{"className":19342,"style":19075},[1764],[772,19344],{},": (15.000, 50.000)",[338,19347,19348,19434],{},[772,19349,19351,19371],{"className":19350},[775],[772,19352,19354],{"className":19353},[779],[781,19355,19356],{"xmlns":783},[785,19357,19358,19368],{},[788,19359,19360],{},[13878,19361,19362,19364,19366],{},[791,19363,14943],{},[1084,19365,2177],{},[980,19367,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19369,19370],{"encoding":797},"P'_5",[772,19372,19374],{"className":19373,"ariaHidden":803},[802],[772,19375,19377,19380],{"className":19376},[807],[772,19378],{"className":19379,"style":19021},[811],[772,19381,19383,19386],{"className":19382},[816],[772,19384,14943],{"className":19385,"style":14964},[816,817],[772,19387,19389],{"className":19388},[1752],[772,19390,19392,19426],{"className":19391},[1756,3556],[772,19393,19395,19423],{"className":19394},[1760],[772,19396,19398,19409],{"className":19397,"style":1805},[1764],[772,19399,19400,19403],{"style":19042},[772,19401],{"className":19402,"style":1773},[1772],[772,19404,19406],{"className":19405},[1777,1778,1779,1780],[772,19407,2177],{"className":19408},[816,1780],[772,19410,19411,19414],{"style":1768},[772,19412],{"className":19413,"style":1773},[1772],[772,19415,19417],{"className":19416},[1777,1778,1779,1780],[772,19418,19420],{"className":19419},[816,1780],[772,19421,1651],{"className":19422},[816,1780],[772,19424,3580],{"className":19425},[3579],[772,19427,19429],{"className":19428},[1760],[772,19430,19432],{"className":19431,"style":19075},[1764],[772,19433],{},": (0.000, 30.000)",[851,19436,19437,20233],{},[854,19438,19439],{},[857,19440,19441,19444,19533,19621,19726,19829,19995,20160],{},[860,19442,19443],{},"Vertex",[860,19445,19446,19532],{},[772,19447,19449,19469],{"className":19448},[775],[772,19450,19452],{"className":19451},[779],[781,19453,19454],{"xmlns":783},[785,19455,19456,19466],{},[788,19457,19458],{},[13878,19459,19460,19462,19464],{},[791,19461,2808],{},[791,19463,13887],{},[980,19465,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19467,19468],{"encoding":797},"E'_i",[772,19470,19472],{"className":19471,"ariaHidden":803},[802],[772,19473,19475,19478],{"className":19474},[807],[772,19476],{"className":19477,"style":18861},[811],[772,19479,19481,19484],{"className":19480},[816],[772,19482,2808],{"className":19483,"style":2823},[816,817],[772,19485,19487],{"className":19486},[1752],[772,19488,19490,19524],{"className":19489},[1756,3556],[772,19491,19493,19521],{"className":19492},[1760],[772,19494,19496,19507],{"className":19495,"style":1805},[1764],[772,19497,19498,19501],{"style":18885},[772,19499],{"className":19500,"style":1773},[1772],[772,19502,19504],{"className":19503},[1777,1778,1779,1780],[772,19505,13887],{"className":19506},[816,817,1780],[772,19508,19509,19512],{"style":1768},[772,19510],{"className":19511,"style":1773},[1772],[772,19513,19515],{"className":19514},[1777,1778,1779,1780],[772,19516,19518],{"className":19517},[816,1780],[772,19519,1651],{"className":19520},[816,1780],[772,19522,3580],{"className":19523},[3579],[772,19525,19527],{"className":19526},[1760],[772,19528,19530],{"className":19529,"style":18918},[1764],[772,19531],{}," (m)",[860,19534,19535,19532],{},[772,19536,19538,19558],{"className":19537},[775],[772,19539,19541],{"className":19540},[779],[781,19542,19543],{"xmlns":783},[785,19544,19545,19555],{},[788,19546,19547],{},[13878,19548,19549,19551,19553],{},[791,19550,2839],{},[791,19552,13887],{},[980,19554,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19556,19557],{"encoding":797},"N'_i",[772,19559,19561],{"className":19560,"ariaHidden":803},[802],[772,19562,19564,19567],{"className":19563},[807],[772,19565],{"className":19566,"style":18861},[811],[772,19568,19570,19573],{"className":19569},[816],[772,19571,2839],{"className":19572,"style":2854},[816,817],[772,19574,19576],{"className":19575},[1752],[772,19577,19579,19613],{"className":19578},[1756,3556],[772,19580,19582,19610],{"className":19581},[1760],[772,19583,19585,19596],{"className":19584,"style":1805},[1764],[772,19586,19587,19590],{"style":18947},[772,19588],{"className":19589,"style":1773},[1772],[772,19591,19593],{"className":19592},[1777,1778,1779,1780],[772,19594,13887],{"className":19595},[816,817,1780],[772,19597,19598,19601],{"style":1768},[772,19599],{"className":19600,"style":1773},[1772],[772,19602,19604],{"className":19603},[1777,1778,1779,1780],[772,19605,19607],{"className":19606},[816,1780],[772,19608,1651],{"className":19609},[816,1780],[772,19611,3580],{"className":19612},[3579],[772,19614,19616],{"className":19615},[1760],[772,19617,19619],{"className":19618,"style":18918},[1764],[772,19620],{},[860,19622,19623,19532],{},[772,19624,19626,19652],{"className":19625},[775],[772,19627,19629],{"className":19628},[779],[781,19630,19631],{"xmlns":783},[785,19632,19633,19649],{},[788,19634,19635],{},[13878,19636,19637,19639,19647],{},[791,19638,2808],{},[788,19640,19641,19643,19645],{},[791,19642,13887],{},[980,19644,1353],{},[1084,19646,2170],{},[980,19648,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19650,19651],{"encoding":797},"E'_{i+1}",[772,19653,19655],{"className":19654,"ariaHidden":803},[802],[772,19656,19658,19662],{"className":19657},[807],[772,19659],{"className":19660,"style":19661},[811],"height:1.0689em;vertical-align:-0.317em;",[772,19663,19665,19668],{"className":19664},[816],[772,19666,2808],{"className":19667,"style":2823},[816,817],[772,19669,19671],{"className":19670},[1752],[772,19672,19674,19717],{"className":19673},[1756,3556],[772,19675,19677,19714],{"className":19676},[1760],[772,19678,19680,19700],{"className":19679,"style":1805},[1764],[772,19681,19682,19685],{"style":18885},[772,19683],{"className":19684,"style":1773},[1772],[772,19686,19688],{"className":19687},[1777,1778,1779,1780],[772,19689,19691,19694,19697],{"className":19690},[816,1780],[772,19692,13887],{"className":19693},[816,817,1780],[772,19695,1353],{"className":19696},[1125,1780],[772,19698,2170],{"className":19699},[816,1780],[772,19701,19702,19705],{"style":1768},[772,19703],{"className":19704,"style":1773},[1772],[772,19706,19708],{"className":19707},[1777,1778,1779,1780],[772,19709,19711],{"className":19710},[816,1780],[772,19712,1651],{"className":19713},[816,1780],[772,19715,3580],{"className":19716},[3579],[772,19718,19720],{"className":19719},[1760],[772,19721,19724],{"className":19722,"style":19723},[1764],"height:0.317em;",[772,19725],{},[860,19727,19728,19532],{},[772,19729,19731,19757],{"className":19730},[775],[772,19732,19734],{"className":19733},[779],[781,19735,19736],{"xmlns":783},[785,19737,19738,19754],{},[788,19739,19740],{},[13878,19741,19742,19744,19752],{},[791,19743,2839],{},[788,19745,19746,19748,19750],{},[791,19747,13887],{},[980,19749,1353],{},[1084,19751,2170],{},[980,19753,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19755,19756],{"encoding":797},"N'_{i+1}",[772,19758,19760],{"className":19759,"ariaHidden":803},[802],[772,19761,19763,19766],{"className":19762},[807],[772,19764],{"className":19765,"style":19661},[811],[772,19767,19769,19772],{"className":19768},[816],[772,19770,2839],{"className":19771,"style":2854},[816,817],[772,19773,19775],{"className":19774},[1752],[772,19776,19778,19821],{"className":19777},[1756,3556],[772,19779,19781,19818],{"className":19780},[1760],[772,19782,19784,19804],{"className":19783,"style":1805},[1764],[772,19785,19786,19789],{"style":18947},[772,19787],{"className":19788,"style":1773},[1772],[772,19790,19792],{"className":19791},[1777,1778,1779,1780],[772,19793,19795,19798,19801],{"className":19794},[816,1780],[772,19796,13887],{"className":19797},[816,817,1780],[772,19799,1353],{"className":19800},[1125,1780],[772,19802,2170],{"className":19803},[816,1780],[772,19805,19806,19809],{"style":1768},[772,19807],{"className":19808,"style":1773},[1772],[772,19810,19812],{"className":19811},[1777,1778,1779,1780],[772,19813,19815],{"className":19814},[816,1780],[772,19816,1651],{"className":19817},[816,1780],[772,19819,3580],{"className":19820},[3579],[772,19822,19824],{"className":19823},[1760],[772,19825,19827],{"className":19826,"style":19723},[1764],[772,19828],{},[860,19830,19831,19994],{},[772,19832,19834,19868],{"className":19833},[775],[772,19835,19837],{"className":19836},[779],[781,19838,19839],{"xmlns":783},[785,19840,19841,19865],{},[788,19842,19843,19851],{},[13878,19844,19845,19847,19849],{},[791,19846,2808],{},[791,19848,13887],{},[980,19850,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[13878,19852,19853,19855,19863],{},[791,19854,2839],{},[788,19856,19857,19859,19861],{},[791,19858,13887],{},[980,19860,1353],{},[1084,19862,2170],{},[980,19864,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,19866,19867],{"encoding":797},"E'_i N'_{i+1}",[772,19869,19871],{"className":19870,"ariaHidden":803},[802],[772,19872,19874,19877,19931],{"className":19873},[807],[772,19875],{"className":19876,"style":19661},[811],[772,19878,19880,19883],{"className":19879},[816],[772,19881,2808],{"className":19882,"style":2823},[816,817],[772,19884,19886],{"className":19885},[1752],[772,19887,19889,19923],{"className":19888},[1756,3556],[772,19890,19892,19920],{"className":19891},[1760],[772,19893,19895,19906],{"className":19894,"style":1805},[1764],[772,19896,19897,19900],{"style":18885},[772,19898],{"className":19899,"style":1773},[1772],[772,19901,19903],{"className":19902},[1777,1778,1779,1780],[772,19904,13887],{"className":19905},[816,817,1780],[772,19907,19908,19911],{"style":1768},[772,19909],{"className":19910,"style":1773},[1772],[772,19912,19914],{"className":19913},[1777,1778,1779,1780],[772,19915,19917],{"className":19916},[816,1780],[772,19918,1651],{"className":19919},[816,1780],[772,19921,3580],{"className":19922},[3579],[772,19924,19926],{"className":19925},[1760],[772,19927,19929],{"className":19928,"style":18918},[1764],[772,19930],{},[772,19932,19934,19937],{"className":19933},[816],[772,19935,2839],{"className":19936,"style":2854},[816,817],[772,19938,19940],{"className":19939},[1752],[772,19941,19943,19986],{"className":19942},[1756,3556],[772,19944,19946,19983],{"className":19945},[1760],[772,19947,19949,19969],{"className":19948,"style":1805},[1764],[772,19950,19951,19954],{"style":18947},[772,19952],{"className":19953,"style":1773},[1772],[772,19955,19957],{"className":19956},[1777,1778,1779,1780],[772,19958,19960,19963,19966],{"className":19959},[816,1780],[772,19961,13887],{"className":19962},[816,817,1780],[772,19964,1353],{"className":19965},[1125,1780],[772,19967,2170],{"className":19968},[816,1780],[772,19970,19971,19974],{"style":1768},[772,19972],{"className":19973,"style":1773},[1772],[772,19975,19977],{"className":19976},[1777,1778,1779,1780],[772,19978,19980],{"className":19979},[816,1780],[772,19981,1651],{"className":19982},[816,1780],[772,19984,3580],{"className":19985},[3579],[772,19987,19989],{"className":19988},[1760],[772,19990,19992],{"className":19991,"style":19723},[1764],[772,19993],{}," (m²)",[860,19996,19997,19994],{},[772,19998,20000,20034],{"className":19999},[775],[772,20001,20003],{"className":20002},[779],[781,20004,20005],{"xmlns":783},[785,20006,20007,20031],{},[788,20008,20009,20017],{},[13878,20010,20011,20013,20015],{},[791,20012,2839],{},[791,20014,13887],{},[980,20016,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[13878,20018,20019,20021,20029],{},[791,20020,2808],{},[788,20022,20023,20025,20027],{},[791,20024,13887],{},[980,20026,1353],{},[1084,20028,2170],{},[980,20030,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[795,20032,20033],{"encoding":797},"N'_i E'_{i+1}",[772,20035,20037],{"className":20036,"ariaHidden":803},[802],[772,20038,20040,20043,20097],{"className":20039},[807],[772,20041],{"className":20042,"style":19661},[811],[772,20044,20046,20049],{"className":20045},[816],[772,20047,2839],{"className":20048,"style":2854},[816,817],[772,20050,20052],{"className":20051},[1752],[772,20053,20055,20089],{"className":20054},[1756,3556],[772,20056,20058,20086],{"className":20057},[1760],[772,20059,20061,20072],{"className":20060,"style":1805},[1764],[772,20062,20063,20066],{"style":18947},[772,20064],{"className":20065,"style":1773},[1772],[772,20067,20069],{"className":20068},[1777,1778,1779,1780],[772,20070,13887],{"className":20071},[816,817,1780],[772,20073,20074,20077],{"style":1768},[772,20075],{"className":20076,"style":1773},[1772],[772,20078,20080],{"className":20079},[1777,1778,1779,1780],[772,20081,20083],{"className":20082},[816,1780],[772,20084,1651],{"className":20085},[816,1780],[772,20087,3580],{"className":20088},[3579],[772,20090,20092],{"className":20091},[1760],[772,20093,20095],{"className":20094,"style":18918},[1764],[772,20096],{},[772,20098,20100,20103],{"className":20099},[816],[772,20101,2808],{"className":20102,"style":2823},[816,817],[772,20104,20106],{"className":20105},[1752],[772,20107,20109,20152],{"className":20108},[1756,3556],[772,20110,20112,20149],{"className":20111},[1760],[772,20113,20115,20135],{"className":20114,"style":1805},[1764],[772,20116,20117,20120],{"style":18885},[772,20118],{"className":20119,"style":1773},[1772],[772,20121,20123],{"className":20122},[1777,1778,1779,1780],[772,20124,20126,20129,20132],{"className":20125},[816,1780],[772,20127,13887],{"className":20128},[816,817,1780],[772,20130,1353],{"className":20131},[1125,1780],[772,20133,2170],{"className":20134},[816,1780],[772,20136,20137,20140],{"style":1768},[772,20138],{"className":20139,"style":1773},[1772],[772,20141,20143],{"className":20142},[1777,1778,1779,1780],[772,20144,20146],{"className":20145},[816,1780],[772,20147,1651],{"className":20148},[816,1780],[772,20150,3580],{"className":20151},[3579],[772,20153,20155],{"className":20154},[1760],[772,20156,20158],{"className":20157,"style":19723},[1764],[772,20159],{},[860,20161,20162,20163,19532],{},"Edge ",[772,20164,20166,20184],{"className":20165},[775],[772,20167,20169],{"className":20168},[779],[781,20170,20171],{"xmlns":783},[785,20172,20173,20181],{},[788,20174,20175],{},[3525,20176,20177,20179],{},[791,20178,16242],{},[791,20180,13887],{},[795,20182,20183],{"encoding":797},"d_i",[772,20185,20187],{"className":20186,"ariaHidden":803},[802],[772,20188,20190,20193],{"className":20189},[807],[772,20191],{"className":20192,"style":5959},[811],[772,20194,20196,20199],{"className":20195},[816],[772,20197,16242],{"className":20198},[816,817],[772,20200,20202],{"className":20201},[1752],[772,20203,20205,20225],{"className":20204},[1756,3556],[772,20206,20208,20222],{"className":20207},[1760],[772,20209,20211],{"className":20210,"style":14137},[1764],[772,20212,20213,20216],{"style":16341},[772,20214],{"className":20215,"style":1773},[1772],[772,20217,20219],{"className":20218},[1777,1778,1779,1780],[772,20220,13887],{"className":20221},[816,817,1780],[772,20223,3580],{"className":20224},[3579],[772,20226,20228],{"className":20227},[1760],[772,20229,20231],{"className":20230,"style":3587},[1764],[772,20232],{},[876,20234,20235,20257,20278,20301,20321,20339],{},[857,20236,20237,20239,20242,20244,20247,20250,20252,20254],{},[881,20238,2170],{},[881,20240,20241],{},"0.000",[881,20243,20241],{},[881,20245,20246],{},"30.000",[881,20248,20249],{},"10.000",[881,20251,20241],{},[881,20253,20241],{},[881,20255,20256],{},"31.623",[857,20258,20259,20261,20263,20265,20268,20270,20273,20276],{},[881,20260,1872],{},[881,20262,20246],{},[881,20264,20249],{},[881,20266,20267],{},"40.000",[881,20269,20267],{},[881,20271,20272],{},"1200.000",[881,20274,20275],{},"400.000",[881,20277,20256],{},[857,20279,20280,20282,20284,20286,20289,20292,20295,20298],{},[881,20281,1845],{},[881,20283,20267],{},[881,20285,20267],{},[881,20287,20288],{},"15.000",[881,20290,20291],{},"50.000",[881,20293,20294],{},"2000.000",[881,20296,20297],{},"600.000",[881,20299,20300],{},"26.926",[857,20302,20303,20305,20307,20309,20311,20313,20316,20318],{},[881,20304,1977],{},[881,20306,20288],{},[881,20308,20291],{},[881,20310,20241],{},[881,20312,20246],{},[881,20314,20315],{},"450.000",[881,20317,20241],{},[881,20319,20320],{},"25.000",[857,20322,20323,20325,20327,20329,20331,20333,20335,20337],{},[881,20324,2177],{},[881,20326,20241],{},[881,20328,20246],{},[881,20330,20241],{},[881,20332,20241],{},[881,20334,20241],{},[881,20336,20241],{},[881,20338,20246],{},[857,20340,20341,20346,20349,20351,20353,20355,20360,20365],{},[881,20342,20343],{},[318,20344,20345],{},"Sum",[881,20347,20348],{},"—",[881,20350,20348],{},[881,20352,20348],{},[881,20354,20348],{},[881,20356,20357],{},[318,20358,20359],{},"3650.000",[881,20361,20362],{},[318,20363,20364],{},"1000.000",[881,20366,20367],{},[318,20368,20369],{},"145.172",[11,20371,20372],{},"Plan area:",[11,20374,20375],{},[772,20376,20378,20464],{"className":20377},[775],[772,20379,20381],{"className":20380},[779],[781,20382,20383],{"xmlns":783},[785,20384,20385,20461],{},[788,20386,20387,20389,20391,20397],{},[791,20388,13735],{},[980,20390,982],{},[5599,20392,20393,20395],{},[1084,20394,2170],{},[1084,20396,1872],{},[788,20398,20399,20401,20403,20405,20413,20427,20429,20431,20433,20435,20443,20457,20459],{},[980,20400,5605],{"fence":803},[980,20402,13882],{},[980,20404,1082],{"stretchy":1081},[13878,20406,20407,20409,20411],{},[791,20408,2808],{},[791,20410,13887],{},[980,20412,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[13878,20414,20415,20417,20425],{},[791,20416,2839],{},[788,20418,20419,20421,20423],{},[791,20420,13887],{},[980,20422,1353],{},[1084,20424,2170],{},[980,20426,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,20428,1096],{"stretchy":1081},[980,20430,1091],{},[980,20432,13882],{},[980,20434,1082],{"stretchy":1081},[13878,20436,20437,20439,20441],{},[791,20438,2839],{},[791,20440,13887],{},[980,20442,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[13878,20444,20445,20447,20455],{},[791,20446,2808],{},[788,20448,20449,20451,20453],{},[791,20450,13887],{},[980,20452,1353],{},[1084,20454,2170],{},[980,20456,1651],{"mathvariant":906,"lspace":1650,"rspace":1650},[980,20458,1096],{"stretchy":1081},[980,20460,5605],{"fence":803},[795,20462,20463],{"encoding":797},"A = \\frac{1}{2} \\left| \\sum (E'_i N'_{i+1}) - \\sum (N'_i E'_{i+1}) \\right|",[772,20465,20467,20485],{"className":20466,"ariaHidden":803},[802],[772,20468,20470,20473,20476,20479,20482],{"className":20469},[807],[772,20471],{"className":20472,"style":2758},[811],[772,20474,13735],{"className":20475},[816,817],[772,20477],{"className":20478,"style":1004},[1003],[772,20480,982],{"className":20481},[1008],[772,20483],{"className":20484,"style":1004},[1003],[772,20486,20488,20491,20559,20562],{"className":20487},[807],[772,20489],{"className":20490,"style":6529},[811],[772,20492,20494,20497,20556],{"className":20493},[816],[772,20495],{"className":20496},[1113,5681],[772,20498,20500],{"className":20499},[5599],[772,20501,20503,20548],{"className":20502},[1756,3556],[772,20504,20506,20545],{"className":20505},[1760],[772,20507,20509,20523,20531],{"className":20508,"style":6558},[1764],[772,20510,20511,20514],{"style":5697},[772,20512],{"className":20513,"style":5422},[1772],[772,20515,20517],{"className":20516},[1777,1778,1779,1780],[772,20518,20520],{"className":20519},[816,1780],[772,20521,1872],{"className":20522},[816,1780],[772,20524,20525,20528],{"style":5719},[772,20526],{"className":20527,"style":5422},[1772],[772,20529],{"className":20530,"style":5727},[5726],[772,20532,20533,20536],{"style":6583},[772,20534],{"className":20535,"style":5422},[1772],[772,20537,20539],{"className":20538},[1777,1778,1779,1780],[772,20540,20542],{"className":20541},[816,1780],[772,20543,2170],{"className":20544},[816,1780],[772,20546,3580],{"className":20547},[3579],[772,20549,20551],{"className":20550},[1760],[772,20552,20554],{"className":20553,"style":6605},[1764],[772,20555],{},[772,20557],{"className":20558},[1141,5681],[772,20560],{"className":20561,"style":3294},[1003],[772,20563,20565,20611,20614,20617,20671,20734,20737,20740,20743,20746,20749,20752,20806,20869,20872],{"className":20564},[5664],[772,20566,20568],{"className":20567},[1113],[772,20569,20572],{"className":20570},[5673,20571],"mult",[772,20573,20575,20602],{"className":20574},[1756,3556],[772,20576,20578,20599],{"className":20577},[1760],[772,20579,20582],{"className":20580,"style":20581},[1764],"height:0.85em;",[772,20583,20585,20588],{"style":20584},"top:-2.85em;",[772,20586],{"className":20587,"style":9459},[1772],[772,20589,20591],{"style":20590},"width:0.333em;height:1.2em;",[5514,20592,20596],{"xmlns":5516,"width":20593,"height":20594,"viewBox":20595},"0.333em","1.2em","0 0 333 1200",[5522,20597],{"d":20598},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z",[772,20600,3580],{"className":20601},[3579],[772,20603,20605],{"className":20604},[1760],[772,20606,20609],{"className":20607,"style":20608},[1764],"height:0.35em;",[772,20610],{},[772,20612,13882],{"className":20613,"style":14054},[3298,14052,14053],[772,20615,1082],{"className":20616},[1113],[772,20618,20620,20623],{"className":20619},[816],[772,20621,2808],{"className":20622,"style":2823},[816,817],[772,20624,20626],{"className":20625},[1752],[772,20627,20629,20663],{"className":20628},[1756,3556],[772,20630,20632,20660],{"className":20631},[1760],[772,20633,20635,20646],{"className":20634,"style":1805},[1764],[772,20636,20637,20640],{"style":18885},[772,20638],{"className":20639,"style":1773},[1772],[772,20641,20643],{"className":20642},[1777,1778,1779,1780],[772,20644,13887],{"className":20645},[816,817,1780],[772,20647,20648,20651],{"style":1768},[772,20649],{"className":20650,"style":1773},[1772],[772,20652,20654],{"className":20653},[1777,1778,1779,1780],[772,20655,20657],{"className":20656},[816,1780],[772,20658,1651],{"className":20659},[816,1780],[772,20661,3580],{"className":20662},[3579],[772,20664,20666],{"className":20665},[1760],[772,20667,20669],{"className":20668,"style":18918},[1764],[772,20670],{},[772,20672,20674,20677],{"className":20673},[816],[772,20675,2839],{"className":20676,"style":2854},[816,817],[772,20678,20680],{"className":20679},[1752],[772,20681,20683,20726],{"className":20682},[1756,3556],[772,20684,20686,20723],{"className":20685},[1760],[772,20687,20689,20709],{"className":20688,"style":1805},[1764],[772,20690,20691,20694],{"style":18947},[772,20692],{"className":20693,"style":1773},[1772],[772,20695,20697],{"className":20696},[1777,1778,1779,1780],[772,20698,20700,20703,20706],{"className":20699},[816,1780],[772,20701,13887],{"className":20702},[816,817,1780],[772,20704,1353],{"className":20705},[1125,1780],[772,20707,2170],{"className":20708},[816,1780],[772,20710,20711,20714],{"style":1768},[772,20712],{"className":20713,"style":1773},[1772],[772,20715,20717],{"className":20716},[1777,1778,1779,1780],[772,20718,20720],{"className":20719},[816,1780],[772,20721,1651],{"className":20722},[816,1780],[772,20724,3580],{"className":20725},[3579],[772,20727,20729],{"className":20728},[1760],[772,20730,20732],{"className":20731,"style":19723},[1764],[772,20733],{},[772,20735,1096],{"className":20736},[1141],[772,20738],{"className":20739,"style":1121},[1003],[772,20741,1091],{"className":20742},[1125],[772,20744],{"className":20745,"style":1121},[1003],[772,20747,13882],{"className":20748,"style":14054},[3298,14052,14053],[772,20750,1082],{"className":20751},[1113],[772,20753,20755,20758],{"className":20754},[816],[772,20756,2839],{"className":20757,"style":2854},[816,817],[772,20759,20761],{"className":20760},[1752],[772,20762,20764,20798],{"className":20763},[1756,3556],[772,20765,20767,20795],{"className":20766},[1760],[772,20768,20770,20781],{"className":20769,"style":1805},[1764],[772,20771,20772,20775],{"style":18947},[772,20773],{"className":20774,"style":1773},[1772],[772,20776,20778],{"className":20777},[1777,1778,1779,1780],[772,20779,13887],{"className":20780},[816,817,1780],[772,20782,20783,20786],{"style":1768},[772,20784],{"className":20785,"style":1773},[1772],[772,20787,20789],{"className":20788},[1777,1778,1779,1780],[772,20790,20792],{"className":20791},[816,1780],[772,20793,1651],{"className":20794},[816,1780],[772,20796,3580],{"className":20797},[3579],[772,20799,20801],{"className":20800},[1760],[772,20802,20804],{"className":20803,"style":18918},[1764],[772,20805],{},[772,20807,20809,20812],{"className":20808},[816],[772,20810,2808],{"className":20811,"style":2823},[816,817],[772,20813,20815],{"className":20814},[1752],[772,20816,20818,20861],{"className":20817},[1756,3556],[772,20819,20821,20858],{"className":20820},[1760],[772,20822,20824,20844],{"className":20823,"style":1805},[1764],[772,20825,20826,20829],{"style":18885},[772,20827],{"className":20828,"style":1773},[1772],[772,20830,20832],{"className":20831},[1777,1778,1779,1780],[772,20833,20835,20838,20841],{"className":20834},[816,1780],[772,20836,13887],{"className":20837},[816,817,1780],[772,20839,1353],{"className":20840},[1125,1780],[772,20842,2170],{"className":20843},[816,1780],[772,20845,20846,20849],{"style":1768},[772,20847],{"className":20848,"style":1773},[1772],[772,20850,20852],{"className":20851},[1777,1778,1779,1780],[772,20853,20855],{"className":20854},[816,1780],[772,20856,1651],{"className":20857},[816,1780],[772,20859,3580],{"className":20860},[3579],[772,20862,20864],{"className":20863},[1760],[772,20865,20867],{"className":20866,"style":19723},[1764],[772,20868],{},[772,20870,1096],{"className":20871},[1141],[772,20873,20875],{"className":20874},[1141],[772,20876,20878],{"className":20877},[5673,20571],[772,20879,20881,20901],{"className":20880},[1756,3556],[772,20882,20884,20898],{"className":20883},[1760],[772,20885,20887],{"className":20886,"style":20581},[1764],[772,20888,20889,20892],{"style":20584},[772,20890],{"className":20891,"style":9459},[1772],[772,20893,20894],{"style":20590},[5514,20895,20896],{"xmlns":5516,"width":20593,"height":20594,"viewBox":20595},[5522,20897],{"d":20598},[772,20899,3580],{"className":20900},[3579],[772,20902,20904],{"className":20903},[1760],[772,20905,20907],{"className":20906,"style":20608},[1764],[772,20908],{},[11,20910,20911],{},[772,20912,20914,20960],{"className":20913},[775],[772,20915,20917],{"className":20916},[779],[781,20918,20919],{"xmlns":783},[785,20920,20921,20957],{},[788,20922,20923,20925,20927,20933,20945,20947,20950],{},[791,20924,13735],{},[980,20926,982],{},[5599,20928,20929,20931],{},[1084,20930,2170],{},[1084,20932,1872],{},[788,20934,20935,20937,20939,20941,20943],{},[980,20936,5605],{"fence":803},[1084,20938,20359],{},[980,20940,1091],{},[1084,20942,20364],{},[980,20944,5605],{"fence":803},[980,20946,982],{},[1084,20948,20949],{},"1325.000",[1636,20951,20952,20955],{},[1625,20953,20954],{}," m",[1084,20956,1872],{},[795,20958,20959],{"encoding":797},"A = \\frac{1}{2} \\left| 3650.000 - 1000.000 \\right| = 1325.000 \\text{ m}^2",[772,20961,20963,20981,21091],{"className":20962,"ariaHidden":803},[802],[772,20964,20966,20969,20972,20975,20978],{"className":20965},[807],[772,20967],{"className":20968,"style":2758},[811],[772,20970,13735],{"className":20971},[816,817],[772,20973],{"className":20974,"style":1004},[1003],[772,20976,982],{"className":20977},[1008],[772,20979],{"className":20980,"style":1004},[1003],[772,20982,20984,20987,21055,21058,21082,21085,21088],{"className":20983},[807],[772,20985],{"className":20986,"style":13968},[811],[772,20988,20990,20993,21052],{"className":20989},[816],[772,20991],{"className":20992},[1113,5681],[772,20994,20996],{"className":20995},[5599],[772,20997,20999,21044],{"className":20998},[1756,3556],[772,21000,21002,21041],{"className":21001},[1760],[772,21003,21005,21019,21027],{"className":21004,"style":6558},[1764],[772,21006,21007,21010],{"style":5697},[772,21008],{"className":21009,"style":5422},[1772],[772,21011,21013],{"className":21012},[1777,1778,1779,1780],[772,21014,21016],{"className":21015},[816,1780],[772,21017,1872],{"className":21018},[816,1780],[772,21020,21021,21024],{"style":5719},[772,21022],{"className":21023,"style":5422},[1772],[772,21025],{"className":21026,"style":5727},[5726],[772,21028,21029,21032],{"style":6583},[772,21030],{"className":21031,"style":5422},[1772],[772,21033,21035],{"className":21034},[1777,1778,1779,1780],[772,21036,21038],{"className":21037},[816,1780],[772,21039,2170],{"className":21040},[816,1780],[772,21042,3580],{"className":21043},[3579],[772,21045,21047],{"className":21046},[1760],[772,21048,21050],{"className":21049,"style":6605},[1764],[772,21051],{},[772,21053],{"className":21054},[1141,5681],[772,21056],{"className":21057,"style":3294},[1003],[772,21059,21061,21064,21067,21070,21073,21076,21079],{"className":21060},[5664],[772,21062,5605],{"className":21063,"style":5669},[1113,5668],[772,21065,20359],{"className":21066},[816],[772,21068],{"className":21069,"style":1121},[1003],[772,21071,1091],{"className":21072},[1125],[772,21074],{"className":21075,"style":1121},[1003],[772,21077,20364],{"className":21078},[816],[772,21080,5605],{"className":21081,"style":5669},[1141,5668],[772,21083],{"className":21084,"style":1004},[1003],[772,21086,982],{"className":21087},[1008],[772,21089],{"className":21090,"style":1004},[1003],[772,21092,21094,21098,21101],{"className":21093},[807],[772,21095],{"className":21096,"style":21097},[811],"height:0.8141em;",[772,21099,20949],{"className":21100},[816],[772,21102,21104,21110],{"className":21103},[816],[772,21105,21107],{"className":21106},[816,1709],[772,21108,20954],{"className":21109},[816],[772,21111,21113],{"className":21112},[1752],[772,21114,21116],{"className":21115},[1756],[772,21117,21119],{"className":21118},[1760],[772,21120,21122],{"className":21121,"style":21097},[1764],[772,21123,21124,21127],{"style":1768},[772,21125],{"className":21126,"style":1773},[1772],[772,21128,21130],{"className":21129},[1777,1778,1779,1780],[772,21131,1872],{"className":21132},[816,1780],[11,21134,21135],{},"In hectares:",[11,21137,21138],{},[772,21139,21141,21164],{"className":21140},[775],[772,21142,21144],{"className":21143},[779],[781,21145,21146],{"xmlns":783},[785,21147,21148,21161],{},[788,21149,21150,21153,21155,21158],{},[1625,21151,21152],{},"Area",[980,21154,982],{},[1084,21156,21157],{},"0.1325",[1625,21159,21160],{}," ha",[795,21162,21163],{"encoding":797},"\\text{Area} = 0.1325 \\text{ ha}",[772,21165,21167,21188],{"className":21166,"ariaHidden":803},[802],[772,21168,21170,21173,21179,21182,21185],{"className":21169},[807],[772,21171],{"className":21172,"style":2758},[811],[772,21174,21176],{"className":21175},[816,1709],[772,21177,21152],{"className":21178},[816],[772,21180],{"className":21181,"style":1004},[1003],[772,21183,982],{"className":21184},[1008],[772,21186],{"className":21187,"style":1004},[1003],[772,21189,21191,21194,21197],{"className":21190},[807],[772,21192],{"className":21193,"style":812},[811],[772,21195,21157],{"className":21196},[816],[772,21198,21200],{"className":21199},[816,1709],[772,21201,21160],{"className":21202},[816],[11,21204,21205],{},"Perimeter:",[11,21207,21208],{},[772,21209,21211,21251],{"className":21210},[775],[772,21212,21214],{"className":21213},[779],[781,21215,21216],{"xmlns":783},[785,21217,21218,21248],{},[788,21219,21220,21222,21224,21226,21228,21230,21232,21234,21236,21238,21240,21242,21244,21246],{},[791,21221,14943],{},[980,21223,982],{},[1084,21225,20256],{},[980,21227,1353],{},[1084,21229,20256],{},[980,21231,1353],{},[1084,21233,20300],{},[980,21235,1353],{},[1084,21237,20320],{},[980,21239,1353],{},[1084,21241,20246],{},[980,21243,982],{},[1084,21245,20369],{},[1625,21247,20954],{},[795,21249,21250],{"encoding":797},"P = 31.623 + 31.623 + 26.926 + 25.000 + 30.000 = 145.172 \\text{ m}",[772,21252,21254,21272,21290,21308,21326,21344,21362],{"className":21253,"ariaHidden":803},[802],[772,21255,21257,21260,21263,21266,21269],{"className":21256},[807],[772,21258],{"className":21259,"style":2758},[811],[772,21261,14943],{"className":21262,"style":14964},[816,817],[772,21264],{"className":21265,"style":1004},[1003],[772,21267,982],{"className":21268},[1008],[772,21270],{"className":21271,"style":1004},[1003],[772,21273,21275,21278,21281,21284,21287],{"className":21274},[807],[772,21276],{"className":21277,"style":7851},[811],[772,21279,20256],{"className":21280},[816],[772,21282],{"className":21283,"style":1121},[1003],[772,21285,1353],{"className":21286},[1125],[772,21288],{"className":21289,"style":1121},[1003],[772,21291,21293,21296,21299,21302,21305],{"className":21292},[807],[772,21294],{"className":21295,"style":7851},[811],[772,21297,20256],{"className":21298},[816],[772,21300],{"className":21301,"style":1121},[1003],[772,21303,1353],{"className":21304},[1125],[772,21306],{"className":21307,"style":1121},[1003],[772,21309,21311,21314,21317,21320,21323],{"className":21310},[807],[772,21312],{"className":21313,"style":7851},[811],[772,21315,20300],{"className":21316},[816],[772,21318],{"className":21319,"style":1121},[1003],[772,21321,1353],{"className":21322},[1125],[772,21324],{"className":21325,"style":1121},[1003],[772,21327,21329,21332,21335,21338,21341],{"className":21328},[807],[772,21330],{"className":21331,"style":7851},[811],[772,21333,20320],{"className":21334},[816],[772,21336],{"className":21337,"style":1121},[1003],[772,21339,1353],{"className":21340},[1125],[772,21342],{"className":21343,"style":1121},[1003],[772,21345,21347,21350,21353,21356,21359],{"className":21346},[807],[772,21348],{"className":21349,"style":5899},[811],[772,21351,20246],{"className":21352},[816],[772,21354],{"className":21355,"style":1004},[1003],[772,21357,982],{"className":21358},[1008],[772,21360],{"className":21361,"style":1004},[1003],[772,21363,21365,21368,21371],{"className":21364},[807],[772,21366],{"className":21367,"style":5899},[811],[772,21369,20369],{"className":21370},[816],[772,21372,21374],{"className":21373},[816,1709],[772,21375,20954],{"className":21376},[816],[11,21378,21379,21380,21382],{},"Enter the local coordinates into the ",[693,21381,13243],{"href":13242}," in vertex order to verify these results.",[323,21384,21386],{"id":21385},"accuracy-pitfalls-and-common-errors","Accuracy, Pitfalls, and Common Errors",[11,21388,21389],{},"The Shoelace algorithm is elegant, but field and office practice requires managing several operational risks.",[706,21391,21393],{"id":21392},"coordinate-precision-and-catastrophic-cancellation","Coordinate precision and catastrophic cancellation",[11,21395,21396,21397,17739,21460,21530,21531,5868],{},"Projected coordinates often use large magnitudes (e.g. ",[772,21398,21400,21419],{"className":21399},[775],[772,21401,21403],{"className":21402},[779],[781,21404,21405],{"xmlns":783},[785,21406,21407,21416],{},[788,21408,21409,21411,21413],{},[791,21410,2808],{},[980,21412,7437],{},[1084,21414,21415],{},"500,000",[795,21417,21418],{"encoding":797},"E \\approx 500{,}000",[772,21420,21422,21440],{"className":21421,"ariaHidden":803},[802],[772,21423,21425,21428,21431,21434,21437],{"className":21424},[807],[772,21426],{"className":21427,"style":2758},[811],[772,21429,2808],{"className":21430,"style":2823},[816,817],[772,21432],{"className":21433,"style":1004},[1003],[772,21435,7437],{"className":21436},[1008],[772,21438],{"className":21439,"style":1004},[1003],[772,21441,21443,21446,21450,21456],{"className":21442},[807],[772,21444],{"className":21445,"style":11620},[811],[772,21447,21449],{"className":21448},[816],"500",[772,21451,21453],{"className":21452},[816],[772,21454,5940],{"className":21455},[6041],[772,21457,21459],{"className":21458},[816],"000",[772,21461,21463,21482],{"className":21462},[775],[772,21464,21466],{"className":21465},[779],[781,21467,21468],{"xmlns":783},[785,21469,21470,21479],{},[788,21471,21472,21474,21476],{},[791,21473,2839],{},[980,21475,7437],{},[1084,21477,21478],{},"5,000,000",[795,21480,21481],{"encoding":797},"N \\approx 5{,}000{,}000",[772,21483,21485,21503],{"className":21484,"ariaHidden":803},[802],[772,21486,21488,21491,21494,21497,21500],{"className":21487},[807],[772,21489],{"className":21490,"style":2758},[811],[772,21492,2839],{"className":21493,"style":2854},[816,817],[772,21495],{"className":21496,"style":1004},[1003],[772,21498,7437],{"className":21499},[1008],[772,21501],{"className":21502,"style":1004},[1003],[772,21504,21506,21509,21512,21518,21521,21527],{"className":21505},[807],[772,21507],{"className":21508,"style":11620},[811],[772,21510,2177],{"className":21511},[816],[772,21513,21515],{"className":21514},[816],[772,21516,5940],{"className":21517},[6041],[772,21519,21459],{"className":21520},[816],[772,21522,21524],{"className":21523},[816],[772,21525,5940],{"className":21526},[6041],[772,21528,21459],{"className":21529},[816]," m). In floating-point arithmetic, subtracting two large, nearly equal products to obtain a small area can lose significant digits — ",[318,21532,21533],{},"catastrophic cancellation",[11,21535,21536,21537,3474,21621,21705],{},"Professional practice translates vertices to a local origin before area computation: subtract ",[772,21538,21540,21560],{"className":21539},[775],[772,21541,21543],{"className":21542},[779],[781,21544,21545],{"xmlns":783},[785,21546,21547,21557],{},[788,21548,21549],{},[3525,21550,21551,21553,21555],{},[791,21552,2808],{},[791,21554,18599],{},[980,21556,3252],{},[795,21558,21559],{"encoding":797},"E_{\\min}",[772,21561,21563],{"className":21562,"ariaHidden":803},[802],[772,21564,21566,21569],{"className":21565},[807],[772,21567],{"className":21568,"style":3543},[811],[772,21570,21572,21575],{"className":21571},[816],[772,21573,2808],{"className":21574,"style":2823},[816,817],[772,21576,21578],{"className":21577},[1752],[772,21579,21581,21613],{"className":21580},[1756,3556],[772,21582,21584,21610],{"className":21583},[1760],[772,21585,21587],{"className":21586,"style":18636},[1764],[772,21588,21589,21592],{"style":3566},[772,21590],{"className":21591,"style":1773},[1772],[772,21593,21595],{"className":21594},[1777,1778,1779,1780],[772,21596,21598],{"className":21597},[816,1780],[772,21599,21601,21604,21607],{"className":21600},[3298,1780],[772,21602,18654],{"className":21603},[1780],[772,21605,13887],{"className":21606},[1780],[772,21608,13833],{"className":21609},[1780],[772,21611,3580],{"className":21612},[3579],[772,21614,21616],{"className":21615},[1760],[772,21617,21619],{"className":21618,"style":3587},[1764],[772,21620],{},[772,21622,21624,21644],{"className":21623},[775],[772,21625,21627],{"className":21626},[779],[781,21628,21629],{"xmlns":783},[785,21630,21631,21641],{},[788,21632,21633],{},[3525,21634,21635,21637,21639],{},[791,21636,2839],{},[791,21638,18599],{},[980,21640,3252],{},[795,21642,21643],{"encoding":797},"N_{\\min}",[772,21645,21647],{"className":21646,"ariaHidden":803},[802],[772,21648,21650,21653],{"className":21649},[807],[772,21651],{"className":21652,"style":3543},[811],[772,21654,21656,21659],{"className":21655},[816],[772,21657,2839],{"className":21658,"style":2854},[816,817],[772,21660,21662],{"className":21661},[1752],[772,21663,21665,21697],{"className":21664},[1756,3556],[772,21666,21668,21694],{"className":21667},[1760],[772,21669,21671],{"className":21670,"style":18636},[1764],[772,21672,21673,21676],{"style":3640},[772,21674],{"className":21675,"style":1773},[1772],[772,21677,21679],{"className":21678},[1777,1778,1779,1780],[772,21680,21682],{"className":21681},[816,1780],[772,21683,21685,21688,21691],{"className":21684},[3298,1780],[772,21686,18654],{"className":21687},[1780],[772,21689,13887],{"className":21690},[1780],[772,21692,13833],{"className":21693},[1780],[772,21695,3580],{"className":21696},[3579],[772,21698,21700],{"className":21699},[1760],[772,21701,21703],{"className":21702,"style":3587},[1764],[772,21704],{},", or use the first vertex as origin. Translation is rigid — shape, area, and perimeter are unchanged — while arithmetic runs on smaller, more stable numbers. For very large grid values, translate coordinates before pasting them into the calculator.",[706,21707,21709],{"id":21708},"bow-tie-and-self-intersecting-polygons","Bow-tie and self-intersecting polygons",[11,21711,21712,21713,21716],{},"The Shoelace formula requires a ",[318,21714,21715],{},"simple"," polygon (edges do not cross). Incorrect vertex order can produce a bow-tie shape. Signed areas of overlapping lobes partially cancel, yielding a degraded or near-zero result. Complex self-intersecting boundaries are out of scope for standard 2D area tools — validate topology in CAD\u002FGIS before calculating area.",[706,21718,21720],{"id":21719},"spurious-vertices","Spurious vertices",[11,21722,21723],{},"Collinear or duplicate points along a straight leg do not change area materially but add computation and can contribute minor rounding noise in very large datasets.",[706,21725,21727],{"id":21726},"measurement-error-propagation","Measurement error propagation",[11,21729,21730],{},"Coordinate uncertainties from GNSS, total stations, or low-accuracy sources propagate into area and perimeter. Perimeter is especially sensitive: every edge length is a positive root",[11,21732,21733],{},[772,21734,21736,21786],{"className":21735},[775],[772,21737,21739],{"className":21738},[779],[781,21740,21741],{"xmlns":783},[785,21742,21743,21783],{},[788,21744,21745,21751,21753],{},[3525,21746,21747,21749],{},[791,21748,16242],{},[791,21750,13887],{},[980,21752,982],{},[5347,21754,21755],{},[788,21756,21757,21759,21761,21763,21769,21771,21773,21775,21777],{},[980,21758,1082],{"stretchy":1081},[791,21760,2963],{"mathvariant":906},[791,21762,2808],{},[1636,21764,21765,21767],{},[980,21766,1096],{"stretchy":1081},[1084,21768,1872],{},[980,21770,1353],{},[980,21772,1082],{"stretchy":1081},[791,21774,2963],{"mathvariant":906},[791,21776,2839],{},[1636,21778,21779,21781],{},[980,21780,1096],{"stretchy":1081},[1084,21782,1872],{},[795,21784,21785],{"encoding":797},"d_i = \\sqrt{(\\Delta E)^2 + (\\Delta N)^2}",[772,21787,21789,21844],{"className":21788,"ariaHidden":803},[802],[772,21790,21792,21795,21835,21838,21841],{"className":21791},[807],[772,21793],{"className":21794,"style":5959},[811],[772,21796,21798,21801],{"className":21797},[816],[772,21799,16242],{"className":21800},[816,817],[772,21802,21804],{"className":21803},[1752],[772,21805,21807,21827],{"className":21806},[1756,3556],[772,21808,21810,21824],{"className":21809},[1760],[772,21811,21813],{"className":21812,"style":14137},[1764],[772,21814,21815,21818],{"style":16341},[772,21816],{"className":21817,"style":1773},[1772],[772,21819,21821],{"className":21820},[1777,1778,1779,1780],[772,21822,13887],{"className":21823},[816,817,1780],[772,21825,3580],{"className":21826},[3579],[772,21828,21830],{"className":21829},[1760],[772,21831,21833],{"className":21832,"style":3587},[1764],[772,21834],{},[772,21836],{"className":21837,"style":1004},[1003],[772,21839,982],{"className":21840},[1008],[772,21842],{"className":21843,"style":1004},[1003],[772,21845,21847,21850],{"className":21846},[807],[772,21848],{"className":21849,"style":9438},[811],[772,21851,21853],{"className":21852},[816,5403],[772,21854,21856,21971],{"className":21855},[1756,3556],[772,21857,21859,21968],{"className":21858},[1760],[772,21860,21862,21956],{"className":21861,"style":9451},[1764],[772,21863,21865,21868],{"className":21864,"style":9455},[5417],[772,21866],{"className":21867,"style":9459},[1772],[772,21869,21871,21874,21877,21880,21909,21912,21915,21918,21921,21924,21927],{"className":21870,"style":9463},[816],[772,21872,1082],{"className":21873},[1113],[772,21875,2963],{"className":21876},[816],[772,21878,2808],{"className":21879,"style":2823},[816,817],[772,21881,21883,21886],{"className":21882},[1141],[772,21884,1096],{"className":21885},[1141],[772,21887,21889],{"className":21888},[1752],[772,21890,21892],{"className":21891},[1756],[772,21893,21895],{"className":21894},[1760],[772,21896,21898],{"className":21897,"style":5448},[1764],[772,21899,21900,21903],{"style":5451},[772,21901],{"className":21902,"style":1773},[1772],[772,21904,21906],{"className":21905},[1777,1778,1779,1780],[772,21907,1872],{"className":21908},[816,1780],[772,21910],{"className":21911,"style":1121},[1003],[772,21913,1353],{"className":21914},[1125],[772,21916],{"className":21917,"style":1121},[1003],[772,21919,1082],{"className":21920},[1113],[772,21922,2963],{"className":21923},[816],[772,21925,2839],{"className":21926,"style":2854},[816,817],[772,21928,21930,21933],{"className":21929},[1141],[772,21931,1096],{"className":21932},[1141],[772,21934,21936],{"className":21935},[1752],[772,21937,21939],{"className":21938},[1756],[772,21940,21942],{"className":21941},[1760],[772,21943,21945],{"className":21944,"style":5448},[1764],[772,21946,21947,21950],{"style":5451},[772,21948],{"className":21949,"style":1773},[1772],[772,21951,21953],{"className":21952},[1777,1778,1779,1780],[772,21954,1872],{"className":21955},[816,1780],[772,21957,21958,21961],{"style":9548},[772,21959],{"className":21960,"style":9459},[1772],[772,21962,21964],{"className":21963,"style":9555},[5511],[5514,21965,21966],{"xmlns":5516,"width":5517,"height":9558,"viewBox":9559,"preserveAspectRatio":5520},[5522,21967],{"d":9562},[772,21969,3580],{"className":21970},[3579],[772,21972,21974],{"className":21973},[1760],[772,21975,21977],{"className":21976,"style":9572},[1764],[772,21978],{},[11,21980,21981],{},"so random noise at intermediate vertices along a straight line can inflate total perimeter. Filter redundant vertices on straight segments when perimeter accuracy matters.",[706,21983,21985],{"id":21984},"common-errors-summary","Common errors summary",[851,21987,21988,22004],{},[854,21989,21990],{},[857,21991,21992,21995,21998,22001],{},[860,21993,21994],{},"Error type",[860,21996,21997],{},"Mechanism",[860,21999,22000],{},"Impact",[860,22002,22003],{},"Prevention",[876,22005,22006,22020,22167,22181],{},[857,22007,22008,22011,22014,22017],{},[881,22009,22010],{},"Out-of-order vertices",[881,22012,22013],{},"Scrambled sequence → crossings",[881,22015,22016],{},"Near-zero or wrong area; spurious perimeter chords",[881,22018,22019],{},"Sort vertices sequentially around the boundary",[857,22021,22022,22025,22028,22031],{},[881,22023,22024],{},"Missing closing leg",[881,22026,22027],{},"Final edge omitted",[881,22029,22030],{},"Incomplete area and perimeter",[881,22032,22033,22034,22166],{},"Use implicit closure (",[772,22035,22037,22062],{"className":22036},[775],[772,22038,22040],{"className":22039},[779],[781,22041,22042],{"xmlns":783},[785,22043,22044,22060],{},[788,22045,22046,22052,22054],{},[3525,22047,22048,22050],{},[791,22049,14943],{},[791,22051,13833],{},[980,22053,17398],{},[3525,22055,22056,22058],{},[791,22057,14943],{},[1084,22059,2170],{},[795,22061,17407],{"encoding":797},[772,22063,22065,22120],{"className":22064,"ariaHidden":803},[802],[772,22066,22068,22071,22111,22114,22117],{"className":22067},[807],[772,22069],{"className":22070,"style":3543},[811],[772,22072,22074,22077],{"className":22073},[816],[772,22075,14943],{"className":22076,"style":14964},[816,817],[772,22078,22080],{"className":22079},[1752],[772,22081,22083,22103],{"className":22082},[1756,3556],[772,22084,22086,22100],{"className":22085},[1760],[772,22087,22089],{"className":22088,"style":15612},[1764],[772,22090,22091,22094],{"style":14979},[772,22092],{"className":22093,"style":1773},[1772],[772,22095,22097],{"className":22096},[1777,1778,1779,1780],[772,22098,13833],{"className":22099},[816,817,1780],[772,22101,3580],{"className":22102},[3579],[772,22104,22106],{"className":22105},[1760],[772,22107,22109],{"className":22108,"style":3587},[1764],[772,22110],{},[772,22112],{"className":22113,"style":1004},[1003],[772,22115,17398],{"className":22116},[1008],[772,22118],{"className":22119,"style":1004},[1003],[772,22121,22123,22126],{"className":22122},[807],[772,22124],{"className":22125,"style":3543},[811],[772,22127,22129,22132],{"className":22128},[816],[772,22130,14943],{"className":22131,"style":14964},[816,817],[772,22133,22135],{"className":22134},[1752],[772,22136,22138,22158],{"className":22137},[1756,3556],[772,22139,22141,22155],{"className":22140},[1760],[772,22142,22144],{"className":22143,"style":3563},[1764],[772,22145,22146,22149],{"style":14979},[772,22147],{"className":22148,"style":1773},[1772],[772,22150,22152],{"className":22151},[1777,1778,1779,1780],[772,22153,2170],{"className":22154},[816,1780],[772,22156,3580],{"className":22157},[3579],[772,22159,22161],{"className":22160},[1760],[772,22162,22164],{"className":22163,"style":3587},[1764],[772,22165],{},") or duplicate first vertex",[857,22168,22169,22172,22175,22178],{},[881,22170,22171],{},"Units mix-up",[881,22173,22174],{},"Feet vs metres mismatch",[881,22176,22177],{},"Area scaled by wrong factor",[881,22179,22180],{},"Confirm CRS units before calculation",[857,22182,22183,22186,22189,22192],{},[881,22184,22185],{},"Catastrophic cancellation",[881,22187,22188],{},"Large coordinates without translation",[881,22190,22191],{},"Noisy or wrong least significant digits",[881,22193,22194],{},"Shift to a local origin first",[323,22196,22198],{"id":22197},"integration-with-cad-and-gis-workflows","Integration with CAD and GIS Workflows",[11,22200,22201,22202,22205],{},"CAD (Civil 3D, MicroStation) and GIS (ArcGIS, QGIS) apply the same underlying math when reporting closed-polygon area and length from vertex arrays. Spatial databases enforce validity: OGC simple-feature rules require non-self-intersecting boundaries. PostGIS functions such as ",[2863,22203,22204],{},"ST_IsValid(geom)"," detect bow-ties; repair tools decompose invalid geometry into valid parts. If the path is self-intersecting, the reported area is not trustworthy.",[323,22207,11205],{"id":11204},[11,22209,22210,22211,22213,22214,22266],{},"Planimetric area and perimeter underpin boundary surveying, land development, and civil design. Understanding the Shoelace formula, winding order, coordinate translation, and topological validity prevents costly errors. To calculate or verify values from field coordinates, use the ",[693,22212,13243],{"href":13242}," — ordered ",[772,22215,22217,22239],{"className":22216},[775],[772,22218,22220],{"className":22219},[779],[781,22221,22222],{"xmlns":783},[785,22223,22224,22236],{},[788,22225,22226,22228,22230,22232,22234],{},[980,22227,1082],{"stretchy":1081},[791,22229,2808],{},[980,22231,5940],{"separator":803},[791,22233,2839],{},[980,22235,1096],{"stretchy":1081},[795,22237,22238],{"encoding":797},"(E, N)",[772,22240,22242],{"className":22241,"ariaHidden":803},[802],[772,22243,22245,22248,22251,22254,22257,22260,22263],{"className":22244},[807],[772,22246],{"className":22247,"style":1109},[811],[772,22249,1082],{"className":22250},[1113],[772,22252,2808],{"className":22253,"style":2823},[816,817],[772,22255,5940],{"className":22256},[6041],[772,22258],{"className":22259,"style":3294},[1003],[772,22261,2839],{"className":22262,"style":2854},[816,817],[772,22264,1096],{"className":22265},[1141]," vertices, implicit closure, absolute area, and summed perimeter.",{"title":15,"searchDepth":16,"depth":16,"links":22268},[22269,22270,22271,22272,22273,22274,22275,22276,22283,22284],{"id":684,"depth":16,"text":685},{"id":13247,"depth":16,"text":13248},{"id":13257,"depth":16,"text":13258},{"id":13600,"depth":16,"text":13601},{"id":13811,"depth":16,"text":13812},{"id":15785,"depth":16,"text":15786},{"id":17596,"depth":16,"text":17597},{"id":21385,"depth":16,"text":21386,"children":22277},[22278,22279,22280,22281,22282],{"id":21392,"depth":73,"text":21393},{"id":21708,"depth":73,"text":21709},{"id":21719,"depth":73,"text":21720},{"id":21726,"depth":73,"text":21727},{"id":21984,"depth":73,"text":21985},{"id":22197,"depth":16,"text":22198},{"id":11204,"depth":16,"text":11205},"Planimetric area and perimeter from ordered E\u002FN vertices — the Shoelace formula, winding order, worked example, and common pitfalls.",{},"\u002Flearn\u002Ftools\u002Fpolygon-area",{"title":13232,"description":22285},{"loc":22287},"learn\u002Ftools\u002Fpolygon-area","H0VqrJsWEARggn-PL4vbZ1FO7ikSDJoBBsPvuftrS6o",{"id":22293,"title":22294,"body":22295,"category":11245,"description":32038,"extension":20,"meta":32039,"navigation":22,"order":73,"path":32040,"seo":32041,"sitemap":32042,"stem":32043,"__hash__":32044},"learn\u002Flearn\u002Ftools\u002Ftienstra-resection.md","Tienstra Resection",{"type":8,"value":22296,"toc":31997},[22297,22301,22304,22345,22349,22352,22356,22359,22363,22488,22671,22675,22739,22957,23561,24153,24254,24258,24261,24265,24452,24514,24916,24919,25049,25053,25135,25138,25390,25394,25649,25655,25659,25663,25674,25678,25807,26011,26014,26294,26470,26474,26477,26481,26485,26604,26781,26999,27003,27204,27421,27634,27849,27881,29042,29046,29050,29099,29131,29518,29620,29667,30475,30510,30514,30671,30842,31204,31266,31270,31455,31461,31465,31469,31656,31660,31790,31794,31798,31891,31895,31983,31987,31990,31992],[323,22298,22300],{"id":22299},"planimetric-positioning-via-the-tienstra-three-point-resection","Planimetric Positioning via the Tienstra Three-Point Resection",[11,22302,22303],{},"In engineering geodesy, establishing high-precision horizontal control is a prerequisite for total station setups and subsequent stakeout or monitoring. GNSS is often blocked in urban canyons, open pits, and forest canopy, forcing crews to rely on classical optical methods. When distances to control cannot be measured — due to hazards, obstacles, or coordinate-only networks — the surveyor must solve position from angles alone.",[11,22305,22306,22307,22310,22311,22339,22340,22344],{},"The three-point angular resection establishes a ",[318,22308,22309],{},"free station"," at an unknown point ",[772,22312,22314,22327],{"className":22313},[775],[772,22315,22317],{"className":22316},[779],[781,22318,22319],{"xmlns":783},[785,22320,22321,22325],{},[788,22322,22323],{},[791,22324,14943],{},[795,22326,14943],{"encoding":797},[772,22328,22330],{"className":22329,"ariaHidden":803},[802],[772,22331,22333,22336],{"className":22332},[807],[772,22334],{"className":22335,"style":2758},[811],[772,22337,14943],{"className":22338,"style":14964},[816,817]," by measuring horizontal angles to three visible, coordinated benchmarks. The ",[693,22341,22343],{"href":22342},"\u002Ftools\u002Ftienstra-resection","Tienstra resection calculator"," implements the cotangent barycentric formulation for rapid coordinate resolution from field inputs. This article covers the geometry, danger circle, field practice, computation, and verification.",[323,22346,22348],{"id":22347},"defining-the-resection-problem-resection-vs-intersection","Defining the Resection Problem: Resection vs. Intersection",[11,22350,22351],{},"Planimetric positioning divides into two primary configurations:",[706,22353,22355],{"id":22354},"intersection","Intersection",[11,22357,22358],{},"The unknown point is an inaccessible target (peak, spire, remote monument). Instruments occupy at least two known stations; bearings from each station intersect at the target. Accurate, but requires multiple setups and access to control.",[706,22360,22362],{"id":22361},"resection","Resection",[11,22364,22365,22366,22369,22370,22398,22399,2896,22427,11896,22457,22487],{},"The surveyor occupies the ",[318,22367,22368],{},"unknown"," point ",[772,22371,22373,22386],{"className":22372},[775],[772,22374,22376],{"className":22375},[779],[781,22377,22378],{"xmlns":783},[785,22379,22380,22384],{},[788,22381,22382],{},[791,22383,14943],{},[795,22385,14943],{"encoding":797},[772,22387,22389],{"className":22388,"ariaHidden":803},[802],[772,22390,22392,22395],{"className":22391},[807],[772,22393],{"className":22394,"style":2758},[811],[772,22396,14943],{"className":22397,"style":14964},[816,817]," and observes horizontal angles between three known stations ",[772,22400,22402,22415],{"className":22401},[775],[772,22403,22405],{"className":22404},[779],[781,22406,22407],{"xmlns":783},[785,22408,22409,22413],{},[788,22410,22411],{},[791,22412,13735],{},[795,22414,13735],{"encoding":797},[772,22416,22418],{"className":22417,"ariaHidden":803},[802],[772,22419,22421,22424],{"className":22420},[807],[772,22422],{"className":22423,"style":2758},[811],[772,22425,13735],{"className":22426},[816,817],[772,22428,22430,22444],{"className":22429},[775],[772,22431,22433],{"className":22432},[779],[781,22434,22435],{"xmlns":783},[785,22436,22437,22442],{},[788,22438,22439],{},[791,22440,22441],{},"B",[795,22443,22441],{"encoding":797},[772,22445,22447],{"className":22446,"ariaHidden":803},[802],[772,22448,22450,22453],{"className":22449},[807],[772,22451],{"className":22452,"style":2758},[811],[772,22454,22441],{"className":22455,"style":22456},[816,817],"margin-right:0.0502em;",[772,22458,22460,22474],{"className":22459},[775],[772,22461,22463],{"className":22462},[779],[781,22464,22465],{"xmlns":783},[785,22466,22467,22472],{},[788,22468,22469],{},[791,22470,22471],{},"C",[795,22473,22471],{"encoding":797},[772,22475,22477],{"className":22476,"ariaHidden":803},[802],[772,22478,22480,22483],{"className":22479},[807],[772,22481],{"className":22482,"style":2758},[811],[772,22484,22471],{"className":22485,"style":22486},[816,817],"margin-right:0.0715em;",". Only one setup is required; control points need not be occupied — only visible. Ideal for construction site free-stationing referencing distant stable benchmarks.",[851,22489,22490,22502],{},[854,22491,22492],{},[857,22493,22494,22496,22499],{},[860,22495,11783],{},[860,22497,22498],{},"Angular intersection",[860,22500,22501],{},"Angular resection",[876,22503,22504,22543,22610,22649,22660],{},[857,22505,22506,22509,22512],{},[881,22507,22508],{},"Occupied station(s)",[881,22510,22511],{},"Multiple known points",[881,22513,22514,22515],{},"Single unknown point ",[772,22516,22518,22531],{"className":22517},[775],[772,22519,22521],{"className":22520},[779],[781,22522,22523],{"xmlns":783},[785,22524,22525,22529],{},[788,22526,22527],{},[791,22528,14943],{},[795,22530,14943],{"encoding":797},[772,22532,22534],{"className":22533,"ariaHidden":803},[802],[772,22535,22537,22540],{"className":22536},[807],[772,22538],{"className":22539,"style":2758},[811],[772,22541,14943],{"className":22542,"style":14964},[816,817],[857,22544,22545,22548,22579],{},[881,22546,22547],{},"Observation type",[881,22549,22550,22551],{},"Bearings from known points to ",[772,22552,22554,22567],{"className":22553},[775],[772,22555,22557],{"className":22556},[779],[781,22558,22559],{"xmlns":783},[785,22560,22561,22565],{},[788,22562,22563],{},[791,22564,14943],{},[795,22566,14943],{"encoding":797},[772,22568,22570],{"className":22569,"ariaHidden":803},[802],[772,22571,22573,22576],{"className":22572},[807],[772,22574],{"className":22575,"style":2758},[811],[772,22577,14943],{"className":22578,"style":14964},[816,817],[881,22580,22581,22582],{},"Angles between known points at ",[772,22583,22585,22598],{"className":22584},[775],[772,22586,22588],{"className":22587},[779],[781,22589,22590],{"xmlns":783},[785,22591,22592,22596],{},[788,22593,22594],{},[791,22595,14943],{},[795,22597,14943],{"encoding":797},[772,22599,22601],{"className":22600,"ariaHidden":803},[802],[772,22602,22604,22607],{"className":22603},[807],[772,22605],{"className":22606,"style":2758},[811],[772,22608,14943],{"className":22609,"style":14964},[816,817],[857,22611,22612,22615,22618],{},[881,22613,22614],{},"Minimum requirements",[881,22616,22617],{},"Two known points, intervisible baseline",[881,22619,22620,22621],{},"Three non-collinear known points visible from ",[772,22622,22624,22637],{"className":22623},[775],[772,22625,22627],{"className":22626},[779],[781,22628,22629],{"xmlns":783},[785,22630,22631,22635],{},[788,22632,22633],{},[791,22634,14943],{},[795,22636,14943],{"encoding":797},[772,22638,22640],{"className":22639,"ariaHidden":803},[802],[772,22641,22643,22646],{"className":22642},[807],[772,22644],{"className":22645,"style":2758},[811],[772,22647,14943],{"className":22648,"style":14964},[816,817],[857,22650,22651,22654,22657],{},[881,22652,22653],{},"Operational advantage",[881,22655,22656],{},"Inaccessible targets",[881,22658,22659],{},"Rapid free-stationing",[857,22661,22662,22665,22668],{},[881,22663,22664],{},"Logistical effort",[881,22666,22667],{},"High (multiple crews\u002Fsetups)",[881,22669,22670],{},"Low (single setup)",[323,22672,22674],{"id":22673},"the-tienstra-three-point-method","The Tienstra Three-Point Method",[11,22676,22677,22678,22681,22682,22710,22711,5868],{},"Classical geometric methods (Cassini, Collins) use auxiliary constructions and are error-prone in manual calculation. The ",[318,22679,22680],{},"Tienstra formula"," — popularized by J.M. Tienstra (1895–1951) at Delft — solves ",[772,22683,22685,22698],{"className":22684},[775],[772,22686,22688],{"className":22687},[779],[781,22689,22690],{"xmlns":783},[785,22691,22692,22696],{},[788,22693,22694],{},[791,22695,14943],{},[795,22697,14943],{"encoding":797},[772,22699,22701],{"className":22700,"ariaHidden":803},[802],[772,22702,22704,22707],{"className":22703},[807],[772,22705],{"className":22706,"style":2758},[811],[772,22708,14943],{"className":22709,"style":14964},[816,817]," directly with barycentric weights from cotangents of control-triangle interior angles and the observed angles at ",[772,22712,22714,22727],{"className":22713},[775],[772,22715,22717],{"className":22716},[779],[781,22718,22719],{"xmlns":783},[785,22720,22721,22725],{},[788,22722,22723],{},[791,22724,14943],{},[795,22726,14943],{"encoding":797},[772,22728,22730],{"className":22729,"ariaHidden":803},[802],[772,22731,22733,22736],{"className":22732},[807],[772,22734],{"className":22735,"style":2758},[811],[772,22737,14943],{"className":22738,"style":14964},[816,817],[11,22740,22741,22742,2896,22816,2896,22886,22956],{},"Once weights ",[772,22743,22745,22764],{"className":22744},[775],[772,22746,22748],{"className":22747},[779],[781,22749,22750],{"xmlns":783},[785,22751,22752,22761],{},[788,22753,22754],{},[3525,22755,22756,22759],{},[791,22757,22758],{},"f",[1084,22760,2170],{},[795,22762,22763],{"encoding":797},"f_1",[772,22765,22767],{"className":22766,"ariaHidden":803},[802],[772,22768,22770,22774],{"className":22769},[807],[772,22771],{"className":22772,"style":22773},[811],"height:0.8889em;vertical-align:-0.1944em;",[772,22775,22777,22781],{"className":22776},[816],[772,22778,22758],{"className":22779,"style":22780},[816,817],"margin-right:0.1076em;",[772,22782,22784],{"className":22783},[1752],[772,22785,22787,22808],{"className":22786},[1756,3556],[772,22788,22790,22805],{"className":22789},[1760],[772,22791,22793],{"className":22792,"style":3563},[1764],[772,22794,22796,22799],{"style":22795},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[772,22797],{"className":22798,"style":1773},[1772],[772,22800,22802],{"className":22801},[1777,1778,1779,1780],[772,22803,2170],{"className":22804},[816,1780],[772,22806,3580],{"className":22807},[3579],[772,22809,22811],{"className":22810},[1760],[772,22812,22814],{"className":22813,"style":3587},[1764],[772,22815],{},[772,22817,22819,22837],{"className":22818},[775],[772,22820,22822],{"className":22821},[779],[781,22823,22824],{"xmlns":783},[785,22825,22826,22834],{},[788,22827,22828],{},[3525,22829,22830,22832],{},[791,22831,22758],{},[1084,22833,1872],{},[795,22835,22836],{"encoding":797},"f_2",[772,22838,22840],{"className":22839,"ariaHidden":803},[802],[772,22841,22843,22846],{"className":22842},[807],[772,22844],{"className":22845,"style":22773},[811],[772,22847,22849,22852],{"className":22848},[816],[772,22850,22758],{"className":22851,"style":22780},[816,817],[772,22853,22855],{"className":22854},[1752],[772,22856,22858,22878],{"className":22857},[1756,3556],[772,22859,22861,22875],{"className":22860},[1760],[772,22862,22864],{"className":22863,"style":3563},[1764],[772,22865,22866,22869],{"style":22795},[772,22867],{"className":22868,"style":1773},[1772],[772,22870,22872],{"className":22871},[1777,1778,1779,1780],[772,22873,1872],{"className":22874},[816,1780],[772,22876,3580],{"className":22877},[3579],[772,22879,22881],{"className":22880},[1760],[772,22882,22884],{"className":22883,"style":3587},[1764],[772,22885],{},[772,22887,22889,22907],{"className":22888},[775],[772,22890,22892],{"className":22891},[779],[781,22893,22894],{"xmlns":783},[785,22895,22896,22904],{},[788,22897,22898],{},[3525,22899,22900,22902],{},[791,22901,22758],{},[1084,22903,1845],{},[795,22905,22906],{"encoding":797},"f_3",[772,22908,22910],{"className":22909,"ariaHidden":803},[802],[772,22911,22913,22916],{"className":22912},[807],[772,22914],{"className":22915,"style":22773},[811],[772,22917,22919,22922],{"className":22918},[816],[772,22920,22758],{"className":22921,"style":22780},[816,817],[772,22923,22925],{"className":22924},[1752],[772,22926,22928,22948],{"className":22927},[1756,3556],[772,22929,22931,22945],{"className":22930},[1760],[772,22932,22934],{"className":22933,"style":3563},[1764],[772,22935,22936,22939],{"style":22795},[772,22937],{"className":22938,"style":1773},[1772],[772,22940,22942],{"className":22941},[1777,1778,1779,1780],[772,22943,1845],{"className":22944},[816,1780],[772,22946,3580],{"className":22947},[3579],[772,22949,22951],{"className":22950},[1760],[772,22952,22954],{"className":22953,"style":3587},[1764],[772,22955],{}," are found:",[11,22958,22959],{},[772,22960,22962,23050],{"className":22961},[775],[772,22963,22965],{"className":22964},[779],[781,22966,22967],{"xmlns":783},[785,22968,22969,23047],{},[788,22970,22971,22977,22979],{},[3525,22972,22973,22975],{},[791,22974,2808],{},[791,22976,14943],{},[980,22978,982],{},[5599,22980,22981,23023],{},[788,22982,22983,22989,22995,22997,23003,23009,23011,23017],{},[3525,22984,22985,22987],{},[791,22986,22758],{},[1084,22988,2170],{},[3525,22990,22991,22993],{},[791,22992,2808],{},[791,22994,13735],{},[980,22996,1353],{},[3525,22998,22999,23001],{},[791,23000,22758],{},[1084,23002,1872],{},[3525,23004,23005,23007],{},[791,23006,2808],{},[791,23008,22441],{},[980,23010,1353],{},[3525,23012,23013,23015],{},[791,23014,22758],{},[1084,23016,1845],{},[3525,23018,23019,23021],{},[791,23020,2808],{},[791,23022,22471],{},[788,23024,23025,23031,23033,23039,23041],{},[3525,23026,23027,23029],{},[791,23028,22758],{},[1084,23030,2170],{},[980,23032,1353],{},[3525,23034,23035,23037],{},[791,23036,22758],{},[1084,23038,1872],{},[980,23040,1353],{},[3525,23042,23043,23045],{},[791,23044,22758],{},[1084,23046,1845],{},[795,23048,23049],{"encoding":797},"E_P = \\frac{f_1 E_A + f_2 E_B + f_3 E_C}{f_1 + f_2 + f_3}",[772,23051,23053,23109],{"className":23052,"ariaHidden":803},[802],[772,23054,23056,23059,23100,23103,23106],{"className":23055},[807],[772,23057],{"className":23058,"style":3543},[811],[772,23060,23062,23065],{"className":23061},[816],[772,23063,2808],{"className":23064,"style":2823},[816,817],[772,23066,23068],{"className":23067},[1752],[772,23069,23071,23092],{"className":23070},[1756,3556],[772,23072,23074,23089],{"className":23073},[1760],[772,23075,23078],{"className":23076,"style":23077},[1764],"height:0.3283em;",[772,23079,23080,23083],{"style":3566},[772,23081],{"className":23082,"style":1773},[1772],[772,23084,23086],{"className":23085},[1777,1778,1779,1780],[772,23087,14943],{"className":23088,"style":14964},[816,817,1780],[772,23090,3580],{"className":23091},[3579],[772,23093,23095],{"className":23094},[1760],[772,23096,23098],{"className":23097,"style":3587},[1764],[772,23099],{},[772,23101],{"className":23102,"style":1004},[1003],[772,23104,982],{"className":23105},[1008],[772,23107],{"className":23108,"style":1004},[1003],[772,23110,23112,23116],{"className":23111},[807],[772,23113],{"className":23114,"style":23115},[811],"height:1.4133em;vertical-align:-0.4811em;",[772,23117,23119,23122,23558],{"className":23118},[816],[772,23120],{"className":23121},[1113,5681],[772,23123,23125],{"className":23124},[5599],[772,23126,23128,23549],{"className":23127},[1756,3556],[772,23129,23131,23546],{"className":23130},[1760],[772,23132,23135,23277,23285],{"className":23133,"style":23134},[1764],"height:0.9322em;",[772,23136,23137,23140],{"style":5697},[772,23138],{"className":23139,"style":5422},[1772],[772,23141,23143],{"className":23142},[1777,1778,1779,1780],[772,23144,23146,23191,23194,23234,23237],{"className":23145},[816,1780],[772,23147,23149,23152],{"className":23148},[816,1780],[772,23150,22758],{"className":23151,"style":22780},[816,817,1780],[772,23153,23155],{"className":23154},[1752],[772,23156,23158,23182],{"className":23157},[1756,3556],[772,23159,23161,23179],{"className":23160},[1760],[772,23162,23165],{"className":23163,"style":23164},[1764],"height:0.3173em;",[772,23166,23168,23172],{"style":23167},"top:-2.357em;margin-left:-0.1076em;margin-right:0.0714em;",[772,23169],{"className":23170,"style":23171},[1772],"height:2.5em;",[772,23173,23176],{"className":23174},[1777,23175,6539,1780],"reset-size3",[772,23177,2170],{"className":23178},[816,1780],[772,23180,3580],{"className":23181},[3579],[772,23183,23185],{"className":23184},[1760],[772,23186,23189],{"className":23187,"style":23188},[1764],"height:0.143em;",[772,23190],{},[772,23192,1353],{"className":23193},[1125,1780],[772,23195,23197,23200],{"className":23196},[816,1780],[772,23198,22758],{"className":23199,"style":22780},[816,817,1780],[772,23201,23203],{"className":23202},[1752],[772,23204,23206,23226],{"className":23205},[1756,3556],[772,23207,23209,23223],{"className":23208},[1760],[772,23210,23212],{"className":23211,"style":23164},[1764],[772,23213,23214,23217],{"style":23167},[772,23215],{"className":23216,"style":23171},[1772],[772,23218,23220],{"className":23219},[1777,23175,6539,1780],[772,23221,1872],{"className":23222},[816,1780],[772,23224,3580],{"className":23225},[3579],[772,23227,23229],{"className":23228},[1760],[772,23230,23232],{"className":23231,"style":23188},[1764],[772,23233],{},[772,23235,1353],{"className":23236},[1125,1780],[772,23238,23240,23243],{"className":23239},[816,1780],[772,23241,22758],{"className":23242,"style":22780},[816,817,1780],[772,23244,23246],{"className":23245},[1752],[772,23247,23249,23269],{"className":23248},[1756,3556],[772,23250,23252,23266],{"className":23251},[1760],[772,23253,23255],{"className":23254,"style":23164},[1764],[772,23256,23257,23260],{"style":23167},[772,23258],{"className":23259,"style":23171},[1772],[772,23261,23263],{"className":23262},[1777,23175,6539,1780],[772,23264,1845],{"className":23265},[816,1780],[772,23267,3580],{"className":23268},[3579],[772,23270,23272],{"className":23271},[1760],[772,23273,23275],{"className":23274,"style":23188},[1764],[772,23276],{},[772,23278,23279,23282],{"style":5719},[772,23280],{"className":23281,"style":5422},[1772],[772,23283],{"className":23284,"style":5727},[5726],[772,23286,23288,23291],{"style":23287},"top:-3.4461em;",[772,23289],{"className":23290,"style":5422},[1772],[772,23292,23294],{"className":23293},[1777,1778,1779,1780],[772,23295,23297,23337,23380,23383,23423,23463,23466,23506],{"className":23296},[816,1780],[772,23298,23300,23303],{"className":23299},[816,1780],[772,23301,22758],{"className":23302,"style":22780},[816,817,1780],[772,23304,23306],{"className":23305},[1752],[772,23307,23309,23329],{"className":23308},[1756,3556],[772,23310,23312,23326],{"className":23311},[1760],[772,23313,23315],{"className":23314,"style":23164},[1764],[772,23316,23317,23320],{"style":23167},[772,23318],{"className":23319,"style":23171},[1772],[772,23321,23323],{"className":23322},[1777,23175,6539,1780],[772,23324,2170],{"className":23325},[816,1780],[772,23327,3580],{"className":23328},[3579],[772,23330,23332],{"className":23331},[1760],[772,23333,23335],{"className":23334,"style":23188},[1764],[772,23336],{},[772,23338,23340,23343],{"className":23339},[816,1780],[772,23341,2808],{"className":23342,"style":2823},[816,817,1780],[772,23344,23346],{"className":23345},[1752],[772,23347,23349,23371],{"className":23348},[1756,3556],[772,23350,23352,23368],{"className":23351},[1760],[772,23353,23356],{"className":23354,"style":23355},[1764],"height:0.3448em;",[772,23357,23359,23362],{"style":23358},"top:-2.3567em;margin-left:-0.0576em;margin-right:0.0714em;",[772,23360],{"className":23361,"style":23171},[1772],[772,23363,23365],{"className":23364},[1777,23175,6539,1780],[772,23366,13735],{"className":23367},[816,817,1780],[772,23369,3580],{"className":23370},[3579],[772,23372,23374],{"className":23373},[1760],[772,23375,23378],{"className":23376,"style":23377},[1764],"height:0.1433em;",[772,23379],{},[772,23381,1353],{"className":23382},[1125,1780],[772,23384,23386,23389],{"className":23385},[816,1780],[772,23387,22758],{"className":23388,"style":22780},[816,817,1780],[772,23390,23392],{"className":23391},[1752],[772,23393,23395,23415],{"className":23394},[1756,3556],[772,23396,23398,23412],{"className":23397},[1760],[772,23399,23401],{"className":23400,"style":23164},[1764],[772,23402,23403,23406],{"style":23167},[772,23404],{"className":23405,"style":23171},[1772],[772,23407,23409],{"className":23408},[1777,23175,6539,1780],[772,23410,1872],{"className":23411},[816,1780],[772,23413,3580],{"className":23414},[3579],[772,23416,23418],{"className":23417},[1760],[772,23419,23421],{"className":23420,"style":23188},[1764],[772,23422],{},[772,23424,23426,23429],{"className":23425},[816,1780],[772,23427,2808],{"className":23428,"style":2823},[816,817,1780],[772,23430,23432],{"className":23431},[1752],[772,23433,23435,23455],{"className":23434},[1756,3556],[772,23436,23438,23452],{"className":23437},[1760],[772,23439,23441],{"className":23440,"style":23355},[1764],[772,23442,23443,23446],{"style":23358},[772,23444],{"className":23445,"style":23171},[1772],[772,23447,23449],{"className":23448},[1777,23175,6539,1780],[772,23450,22441],{"className":23451,"style":22456},[816,817,1780],[772,23453,3580],{"className":23454},[3579],[772,23456,23458],{"className":23457},[1760],[772,23459,23461],{"className":23460,"style":23377},[1764],[772,23462],{},[772,23464,1353],{"className":23465},[1125,1780],[772,23467,23469,23472],{"className":23468},[816,1780],[772,23470,22758],{"className":23471,"style":22780},[816,817,1780],[772,23473,23475],{"className":23474},[1752],[772,23476,23478,23498],{"className":23477},[1756,3556],[772,23479,23481,23495],{"className":23480},[1760],[772,23482,23484],{"className":23483,"style":23164},[1764],[772,23485,23486,23489],{"style":23167},[772,23487],{"className":23488,"style":23171},[1772],[772,23490,23492],{"className":23491},[1777,23175,6539,1780],[772,23493,1845],{"className":23494},[816,1780],[772,23496,3580],{"className":23497},[3579],[772,23499,23501],{"className":23500},[1760],[772,23502,23504],{"className":23503,"style":23188},[1764],[772,23505],{},[772,23507,23509,23512],{"className":23508},[816,1780],[772,23510,2808],{"className":23511,"style":2823},[816,817,1780],[772,23513,23515],{"className":23514},[1752],[772,23516,23518,23538],{"className":23517},[1756,3556],[772,23519,23521,23535],{"className":23520},[1760],[772,23522,23524],{"className":23523,"style":23355},[1764],[772,23525,23526,23529],{"style":23358},[772,23527],{"className":23528,"style":23171},[1772],[772,23530,23532],{"className":23531},[1777,23175,6539,1780],[772,23533,22471],{"className":23534,"style":22486},[816,817,1780],[772,23536,3580],{"className":23537},[3579],[772,23539,23541],{"className":23540},[1760],[772,23542,23544],{"className":23543,"style":23377},[1764],[772,23545],{},[772,23547,3580],{"className":23548},[3579],[772,23550,23552],{"className":23551},[1760],[772,23553,23556],{"className":23554,"style":23555},[1764],"height:0.4811em;",[772,23557],{},[772,23559],{"className":23560},[1141,5681],[11,23562,23563],{},[772,23564,23566,23654],{"className":23565},[775],[772,23567,23569],{"className":23568},[779],[781,23570,23571],{"xmlns":783},[785,23572,23573,23651],{},[788,23574,23575,23581,23583],{},[3525,23576,23577,23579],{},[791,23578,2839],{},[791,23580,14943],{},[980,23582,982],{},[5599,23584,23585,23627],{},[788,23586,23587,23593,23599,23601,23607,23613,23615,23621],{},[3525,23588,23589,23591],{},[791,23590,22758],{},[1084,23592,2170],{},[3525,23594,23595,23597],{},[791,23596,2839],{},[791,23598,13735],{},[980,23600,1353],{},[3525,23602,23603,23605],{},[791,23604,22758],{},[1084,23606,1872],{},[3525,23608,23609,23611],{},[791,23610,2839],{},[791,23612,22441],{},[980,23614,1353],{},[3525,23616,23617,23619],{},[791,23618,22758],{},[1084,23620,1845],{},[3525,23622,23623,23625],{},[791,23624,2839],{},[791,23626,22471],{},[788,23628,23629,23635,23637,23643,23645],{},[3525,23630,23631,23633],{},[791,23632,22758],{},[1084,23634,2170],{},[980,23636,1353],{},[3525,23638,23639,23641],{},[791,23640,22758],{},[1084,23642,1872],{},[980,23644,1353],{},[3525,23646,23647,23649],{},[791,23648,22758],{},[1084,23650,1845],{},[795,23652,23653],{"encoding":797},"N_P = \\frac{f_1 N_A + f_2 N_B + f_3 N_C}{f_1 + f_2 + f_3}",[772,23655,23657,23712],{"className":23656,"ariaHidden":803},[802],[772,23658,23660,23663,23703,23706,23709],{"className":23659},[807],[772,23661],{"className":23662,"style":3543},[811],[772,23664,23666,23669],{"className":23665},[816],[772,23667,2839],{"className":23668,"style":2854},[816,817],[772,23670,23672],{"className":23671},[1752],[772,23673,23675,23695],{"className":23674},[1756,3556],[772,23676,23678,23692],{"className":23677},[1760],[772,23679,23681],{"className":23680,"style":23077},[1764],[772,23682,23683,23686],{"style":3640},[772,23684],{"className":23685,"style":1773},[1772],[772,23687,23689],{"className":23688},[1777,1778,1779,1780],[772,23690,14943],{"className":23691,"style":14964},[816,817,1780],[772,23693,3580],{"className":23694},[3579],[772,23696,23698],{"className":23697},[1760],[772,23699,23701],{"className":23700,"style":3587},[1764],[772,23702],{},[772,23704],{"className":23705,"style":1004},[1003],[772,23707,982],{"className":23708},[1008],[772,23710],{"className":23711,"style":1004},[1003],[772,23713,23715,23718],{"className":23714},[807],[772,23716],{"className":23717,"style":23115},[811],[772,23719,23721,23724,24150],{"className":23720},[816],[772,23722],{"className":23723},[1113,5681],[772,23725,23727],{"className":23726},[5599],[772,23728,23730,24142],{"className":23729},[1756,3556],[772,23731,23733,24139],{"className":23732},[1760],[772,23734,23736,23873,23881],{"className":23735,"style":23134},[1764],[772,23737,23738,23741],{"style":5697},[772,23739],{"className":23740,"style":5422},[1772],[772,23742,23744],{"className":23743},[1777,1778,1779,1780],[772,23745,23747,23787,23790,23830,23833],{"className":23746},[816,1780],[772,23748,23750,23753],{"className":23749},[816,1780],[772,23751,22758],{"className":23752,"style":22780},[816,817,1780],[772,23754,23756],{"className":23755},[1752],[772,23757,23759,23779],{"className":23758},[1756,3556],[772,23760,23762,23776],{"className":23761},[1760],[772,23763,23765],{"className":23764,"style":23164},[1764],[772,23766,23767,23770],{"style":23167},[772,23768],{"className":23769,"style":23171},[1772],[772,23771,23773],{"className":23772},[1777,23175,6539,1780],[772,23774,2170],{"className":23775},[816,1780],[772,23777,3580],{"className":23778},[3579],[772,23780,23782],{"className":23781},[1760],[772,23783,23785],{"className":23784,"style":23188},[1764],[772,23786],{},[772,23788,1353],{"className":23789},[1125,1780],[772,23791,23793,23796],{"className":23792},[816,1780],[772,23794,22758],{"className":23795,"style":22780},[816,817,1780],[772,23797,23799],{"className":23798},[1752],[772,23800,23802,23822],{"className":23801},[1756,3556],[772,23803,23805,23819],{"className":23804},[1760],[772,23806,23808],{"className":23807,"style":23164},[1764],[772,23809,23810,23813],{"style":23167},[772,23811],{"className":23812,"style":23171},[1772],[772,23814,23816],{"className":23815},[1777,23175,6539,1780],[772,23817,1872],{"className":23818},[816,1780],[772,23820,3580],{"className":23821},[3579],[772,23823,23825],{"className":23824},[1760],[772,23826,23828],{"className":23827,"style":23188},[1764],[772,23829],{},[772,23831,1353],{"className":23832},[1125,1780],[772,23834,23836,23839],{"className":23835},[816,1780],[772,23837,22758],{"className":23838,"style":22780},[816,817,1780],[772,23840,23842],{"className":23841},[1752],[772,23843,23845,23865],{"className":23844},[1756,3556],[772,23846,23848,23862],{"className":23847},[1760],[772,23849,23851],{"className":23850,"style":23164},[1764],[772,23852,23853,23856],{"style":23167},[772,23854],{"className":23855,"style":23171},[1772],[772,23857,23859],{"className":23858},[1777,23175,6539,1780],[772,23860,1845],{"className":23861},[816,1780],[772,23863,3580],{"className":23864},[3579],[772,23866,23868],{"className":23867},[1760],[772,23869,23871],{"className":23870,"style":23188},[1764],[772,23872],{},[772,23874,23875,23878],{"style":5719},[772,23876],{"className":23877,"style":5422},[1772],[772,23879],{"className":23880,"style":5727},[5726],[772,23882,23883,23886],{"style":23287},[772,23884],{"className":23885,"style":5422},[1772],[772,23887,23889],{"className":23888},[1777,1778,1779,1780],[772,23890,23892,23932,23973,23976,24016,24056,24059,24099],{"className":23891},[816,1780],[772,23893,23895,23898],{"className":23894},[816,1780],[772,23896,22758],{"className":23897,"style":22780},[816,817,1780],[772,23899,23901],{"className":23900},[1752],[772,23902,23904,23924],{"className":23903},[1756,3556],[772,23905,23907,23921],{"className":23906},[1760],[772,23908,23910],{"className":23909,"style":23164},[1764],[772,23911,23912,23915],{"style":23167},[772,23913],{"className":23914,"style":23171},[1772],[772,23916,23918],{"className":23917},[1777,23175,6539,1780],[772,23919,2170],{"className":23920},[816,1780],[772,23922,3580],{"className":23923},[3579],[772,23925,23927],{"className":23926},[1760],[772,23928,23930],{"className":23929,"style":23188},[1764],[772,23931],{},[772,23933,23935,23938],{"className":23934},[816,1780],[772,23936,2839],{"className":23937,"style":2854},[816,817,1780],[772,23939,23941],{"className":23940},[1752],[772,23942,23944,23965],{"className":23943},[1756,3556],[772,23945,23947,23962],{"className":23946},[1760],[772,23948,23950],{"className":23949,"style":23355},[1764],[772,23951,23953,23956],{"style":23952},"top:-2.3567em;margin-left:-0.109em;margin-right:0.0714em;",[772,23954],{"className":23955,"style":23171},[1772],[772,23957,23959],{"className":23958},[1777,23175,6539,1780],[772,23960,13735],{"className":23961},[816,817,1780],[772,23963,3580],{"className":23964},[3579],[772,23966,23968],{"className":23967},[1760],[772,23969,23971],{"className":23970,"style":23377},[1764],[772,23972],{},[772,23974,1353],{"className":23975},[1125,1780],[772,23977,23979,23982],{"className":23978},[816,1780],[772,23980,22758],{"className":23981,"style":22780},[816,817,1780],[772,23983,23985],{"className":23984},[1752],[772,23986,23988,24008],{"className":23987},[1756,3556],[772,23989,23991,24005],{"className":23990},[1760],[772,23992,23994],{"className":23993,"style":23164},[1764],[772,23995,23996,23999],{"style":23167},[772,23997],{"className":23998,"style":23171},[1772],[772,24000,24002],{"className":24001},[1777,23175,6539,1780],[772,24003,1872],{"className":24004},[816,1780],[772,24006,3580],{"className":24007},[3579],[772,24009,24011],{"className":24010},[1760],[772,24012,24014],{"className":24013,"style":23188},[1764],[772,24015],{},[772,24017,24019,24022],{"className":24018},[816,1780],[772,24020,2839],{"className":24021,"style":2854},[816,817,1780],[772,24023,24025],{"className":24024},[1752],[772,24026,24028,24048],{"className":24027},[1756,3556],[772,24029,24031,24045],{"className":24030},[1760],[772,24032,24034],{"className":24033,"style":23355},[1764],[772,24035,24036,24039],{"style":23952},[772,24037],{"className":24038,"style":23171},[1772],[772,24040,24042],{"className":24041},[1777,23175,6539,1780],[772,24043,22441],{"className":24044,"style":22456},[816,817,1780],[772,24046,3580],{"className":24047},[3579],[772,24049,24051],{"className":24050},[1760],[772,24052,24054],{"className":24053,"style":23377},[1764],[772,24055],{},[772,24057,1353],{"className":24058},[1125,1780],[772,24060,24062,24065],{"className":24061},[816,1780],[772,24063,22758],{"className":24064,"style":22780},[816,817,1780],[772,24066,24068],{"className":24067},[1752],[772,24069,24071,24091],{"className":24070},[1756,3556],[772,24072,24074,24088],{"className":24073},[1760],[772,24075,24077],{"className":24076,"style":23164},[1764],[772,24078,24079,24082],{"style":23167},[772,24080],{"className":24081,"style":23171},[1772],[772,24083,24085],{"className":24084},[1777,23175,6539,1780],[772,24086,1845],{"className":24087},[816,1780],[772,24089,3580],{"className":24090},[3579],[772,24092,24094],{"className":24093},[1760],[772,24095,24097],{"className":24096,"style":23188},[1764],[772,24098],{},[772,24100,24102,24105],{"className":24101},[816,1780],[772,24103,2839],{"className":24104,"style":2854},[816,817,1780],[772,24106,24108],{"className":24107},[1752],[772,24109,24111,24131],{"className":24110},[1756,3556],[772,24112,24114,24128],{"className":24113},[1760],[772,24115,24117],{"className":24116,"style":23355},[1764],[772,24118,24119,24122],{"style":23952},[772,24120],{"className":24121,"style":23171},[1772],[772,24123,24125],{"className":24124},[1777,23175,6539,1780],[772,24126,22471],{"className":24127,"style":22486},[816,817,1780],[772,24129,3580],{"className":24130},[3579],[772,24132,24134],{"className":24133},[1760],[772,24135,24137],{"className":24136,"style":23377},[1764],[772,24138],{},[772,24140,3580],{"className":24141},[3579],[772,24143,24145],{"className":24144},[1760],[772,24146,24148],{"className":24147,"style":23555},[1764],[772,24149],{},[772,24151],{"className":24152},[1141,5681],[11,24154,24155,24156,24184,24185,24224,24225,24253],{},"Point ",[772,24157,24159,24172],{"className":24158},[775],[772,24160,24162],{"className":24161},[779],[781,24163,24164],{"xmlns":783},[785,24165,24166,24170],{},[788,24167,24168],{},[791,24169,14943],{},[795,24171,14943],{"encoding":797},[772,24173,24175],{"className":24174,"ariaHidden":803},[802],[772,24176,24178,24181],{"className":24177},[807],[772,24179],{"className":24180,"style":2758},[811],[772,24182,14943],{"className":24183,"style":14964},[816,817]," is the weighted balance of triangle ",[772,24186,24188,24206],{"className":24187},[775],[772,24189,24191],{"className":24190},[779],[781,24192,24193],{"xmlns":783},[785,24194,24195,24203],{},[788,24196,24197,24199,24201],{},[791,24198,13735],{},[791,24200,22441],{},[791,24202,22471],{},[795,24204,24205],{"encoding":797},"ABC",[772,24207,24209],{"className":24208,"ariaHidden":803},[802],[772,24210,24212,24215,24218,24221],{"className":24211},[807],[772,24213],{"className":24214,"style":2758},[811],[772,24216,13735],{"className":24217},[816,817],[772,24219,22441],{"className":24220,"style":22456},[816,817],[772,24222,22471],{"className":24223,"style":22486},[816,817]," — the formula handles ",[772,24226,24228,24241],{"className":24227},[775],[772,24229,24231],{"className":24230},[779],[781,24232,24233],{"xmlns":783},[785,24234,24235,24239],{},[788,24236,24237],{},[791,24238,14943],{},[795,24240,14943],{"encoding":797},[772,24242,24244],{"className":24243,"ariaHidden":803},[802],[772,24245,24247,24250],{"className":24246},[807],[772,24248],{"className":24249,"style":2758},[811],[772,24251,14943],{"className":24252,"style":14964},[816,817]," inside or outside the reference triangle.",[323,24255,24257],{"id":24256},"control-geometry-and-the-danger-circle","Control Geometry and the Danger Circle",[11,24259,24260],{},"Resection precision depends on the spatial arrangement of control and the unknown station. Poor geometry magnifies angular errors or causes failure.",[706,24262,24264],{"id":24263},"point-ordering-and-angle-closure","Point ordering and angle closure",[11,24266,24267,24268,2896,24296,2896,24324,12019,24352,24355,24356,24394,24395,24423,24424,2925],{},"Sequence control points ",[772,24269,24271,24284],{"className":24270},[775],[772,24272,24274],{"className":24273},[779],[781,24275,24276],{"xmlns":783},[785,24277,24278,24282],{},[788,24279,24280],{},[791,24281,13735],{},[795,24283,13735],{"encoding":797},[772,24285,24287],{"className":24286,"ariaHidden":803},[802],[772,24288,24290,24293],{"className":24289},[807],[772,24291],{"className":24292,"style":2758},[811],[772,24294,13735],{"className":24295},[816,817],[772,24297,24299,24312],{"className":24298},[775],[772,24300,24302],{"className":24301},[779],[781,24303,24304],{"xmlns":783},[785,24305,24306,24310],{},[788,24307,24308],{},[791,24309,22441],{},[795,24311,22441],{"encoding":797},[772,24313,24315],{"className":24314,"ariaHidden":803},[802],[772,24316,24318,24321],{"className":24317},[807],[772,24319],{"className":24320,"style":2758},[811],[772,24322,22441],{"className":24323,"style":22456},[816,817],[772,24325,24327,24340],{"className":24326},[775],[772,24328,24330],{"className":24329},[779],[781,24331,24332],{"xmlns":783},[785,24333,24334,24338],{},[788,24335,24336],{},[791,24337,22471],{},[795,24339,22471],{"encoding":797},[772,24341,24343],{"className":24342,"ariaHidden":803},[802],[772,24344,24346,24349],{"className":24345},[807],[772,24347],{"className":24348,"style":2758},[811],[772,24350,22471],{"className":24351,"style":22486},[816,817],[318,24353,24354],{},"counterclockwise"," around triangle ",[772,24357,24359,24376],{"className":24358},[775],[772,24360,24362],{"className":24361},[779],[781,24363,24364],{"xmlns":783},[785,24365,24366,24374],{},[788,24367,24368,24370,24372],{},[791,24369,13735],{},[791,24371,22441],{},[791,24373,22471],{},[795,24375,24205],{"encoding":797},[772,24377,24379],{"className":24378,"ariaHidden":803},[802],[772,24380,24382,24385,24388,24391],{"className":24381},[807],[772,24383],{"className":24384,"style":2758},[811],[772,24386,13735],{"className":24387},[816,817],[772,24389,22441],{"className":24390,"style":22456},[816,817],[772,24392,22471],{"className":24393,"style":22486},[816,817]," (right-handed grid: easting = ",[772,24396,24398,24411],{"className":24397},[775],[772,24399,24401],{"className":24400},[779],[781,24402,24403],{"xmlns":783},[785,24404,24405,24409],{},[788,24406,24407],{},[791,24408,11099],{},[795,24410,11099],{"encoding":797},[772,24412,24414],{"className":24413,"ariaHidden":803},[802],[772,24415,24417,24420],{"className":24416},[807],[772,24418],{"className":24419,"style":845},[811],[772,24421,11099],{"className":24422},[816,817],", northing = ",[772,24425,24427,24440],{"className":24426},[775],[772,24428,24430],{"className":24429},[779],[781,24431,24432],{"xmlns":783},[785,24433,24434,24438],{},[788,24435,24436],{},[791,24437,11129],{},[795,24439,11129],{"encoding":797},[772,24441,24443],{"className":24442,"ariaHidden":803},[802],[772,24444,24446,24449],{"className":24445},[807],[772,24447],{"className":24448,"style":11141},[811],[772,24450,11129],{"className":24451,"style":6125},[816,817],[11,24453,24454,24455,24483,24484,4135],{},"At unknown station ",[772,24456,24458,24471],{"className":24457},[775],[772,24459,24461],{"className":24460},[779],[781,24462,24463],{"xmlns":783},[785,24464,24465,24469],{},[788,24466,24467],{},[791,24468,14943],{},[795,24470,14943],{"encoding":797},[772,24472,24474],{"className":24473,"ariaHidden":803},[802],[772,24475,24477,24480],{"className":24476},[807],[772,24478],{"className":24479,"style":2758},[811],[772,24481,14943],{"className":24482,"style":14964},[816,817],", measure horizon angles clockwise around the full ",[772,24485,24487,24502],{"className":24486},[775],[772,24488,24490],{"className":24489},[779],[781,24491,24492],{"xmlns":783},[785,24493,24494,24500],{},[788,24495,24496,24498],{},[1084,24497,1468],{},[791,24499,907],{"mathvariant":906},[795,24501,1495],{"encoding":797},[772,24503,24505],{"className":24504,"ariaHidden":803},[802],[772,24506,24508,24511],{"className":24507},[807],[772,24509],{"className":24510,"style":812},[811],[772,24512,1495],{"className":24513},[816],[335,24515,24516,24650,24783],{},[338,24517,24518,24546,24547,24593,24594,6126,24622,1096],{},[772,24519,24521,24534],{"className":24520},[775],[772,24522,24524],{"className":24523},[779],[781,24525,24526],{"xmlns":783},[785,24527,24528,24532],{},[788,24529,24530],{},[791,24531,5554],{},[795,24533,5557],{"encoding":797},[772,24535,24537],{"className":24536,"ariaHidden":803},[802],[772,24538,24540,24543],{"className":24539},[807],[772,24541],{"className":24542,"style":845},[811],[772,24544,5554],{"className":24545,"style":5570},[816,817]," = ",[772,24548,24550,24571],{"className":24549},[775],[772,24551,24553],{"className":24552},[779],[781,24554,24555],{"xmlns":783},[785,24556,24557,24568],{},[788,24558,24559,24562,24564,24566],{},[791,24560,24561],{"mathvariant":906},"∠",[791,24563,13735],{},[791,24565,14943],{},[791,24567,22441],{},[795,24569,24570],{"encoding":797},"\\angle APB",[772,24572,24574],{"className":24573,"ariaHidden":803},[802],[772,24575,24577,24581,24584,24587,24590],{"className":24576},[807],[772,24578],{"className":24579,"style":24580},[811],"height:0.6922em;",[772,24582,24561],{"className":24583},[816],[772,24585,13735],{"className":24586},[816,817],[772,24588,14943],{"className":24589,"style":14964},[816,817],[772,24591,22441],{"className":24592,"style":22456},[816,817]," (from ",[772,24595,24597,24610],{"className":24596},[775],[772,24598,24600],{"className":24599},[779],[781,24601,24602],{"xmlns":783},[785,24603,24604,24608],{},[788,24605,24606],{},[791,24607,13735],{},[795,24609,13735],{"encoding":797},[772,24611,24613],{"className":24612,"ariaHidden":803},[802],[772,24614,24616,24619],{"className":24615},[807],[772,24617],{"className":24618,"style":2758},[811],[772,24620,13735],{"className":24621},[816,817],[772,24623,24625,24638],{"className":24624},[775],[772,24626,24628],{"className":24627},[779],[781,24629,24630],{"xmlns":783},[785,24631,24632,24636],{},[788,24633,24634],{},[791,24635,22441],{},[795,24637,22441],{"encoding":797},[772,24639,24641],{"className":24640,"ariaHidden":803},[802],[772,24642,24644,24647],{"className":24643},[807],[772,24645],{"className":24646,"style":2758},[811],[772,24648,22441],{"className":24649,"style":22456},[816,817],[338,24651,24652,24546,24683,24593,24727,6126,24755,1096],{},[772,24653,24655,24670],{"className":24654},[775],[772,24656,24658],{"className":24657},[779],[781,24659,24660],{"xmlns":783},[785,24661,24662,24667],{},[788,24663,24664],{},[791,24665,24666],{},"β",[795,24668,24669],{"encoding":797},"\\beta",[772,24671,24673],{"className":24672,"ariaHidden":803},[802],[772,24674,24676,24679],{"className":24675},[807],[772,24677],{"className":24678,"style":22773},[811],[772,24680,24666],{"className":24681,"style":24682},[816,817],"margin-right:0.0528em;",[772,24684,24686,24706],{"className":24685},[775],[772,24687,24689],{"className":24688},[779],[781,24690,24691],{"xmlns":783},[785,24692,24693,24703],{},[788,24694,24695,24697,24699,24701],{},[791,24696,24561],{"mathvariant":906},[791,24698,22441],{},[791,24700,14943],{},[791,24702,22471],{},[795,24704,24705],{"encoding":797},"\\angle BPC",[772,24707,24709],{"className":24708,"ariaHidden":803},[802],[772,24710,24712,24715,24718,24721,24724],{"className":24711},[807],[772,24713],{"className":24714,"style":24580},[811],[772,24716,24561],{"className":24717},[816],[772,24719,22441],{"className":24720,"style":22456},[816,817],[772,24722,14943],{"className":24723,"style":14964},[816,817],[772,24725,22471],{"className":24726,"style":22486},[816,817],[772,24728,24730,24743],{"className":24729},[775],[772,24731,24733],{"className":24732},[779],[781,24734,24735],{"xmlns":783},[785,24736,24737,24741],{},[788,24738,24739],{},[791,24740,22441],{},[795,24742,22441],{"encoding":797},[772,24744,24746],{"className":24745,"ariaHidden":803},[802],[772,24747,24749,24752],{"className":24748},[807],[772,24750],{"className":24751,"style":2758},[811],[772,24753,22441],{"className":24754,"style":22456},[816,817],[772,24756,24758,24771],{"className":24757},[775],[772,24759,24761],{"className":24760},[779],[781,24762,24763],{"xmlns":783},[785,24764,24765,24769],{},[788,24766,24767],{},[791,24768,22471],{},[795,24770,22471],{"encoding":797},[772,24772,24774],{"className":24773,"ariaHidden":803},[802],[772,24775,24777,24780],{"className":24776},[807],[772,24778],{"className":24779,"style":2758},[811],[772,24781,22471],{"className":24782,"style":22486},[816,817],[338,24784,24785,24546,24816,24593,24860,6126,24888,1096],{},[772,24786,24788,24803],{"className":24787},[775],[772,24789,24791],{"className":24790},[779],[781,24792,24793],{"xmlns":783},[785,24794,24795,24800],{},[788,24796,24797],{},[791,24798,24799],{},"γ",[795,24801,24802],{"encoding":797},"\\gamma",[772,24804,24806],{"className":24805,"ariaHidden":803},[802],[772,24807,24809,24812],{"className":24808},[807],[772,24810],{"className":24811,"style":11141},[811],[772,24813,24799],{"className":24814,"style":24815},[816,817],"margin-right:0.0556em;",[772,24817,24819,24839],{"className":24818},[775],[772,24820,24822],{"className":24821},[779],[781,24823,24824],{"xmlns":783},[785,24825,24826,24836],{},[788,24827,24828,24830,24832,24834],{},[791,24829,24561],{"mathvariant":906},[791,24831,22471],{},[791,24833,14943],{},[791,24835,13735],{},[795,24837,24838],{"encoding":797},"\\angle CPA",[772,24840,24842],{"className":24841,"ariaHidden":803},[802],[772,24843,24845,24848,24851,24854,24857],{"className":24844},[807],[772,24846],{"className":24847,"style":24580},[811],[772,24849,24561],{"className":24850},[816],[772,24852,22471],{"className":24853,"style":22486},[816,817],[772,24855,14943],{"className":24856,"style":14964},[816,817],[772,24858,13735],{"className":24859},[816,817],[772,24861,24863,24876],{"className":24862},[775],[772,24864,24866],{"className":24865},[779],[781,24867,24868],{"xmlns":783},[785,24869,24870,24874],{},[788,24871,24872],{},[791,24873,22471],{},[795,24875,22471],{"encoding":797},[772,24877,24879],{"className":24878,"ariaHidden":803},[802],[772,24880,24882,24885],{"className":24881},[807],[772,24883],{"className":24884,"style":2758},[811],[772,24886,22471],{"className":24887,"style":22486},[816,817],[772,24889,24891,24904],{"className":24890},[775],[772,24892,24894],{"className":24893},[779],[781,24895,24896],{"xmlns":783},[785,24897,24898,24902],{},[788,24899,24900],{},[791,24901,13735],{},[795,24903,13735],{"encoding":797},[772,24905,24907],{"className":24906,"ariaHidden":803},[802],[772,24908,24910,24913],{"className":24909},[807],[772,24911],{"className":24912,"style":2758},[811],[772,24914,13735],{"className":24915},[816,817],[11,24917,24918],{},"Closure requirement:",[11,24920,24921],{},[772,24922,24924,24954],{"className":24923},[775],[772,24925,24927],{"className":24926},[779],[781,24928,24929],{"xmlns":783},[785,24930,24931,24951],{},[788,24932,24933,24935,24937,24939,24941,24943,24945],{},[791,24934,5554],{},[980,24936,1353],{},[791,24938,24666],{},[980,24940,1353],{},[791,24942,24799],{},[980,24944,982],{},[1636,24946,24947,24949],{},[1084,24948,1468],{},[980,24950,1642],{},[795,24952,24953],{"encoding":797},"\\alpha + \\beta + \\gamma = 360^\\circ",[772,24955,24957,24975,24993,25011],{"className":24956,"ariaHidden":803},[802],[772,24958,24960,24963,24966,24969,24972],{"className":24959},[807],[772,24961],{"className":24962,"style":6118},[811],[772,24964,5554],{"className":24965,"style":5570},[816,817],[772,24967],{"className":24968,"style":1121},[1003],[772,24970,1353],{"className":24971},[1125],[772,24973],{"className":24974,"style":1121},[1003],[772,24976,24978,24981,24984,24987,24990],{"className":24977},[807],[772,24979],{"className":24980,"style":22773},[811],[772,24982,24666],{"className":24983,"style":24682},[816,817],[772,24985],{"className":24986,"style":1121},[1003],[772,24988,1353],{"className":24989},[1125],[772,24991],{"className":24992,"style":1121},[1003],[772,24994,24996,24999,25002,25005,25008],{"className":24995},[807],[772,24997],{"className":24998,"style":11141},[811],[772,25000,24799],{"className":25001,"style":24815},[816,817],[772,25003],{"className":25004,"style":1004},[1003],[772,25006,982],{"className":25007},[1008],[772,25009],{"className":25010,"style":1004},[1003],[772,25012,25014,25017,25020],{"className":25013},[807],[772,25015],{"className":25016,"style":1765},[811],[772,25018,1681],{"className":25019},[816],[772,25021,25023,25026],{"className":25022},[816],[772,25024,1748],{"className":25025},[816],[772,25027,25029],{"className":25028},[1752],[772,25030,25032],{"className":25031},[1756],[772,25033,25035],{"className":25034},[1760],[772,25036,25038],{"className":25037,"style":1765},[1764],[772,25039,25040,25043],{"style":1768},[772,25041],{"className":25042,"style":1773},[1772],[772,25044,25046],{"className":25045},[1777,1778,1779,1780],[772,25047,1642],{"className":25048},[1125,1780],[706,25050,25052],{"id":25051},"the-danger-circle","The danger circle",[11,25054,25055,25056,25059,25060,25105,25106,25134],{},"The critical singularity is the ",[318,25057,25058],{},"danger circle"," (circumcircle of ",[772,25061,25063,25084],{"className":25062},[775],[772,25064,25066],{"className":25065},[779],[781,25067,25068],{"xmlns":783},[785,25069,25070,25081],{},[788,25071,25072,25075,25077,25079],{},[791,25073,25074],{"mathvariant":906},"△",[791,25076,13735],{},[791,25078,22441],{},[791,25080,22471],{},[795,25082,25083],{"encoding":797},"\\triangle ABC",[772,25085,25087],{"className":25086,"ariaHidden":803},[802],[772,25088,25090,25093,25096,25099,25102],{"className":25089},[807],[772,25091],{"className":25092,"style":22773},[811],[772,25094,25074],{"className":25095},[816],[772,25097,13735],{"className":25098},[816,817],[772,25100,22441],{"className":25101,"style":22456},[816,817],[772,25103,22471],{"className":25104,"style":22486},[816,817],"). If ",[772,25107,25109,25122],{"className":25108},[775],[772,25110,25112],{"className":25111},[779],[781,25113,25114],{"xmlns":783},[785,25115,25116,25120],{},[788,25117,25118],{},[791,25119,14943],{},[795,25121,14943],{"encoding":797},[772,25123,25125],{"className":25124,"ariaHidden":803},[802],[772,25126,25128,25131],{"className":25127},[807],[772,25129],{"className":25130,"style":2758},[811],[772,25132,14943],{"className":25133,"style":14964},[816,817]," lies on that circle, infinitely many positions produce the same observed angles (inscribed angle theorem).",[11,25136,25137],{},"Algebraically, barycentric denominators approach zero:",[11,25139,25140],{},[772,25141,25143,25214],{"className":25142},[775],[772,25144,25146],{"className":25145},[779],[781,25147,25148],{"xmlns":783},[785,25149,25150,25211],{},[788,25151,25152,25155,25157,25159,25161,25163,25165,25167,25169,25171,25173,25175,25177,25179,25181,25183,25185,25187,25189,25191,25193,25195,25197,25199,25201,25203,25205,25207,25209],{},[791,25153,25154],{},"cot",[980,25156,3252],{},[791,25158,13735],{},[980,25160,1091],{},[791,25162,25154],{},[980,25164,3252],{},[791,25166,24666],{},[980,25168,17398],{},[1084,25170,1748],{},[1003,25172],{"width":12337},[791,25174,25154],{},[980,25176,3252],{},[791,25178,22441],{},[980,25180,1091],{},[791,25182,25154],{},[980,25184,3252],{},[791,25186,24799],{},[980,25188,17398],{},[1084,25190,1748],{},[1003,25192],{"width":12337},[791,25194,25154],{},[980,25196,3252],{},[791,25198,22471],{},[980,25200,1091],{},[791,25202,25154],{},[980,25204,3252],{},[791,25206,5554],{},[980,25208,17398],{},[1084,25210,1748],{},[795,25212,25213],{"encoding":797},"\\cot A - \\cot\\beta \\to 0 \\qquad \\cot B - \\cot\\gamma \\to 0 \\qquad \\cot C - \\cot\\alpha \\to 0",[772,25215,25217,25241,25265,25298,25323,25356,25381],{"className":25216,"ariaHidden":803},[802],[772,25218,25220,25223,25226,25229,25232,25235,25238],{"className":25219},[807],[772,25221],{"className":25222,"style":11398},[811],[772,25224,25154],{"className":25225},[3298],[772,25227],{"className":25228,"style":3294},[1003],[772,25230,13735],{"className":25231},[816,817],[772,25233],{"className":25234,"style":1121},[1003],[772,25236,1091],{"className":25237},[1125],[772,25239],{"className":25240,"style":1121},[1003],[772,25242,25244,25247,25250,25253,25256,25259,25262],{"className":25243},[807],[772,25245],{"className":25246,"style":22773},[811],[772,25248,25154],{"className":25249},[3298],[772,25251],{"className":25252,"style":3294},[1003],[772,25254,24666],{"className":25255,"style":24682},[816,817],[772,25257],{"className":25258,"style":1004},[1003],[772,25260,17398],{"className":25261},[1008],[772,25263],{"className":25264,"style":1004},[1003],[772,25266,25268,25271,25274,25277,25280,25283,25286,25289,25292,25295],{"className":25267},[807],[772,25269],{"className":25270,"style":11398},[811],[772,25272,1748],{"className":25273},[816],[772,25275],{"className":25276,"style":12489},[1003],[772,25278],{"className":25279,"style":3294},[1003],[772,25281,25154],{"className":25282},[3298],[772,25284],{"className":25285,"style":3294},[1003],[772,25287,22441],{"className":25288,"style":22456},[816,817],[772,25290],{"className":25291,"style":1121},[1003],[772,25293,1091],{"className":25294},[1125],[772,25296],{"className":25297,"style":1121},[1003],[772,25299,25301,25305,25308,25311,25314,25317,25320],{"className":25300},[807],[772,25302],{"className":25303,"style":25304},[811],"height:0.8095em;vertical-align:-0.1944em;",[772,25306,25154],{"className":25307},[3298],[772,25309],{"className":25310,"style":3294},[1003],[772,25312,24799],{"className":25313,"style":24815},[816,817],[772,25315],{"className":25316,"style":1004},[1003],[772,25318,17398],{"className":25319},[1008],[772,25321],{"className":25322,"style":1004},[1003],[772,25324,25326,25329,25332,25335,25338,25341,25344,25347,25350,25353],{"className":25325},[807],[772,25327],{"className":25328,"style":11398},[811],[772,25330,1748],{"className":25331},[816],[772,25333],{"className":25334,"style":12489},[1003],[772,25336],{"className":25337,"style":3294},[1003],[772,25339,25154],{"className":25340},[3298],[772,25342],{"className":25343,"style":3294},[1003],[772,25345,22471],{"className":25346,"style":22486},[816,817],[772,25348],{"className":25349,"style":1121},[1003],[772,25351,1091],{"className":25352},[1125],[772,25354],{"className":25355,"style":1121},[1003],[772,25357,25359,25363,25366,25369,25372,25375,25378],{"className":25358},[807],[772,25360],{"className":25361,"style":25362},[811],"height:0.6151em;",[772,25364,25154],{"className":25365},[3298],[772,25367],{"className":25368,"style":3294},[1003],[772,25370,5554],{"className":25371,"style":5570},[816,817],[772,25373],{"className":25374,"style":1004},[1003],[772,25376,17398],{"className":25377},[1008],[772,25379],{"className":25380,"style":1004},[1003],[772,25382,25384,25387],{"className":25383},[807],[772,25385],{"className":25386,"style":5899},[811],[772,25388,1748],{"className":25389},[816],[706,25391,25393],{"id":25392},"ensuring-geometric-strength","Ensuring geometric strength",[335,25395,25396,25474,25643],{},[338,25397,25398,12019,25401,25429,25430,25473],{},[318,25399,25400],{},"Enclosure:",[772,25402,25404,25417],{"className":25403},[775],[772,25405,25407],{"className":25406},[779],[781,25408,25409],{"xmlns":783},[785,25410,25411,25415],{},[788,25412,25413],{},[791,25414,14943],{},[795,25416,14943],{"encoding":797},[772,25418,25420],{"className":25419,"ariaHidden":803},[802],[772,25421,25423,25426],{"className":25422},[807],[772,25424],{"className":25425,"style":2758},[811],[772,25427,14943],{"className":25428,"style":14964},[816,817]," inside ",[772,25431,25433,25452],{"className":25432},[775],[772,25434,25436],{"className":25435},[779],[781,25437,25438],{"xmlns":783},[785,25439,25440,25450],{},[788,25441,25442,25444,25446,25448],{},[791,25443,25074],{"mathvariant":906},[791,25445,13735],{},[791,25447,22441],{},[791,25449,22471],{},[795,25451,25083],{"encoding":797},[772,25453,25455],{"className":25454,"ariaHidden":803},[802],[772,25456,25458,25461,25464,25467,25470],{"className":25457},[807],[772,25459],{"className":25460,"style":22773},[811],[772,25462,25074],{"className":25463},[816],[772,25465,13735],{"className":25466},[816,817],[772,25468,22441],{"className":25469,"style":22456},[816,817],[772,25471,22471],{"className":25472,"style":22486},[816,817]," is most stable (positive weights, robust to noise).",[338,25475,25476,25479,25480,25508,25509,25537,25538,25566,25567,25595,25596,25642],{},[318,25477,25478],{},"Outside triangle:"," avoid placing ",[772,25481,25483,25496],{"className":25482},[775],[772,25484,25486],{"className":25485},[779],[781,25487,25488],{"xmlns":783},[785,25489,25490,25494],{},[788,25491,25492],{},[791,25493,14943],{},[795,25495,14943],{"encoding":797},[772,25497,25499],{"className":25498,"ariaHidden":803},[802],[772,25500,25502,25505],{"className":25501},[807],[772,25503],{"className":25504,"style":2758},[811],[772,25506,14943],{"className":25507,"style":14964},[816,817]," on or near the circumcircle. A quick check: if interior angle at ",[772,25510,25512,25525],{"className":25511},[775],[772,25513,25515],{"className":25514},[779],[781,25516,25517],{"xmlns":783},[785,25518,25519,25523],{},[788,25520,25521],{},[791,25522,22441],{},[795,25524,22441],{"encoding":797},[772,25526,25528],{"className":25527,"ariaHidden":803},[802],[772,25529,25531,25534],{"className":25530},[807],[772,25532],{"className":25533,"style":2758},[811],[772,25535,22441],{"className":25536,"style":22456},[816,817]," plus observed ",[772,25539,25541,25554],{"className":25540},[775],[772,25542,25544],{"className":25543},[779],[781,25545,25546],{"xmlns":783},[785,25547,25548,25552],{},[788,25549,25550],{},[791,25551,24799],{},[795,25553,24802],{"encoding":797},[772,25555,25557],{"className":25556,"ariaHidden":803},[802],[772,25558,25560,25563],{"className":25559},[807],[772,25561],{"className":25562,"style":11141},[811],[772,25564,24799],{"className":25565,"style":24815},[816,817]," (opposite ",[772,25568,25570,25583],{"className":25569},[775],[772,25571,25573],{"className":25572},[779],[781,25574,25575],{"xmlns":783},[785,25576,25577,25581],{},[788,25578,25579],{},[791,25580,22441],{},[795,25582,22441],{"encoding":797},[772,25584,25586],{"className":25585,"ariaHidden":803},[802],[772,25587,25589,25592],{"className":25588},[807],[772,25590],{"className":25591,"style":2758},[811],[772,25593,22441],{"className":25594,"style":22456},[816,817],") ",[772,25597,25599,25617],{"className":25598},[775],[772,25600,25602],{"className":25601},[779],[781,25603,25604],{"xmlns":783},[785,25605,25606,25614],{},[788,25607,25608,25610,25612],{},[980,25609,7437],{},[1084,25611,1086],{},[791,25613,907],{"mathvariant":906},[795,25615,25616],{"encoding":797},"\\approx 180°",[772,25618,25620,25633],{"className":25619,"ariaHidden":803},[802],[772,25621,25623,25627,25630],{"className":25622},[807],[772,25624],{"className":25625,"style":25626},[811],"height:0.4831em;",[772,25628,7437],{"className":25629},[1008],[772,25631],{"className":25632,"style":1004},[1003],[772,25634,25636,25639],{"className":25635},[807],[772,25637],{"className":25638,"style":812},[811],[772,25640,1117],{"className":25641},[816],", you are near the danger circle.",[338,25644,25645,25648],{},[318,25646,25647],{},"Collinearity:"," three collinear control points give degenerate geometry — extremely unstable.",[2857,25650,25653],{"className":25651,"code":25652,"language":1709},[2860],"              A -------- B\n             \u002F    *P?    \\\n            \u002F   (danger    \\\n           C --- circle) --- \n",[2863,25654,25652],{"__ignoreMap":15},[323,25656,25658],{"id":25657},"field-observation-practice","Field Observation Practice",[706,25660,25662],{"id":25661},"face-i-and-face-ii","Face I and Face II",[11,25664,25665,25666,25669,25670,25673],{},"Eliminate collimation, tilt-axis, and eccentricity errors by observing each target in ",[318,25667,25668],{},"Face I"," (direct) and ",[318,25671,25672],{},"Face II"," (reverse) and averaging.",[706,25675,25677],{"id":25676},"horizon-closure","Horizon closure",[11,25679,25680,25681,25776,25777,4135],{},"After sighting ",[772,25682,25684,25710],{"className":25683},[775],[772,25685,25687],{"className":25686},[779],[781,25688,25689],{"xmlns":783},[785,25690,25691,25707],{},[788,25692,25693,25695,25697,25699,25701,25703,25705],{},[791,25694,13735],{},[980,25696,17398],{},[791,25698,22441],{},[980,25700,17398],{},[791,25702,22471],{},[980,25704,17398],{},[791,25706,13735],{},[795,25708,25709],{"encoding":797},"A \\to B \\to C \\to A",[772,25711,25713,25731,25749,25767],{"className":25712,"ariaHidden":803},[802],[772,25714,25716,25719,25722,25725,25728],{"className":25715},[807],[772,25717],{"className":25718,"style":2758},[811],[772,25720,13735],{"className":25721},[816,817],[772,25723],{"className":25724,"style":1004},[1003],[772,25726,17398],{"className":25727},[1008],[772,25729],{"className":25730,"style":1004},[1003],[772,25732,25734,25737,25740,25743,25746],{"className":25733},[807],[772,25735],{"className":25736,"style":2758},[811],[772,25738,22441],{"className":25739,"style":22456},[816,817],[772,25741],{"className":25742,"style":1004},[1003],[772,25744,17398],{"className":25745},[1008],[772,25747],{"className":25748,"style":1004},[1003],[772,25750,25752,25755,25758,25761,25764],{"className":25751},[807],[772,25753],{"className":25754,"style":2758},[811],[772,25756,22471],{"className":25757,"style":22486},[816,817],[772,25759],{"className":25760,"style":1004},[1003],[772,25762,17398],{"className":25763},[1008],[772,25765],{"className":25766,"style":1004},[1003],[772,25768,25770,25773],{"className":25769},[807],[772,25771],{"className":25772,"style":2758},[811],[772,25774,13735],{"className":25775},[816,817],", compare the angle sum to ",[772,25778,25780,25795],{"className":25779},[775],[772,25781,25783],{"className":25782},[779],[781,25784,25785],{"xmlns":783},[785,25786,25787,25793],{},[788,25788,25789,25791],{},[1084,25790,1468],{},[791,25792,907],{"mathvariant":906},[795,25794,1495],{"encoding":797},[772,25796,25798],{"className":25797,"ariaHidden":803},[802],[772,25799,25801,25804],{"className":25800},[807],[772,25802],{"className":25803,"style":812},[811],[772,25805,1495],{"className":25806},[816],[11,25808,25809],{},[772,25810,25812,25854],{"className":25811},[775],[772,25813,25815],{"className":25814},[779],[781,25816,25817],{"xmlns":783},[785,25818,25819,25851],{},[788,25820,25821,25827,25829,25835,25837,25839,25841,25843,25845,25847,25849],{},[3525,25822,25823,25825],{},[791,25824,10305],{},[791,25826,793],{},[980,25828,982],{},[1636,25830,25831,25833],{},[1084,25832,1468],{},[980,25834,1642],{},[980,25836,1091],{},[980,25838,1082],{"stretchy":1081},[791,25840,5554],{},[980,25842,1353],{},[791,25844,24666],{},[980,25846,1353],{},[791,25848,24799],{},[980,25850,1096],{"stretchy":1081},[795,25852,25853],{"encoding":797},"\\delta_\\theta = 360^\\circ - (\\alpha + \\beta + \\gamma)",[772,25855,25857,25913,25960,25981,25999],{"className":25856,"ariaHidden":803},[802],[772,25858,25860,25863,25904,25907,25910],{"className":25859},[807],[772,25861],{"className":25862,"style":5959},[811],[772,25864,25866,25869],{"className":25865},[816],[772,25867,10305],{"className":25868,"style":10352},[816,817],[772,25870,25872],{"className":25871},[1752],[772,25873,25875,25896],{"className":25874},[1756,3556],[772,25876,25878,25893],{"className":25877},[1760],[772,25879,25881],{"className":25880,"style":5978},[1764],[772,25882,25884,25887],{"style":25883},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[772,25885],{"className":25886,"style":1773},[1772],[772,25888,25890],{"className":25889},[1777,1778,1779,1780],[772,25891,793],{"className":25892,"style":818},[816,817,1780],[772,25894,3580],{"className":25895},[3579],[772,25897,25899],{"className":25898},[1760],[772,25900,25902],{"className":25901,"style":3587},[1764],[772,25903],{},[772,25905],{"className":25906,"style":1004},[1003],[772,25908,982],{"className":25909},[1008],[772,25911],{"className":25912,"style":1004},[1003],[772,25914,25916,25919,25922,25951,25954,25957],{"className":25915},[807],[772,25917],{"className":25918,"style":7514},[811],[772,25920,1681],{"className":25921},[816],[772,25923,25925,25928],{"className":25924},[816],[772,25926,1748],{"className":25927},[816],[772,25929,25931],{"className":25930},[1752],[772,25932,25934],{"className":25933},[1756],[772,25935,25937],{"className":25936},[1760],[772,25938,25940],{"className":25939,"style":1765},[1764],[772,25941,25942,25945],{"style":1768},[772,25943],{"className":25944,"style":1773},[1772],[772,25946,25948],{"className":25947},[1777,1778,1779,1780],[772,25949,1642],{"className":25950},[1125,1780],[772,25952],{"className":25953,"style":1121},[1003],[772,25955,1091],{"className":25956},[1125],[772,25958],{"className":25959,"style":1121},[1003],[772,25961,25963,25966,25969,25972,25975,25978],{"className":25962},[807],[772,25964],{"className":25965,"style":1109},[811],[772,25967,1082],{"className":25968},[1113],[772,25970,5554],{"className":25971,"style":5570},[816,817],[772,25973],{"className":25974,"style":1121},[1003],[772,25976,1353],{"className":25977},[1125],[772,25979],{"className":25980,"style":1121},[1003],[772,25982,25984,25987,25990,25993,25996],{"className":25983},[807],[772,25985],{"className":25986,"style":22773},[811],[772,25988,24666],{"className":25989,"style":24682},[816,817],[772,25991],{"className":25992,"style":1121},[1003],[772,25994,1353],{"className":25995},[1125],[772,25997],{"className":25998,"style":1121},[1003],[772,26000,26002,26005,26008],{"className":26001},[807],[772,26003],{"className":26004,"style":1109},[811],[772,26006,24799],{"className":26007,"style":24815},[816,817],[772,26009,1096],{"className":26010},[1141],[11,26012,26013],{},"If within instrument tolerance, distribute evenly:",[11,26015,26016],{},[772,26017,26019,26067],{"className":26018},[775],[772,26020,26022],{"className":26021},[779],[781,26023,26024],{"xmlns":783},[785,26025,26026,26064],{},[788,26027,26028,26035,26037,26039,26041,26051,26053,26056,26058,26060,26062],{},[3525,26029,26030,26032],{},[791,26031,5554],{},[1625,26033,26034],{},"corr",[980,26036,982],{},[791,26038,5554],{},[980,26040,1353],{},[5599,26042,26043,26049],{},[3525,26044,26045,26047],{},[791,26046,10305],{},[791,26048,793],{},[1084,26050,1845],{},[1003,26052],{"width":12337},[1625,26054,26055],{},"(and similarly for ",[791,26057,24666],{},[980,26059,5940],{"separator":803},[791,26061,24799],{},[1625,26063,1096],{},[795,26065,26066],{"encoding":797},"\\alpha_{\\text{corr}} = \\alpha + \\frac{\\delta_\\theta}{3} \\qquad \\text{(and similarly for } \\beta, \\gamma \\text{)}",[772,26068,26070,26133,26151],{"className":26069,"ariaHidden":803},[802],[772,26071,26073,26077,26124,26127,26130],{"className":26072},[807],[772,26074],{"className":26075,"style":26076},[811],"height:0.5806em;vertical-align:-0.15em;",[772,26078,26080,26083],{"className":26079},[816],[772,26081,5554],{"className":26082,"style":5570},[816,817],[772,26084,26086],{"className":26085},[1752],[772,26087,26089,26116],{"className":26088},[1756,3556],[772,26090,26092,26113],{"className":26091},[1760],[772,26093,26095],{"className":26094,"style":15612},[1764],[772,26096,26098,26101],{"style":26097},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[772,26099],{"className":26100,"style":1773},[1772],[772,26102,26104],{"className":26103},[1777,1778,1779,1780],[772,26105,26107],{"className":26106},[816,1780],[772,26108,26110],{"className":26109},[816,1709,1780],[772,26111,26034],{"className":26112},[816,1780],[772,26114,3580],{"className":26115},[3579],[772,26117,26119],{"className":26118},[1760],[772,26120,26122],{"className":26121,"style":3587},[1764],[772,26123],{},[772,26125],{"className":26126,"style":1004},[1003],[772,26128,982],{"className":26129},[1008],[772,26131],{"className":26132,"style":1004},[1003],[772,26134,26136,26139,26142,26145,26148],{"className":26135},[807],[772,26137],{"className":26138,"style":6118},[811],[772,26140,5554],{"className":26141,"style":5570},[816,817],[772,26143],{"className":26144,"style":1121},[1003],[772,26146,1353],{"className":26147},[1125],[772,26149],{"className":26150,"style":1121},[1003],[772,26152,26154,26158,26267,26270,26276,26279,26282,26285,26288],{"className":26153},[807],[772,26155],{"className":26156,"style":26157},[811],"height:1.247em;vertical-align:-0.345em;",[772,26159,26161,26164,26264],{"className":26160},[816],[772,26162],{"className":26163},[1113,5681],[772,26165,26167],{"className":26166},[5599],[772,26168,26170,26256],{"className":26169},[1756,3556],[772,26171,26173,26253],{"className":26172},[1760],[772,26174,26177,26191,26199],{"className":26175,"style":26176},[1764],"height:0.902em;",[772,26178,26179,26182],{"style":5697},[772,26180],{"className":26181,"style":5422},[1772],[772,26183,26185],{"className":26184},[1777,1778,1779,1780],[772,26186,26188],{"className":26187},[816,1780],[772,26189,1845],{"className":26190},[816,1780],[772,26192,26193,26196],{"style":5719},[772,26194],{"className":26195,"style":5422},[1772],[772,26197],{"className":26198,"style":5727},[5726],[772,26200,26202,26205],{"style":26201},"top:-3.4159em;",[772,26203],{"className":26204,"style":5422},[1772],[772,26206,26208],{"className":26207},[1777,1778,1779,1780],[772,26209,26211],{"className":26210},[816,1780],[772,26212,26214,26217],{"className":26213},[816,1780],[772,26215,10305],{"className":26216,"style":10352},[816,817,1780],[772,26218,26220],{"className":26219},[1752],[772,26221,26223,26244],{"className":26222},[1756,3556],[772,26224,26226,26241],{"className":26225},[1760],[772,26227,26229],{"className":26228,"style":23355},[1764],[772,26230,26232,26235],{"style":26231},"top:-2.3488em;margin-left:-0.0379em;margin-right:0.0714em;",[772,26233],{"className":26234,"style":23171},[1772],[772,26236,26238],{"className":26237},[1777,23175,6539,1780],[772,26239,793],{"className":26240,"style":818},[816,817,1780],[772,26242,3580],{"className":26243},[3579],[772,26245,26247],{"className":26246},[1760],[772,26248,26251],{"className":26249,"style":26250},[1764],"height:0.1512em;",[772,26252],{},[772,26254,3580],{"className":26255},[3579],[772,26257,26259],{"className":26258},[1760],[772,26260,26262],{"className":26261,"style":6605},[1764],[772,26263],{},[772,26265],{"className":26266},[1141,5681],[772,26268],{"className":26269,"style":12489},[1003],[772,26271,26273],{"className":26272},[816,1709],[772,26274,26055],{"className":26275},[816],[772,26277,24666],{"className":26278,"style":24682},[816,817],[772,26280,5940],{"className":26281},[6041],[772,26283],{"className":26284,"style":3294},[1003],[772,26286,24799],{"className":26287,"style":24815},[816,817],[772,26289,26291],{"className":26290},[816,1709],[772,26292,1096],{"className":26293},[816],[11,26295,26296,26297,26388,26389,26417,26418,26469],{},"Typical tolerance: ",[772,26298,26300,26322],{"className":26299},[775],[772,26301,26303],{"className":26302},[779],[781,26304,26305],{"xmlns":783},[785,26306,26307,26319],{},[788,26308,26309,26312,26315],{},[980,26310,26311],{},"±",[791,26313,26314],{},"c",[5347,26316,26317],{},[791,26318,13833],{},[795,26320,26321],{"encoding":797},"\\pm c\\sqrt{n}",[772,26323,26325],{"className":26324,"ariaHidden":803},[802],[772,26326,26328,26332,26335,26338],{"className":26327},[807],[772,26329],{"className":26330,"style":26331},[811],"height:1.04em;vertical-align:-0.2397em;",[772,26333,26311],{"className":26334},[816],[772,26336,26314],{"className":26337},[816,817],[772,26339,26341],{"className":26340},[816,5403],[772,26342,26344,26379],{"className":26343},[1756,3556],[772,26345,26347,26376],{"className":26346},[1760],[772,26348,26351,26363],{"className":26349,"style":26350},[1764],"height:0.8003em;",[772,26352,26354,26357],{"className":26353,"style":5418},[5417],[772,26355],{"className":26356,"style":5422},[1772],[772,26358,26360],{"className":26359,"style":5426},[816],[772,26361,13833],{"className":26362},[816,817],[772,26364,26366,26369],{"style":26365},"top:-2.7603em;",[772,26367],{"className":26368,"style":5422},[1772],[772,26370,26372],{"className":26371,"style":5512},[5511],[5514,26373,26374],{"xmlns":5516,"width":5517,"height":5518,"viewBox":5519,"preserveAspectRatio":5520},[5522,26375],{"d":5524},[772,26377,3580],{"className":26378},[3579],[772,26380,26382],{"className":26381},[1760],[772,26383,26386],{"className":26384,"style":26385},[1764],"height:0.2397em;",[772,26387],{}," where ",[772,26390,26392,26405],{"className":26391},[775],[772,26393,26395],{"className":26394},[779],[781,26396,26397],{"xmlns":783},[785,26398,26399,26403],{},[788,26400,26401],{},[791,26402,26314],{},[795,26404,26314],{"encoding":797},[772,26406,26408],{"className":26407,"ariaHidden":803},[802],[772,26409,26411,26414],{"className":26410},[807],[772,26412],{"className":26413,"style":845},[811],[772,26415,26314],{"className":26416},[816,817]," is instrument accuracy (seconds) and ",[772,26419,26421,26439],{"className":26420},[775],[772,26422,26424],{"className":26423},[779],[781,26425,26426],{"xmlns":783},[785,26427,26428,26436],{},[788,26429,26430,26432,26434],{},[791,26431,13833],{},[980,26433,982],{},[1084,26435,1845],{},[795,26437,26438],{"encoding":797},"n = 3",[772,26440,26442,26460],{"className":26441,"ariaHidden":803},[802],[772,26443,26445,26448,26451,26454,26457],{"className":26444},[807],[772,26446],{"className":26447,"style":845},[811],[772,26449,13833],{"className":26450},[816,817],[772,26452],{"className":26453,"style":1004},[1003],[772,26455,982],{"className":26456},[1008],[772,26458],{"className":26459,"style":1004},[1003],[772,26461,26463,26466],{"className":26462},[807],[772,26464],{"className":26465,"style":5899},[811],[772,26467,1845],{"className":26468},[816]," angles.",[706,26471,26473],{"id":26472},"height","Height",[11,26475,26476],{},"Standard Tienstra resection is planimetric (2D). Record instrument and target heights for pairing with later vertical observations.",[323,26478,26480],{"id":26479},"computation-walkthrough","Computation Walkthrough",[706,26482,26484],{"id":26483},"step-1-interior-angles-of-the-control-triangle","Step 1: Interior angles of the control triangle",[11,26486,26487,26488,26490,26491,2896,26519,2896,26547,26575,26576,4135],{},"Compute clockwise grid bearings between control points (use the ",[693,26489,696],{"href":695}," for inverse checks). For CCW-ordered ",[772,26492,26494,26507],{"className":26493},[775],[772,26495,26497],{"className":26496},[779],[781,26498,26499],{"xmlns":783},[785,26500,26501,26505],{},[788,26502,26503],{},[791,26504,13735],{},[795,26506,13735],{"encoding":797},[772,26508,26510],{"className":26509,"ariaHidden":803},[802],[772,26511,26513,26516],{"className":26512},[807],[772,26514],{"className":26515,"style":2758},[811],[772,26517,13735],{"className":26518},[816,817],[772,26520,26522,26535],{"className":26521},[775],[772,26523,26525],{"className":26524},[779],[781,26526,26527],{"xmlns":783},[785,26528,26529,26533],{},[788,26530,26531],{},[791,26532,22441],{},[795,26534,22441],{"encoding":797},[772,26536,26538],{"className":26537,"ariaHidden":803},[802],[772,26539,26541,26544],{"className":26540},[807],[772,26542],{"className":26543,"style":2758},[811],[772,26545,22441],{"className":26546,"style":22456},[816,817],[772,26548,26550,26563],{"className":26549},[775],[772,26551,26553],{"className":26552},[779],[781,26554,26555],{"xmlns":783},[785,26556,26557,26561],{},[788,26558,26559],{},[791,26560,22471],{},[795,26562,22471],{"encoding":797},[772,26564,26566],{"className":26565,"ariaHidden":803},[802],[772,26567,26569,26572],{"className":26568},[807],[772,26570],{"className":26571,"style":2758},[811],[772,26573,22471],{"className":26574,"style":22486},[816,817],", interior angle at ",[772,26577,26579,26592],{"className":26578},[775],[772,26580,26582],{"className":26581},[779],[781,26583,26584],{"xmlns":783},[785,26585,26586,26590],{},[788,26587,26588],{},[791,26589,13735],{},[795,26591,13735],{"encoding":797},[772,26593,26595],{"className":26594,"ariaHidden":803},[802],[772,26596,26598,26601],{"className":26597},[807],[772,26599],{"className":26600,"style":2758},[811],[772,26602,13735],{"className":26603},[816,817],[11,26605,26606],{},[772,26607,26609,26647],{"className":26608},[775],[772,26610,26612],{"className":26611},[779],[781,26613,26614],{"xmlns":783},[785,26615,26616,26644],{},[788,26617,26618,26620,26622,26632,26634],{},[791,26619,13735],{},[980,26621,982],{},[3525,26623,26624,26626],{},[791,26625,793],{},[788,26627,26628,26630],{},[791,26629,13735],{},[791,26631,22441],{},[980,26633,1091],{},[3525,26635,26636,26638],{},[791,26637,793],{},[788,26639,26640,26642],{},[791,26641,13735],{},[791,26643,22471],{},[795,26645,26646],{"encoding":797},"A = \\theta_{AB} - \\theta_{AC}",[772,26648,26650,26668,26729],{"className":26649,"ariaHidden":803},[802],[772,26651,26653,26656,26659,26662,26665],{"className":26652},[807],[772,26654],{"className":26655,"style":2758},[811],[772,26657,13735],{"className":26658},[816,817],[772,26660],{"className":26661,"style":1004},[1003],[772,26663,982],{"className":26664},[1008],[772,26666],{"className":26667,"style":1004},[1003],[772,26669,26671,26674,26720,26723,26726],{"className":26670},[807],[772,26672],{"className":26673,"style":5959},[811],[772,26675,26677,26680],{"className":26676},[816],[772,26678,793],{"className":26679,"style":818},[816,817],[772,26681,26683],{"className":26682},[1752],[772,26684,26686,26712],{"className":26685},[1756,3556],[772,26687,26689,26709],{"className":26688},[1760],[772,26690,26692],{"className":26691,"style":23077},[1764],[772,26693,26694,26697],{"style":5981},[772,26695],{"className":26696,"style":1773},[1772],[772,26698,26700],{"className":26699},[1777,1778,1779,1780],[772,26701,26703,26706],{"className":26702},[816,1780],[772,26704,13735],{"className":26705},[816,817,1780],[772,26707,22441],{"className":26708,"style":22456},[816,817,1780],[772,26710,3580],{"className":26711},[3579],[772,26713,26715],{"className":26714},[1760],[772,26716,26718],{"className":26717,"style":3587},[1764],[772,26719],{},[772,26721],{"className":26722,"style":1121},[1003],[772,26724,1091],{"className":26725},[1125],[772,26727],{"className":26728,"style":1121},[1003],[772,26730,26732,26735],{"className":26731},[807],[772,26733],{"className":26734,"style":5959},[811],[772,26736,26738,26741],{"className":26737},[816],[772,26739,793],{"className":26740,"style":818},[816,817],[772,26742,26744],{"className":26743},[1752],[772,26745,26747,26773],{"className":26746},[1756,3556],[772,26748,26750,26770],{"className":26749},[1760],[772,26751,26753],{"className":26752,"style":23077},[1764],[772,26754,26755,26758],{"style":5981},[772,26756],{"className":26757,"style":1773},[1772],[772,26759,26761],{"className":26760},[1777,1778,1779,1780],[772,26762,26764,26767],{"className":26763},[816,1780],[772,26765,13735],{"className":26766},[816,817,1780],[772,26768,22471],{"className":26769,"style":22486},[816,817,1780],[772,26771,3580],{"className":26772},[3579],[772,26774,26776],{"className":26775},[1760],[772,26777,26779],{"className":26778,"style":3587},[1764],[772,26780],{},[11,26782,12912,26783,26813,26814,26844,26845,3474,26873,26901,26902,5868],{},[772,26784,26786,26801],{"className":26785},[775],[772,26787,26789],{"className":26788},[779],[781,26790,26791],{"xmlns":783},[785,26792,26793,26799],{},[788,26794,26795,26797],{},[1084,26796,1748],{},[791,26798,907],{"mathvariant":906},[795,26800,12955],{"encoding":797},[772,26802,26804],{"className":26803,"ariaHidden":803},[802],[772,26805,26807,26810],{"className":26806},[807],[772,26808],{"className":26809,"style":812},[811],[772,26811,12955],{"className":26812},[816],"–",[772,26815,26817,26832],{"className":26816},[775],[772,26818,26820],{"className":26819},[779],[781,26821,26822],{"xmlns":783},[785,26823,26824,26830],{},[788,26825,26826,26828],{},[1084,26827,1468],{},[791,26829,907],{"mathvariant":906},[795,26831,1495],{"encoding":797},[772,26833,26835],{"className":26834,"ariaHidden":803},[802],[772,26836,26838,26841],{"className":26837},[807],[772,26839],{"className":26840,"style":812},[811],[772,26842,1495],{"className":26843},[816],"). Similarly find ",[772,26846,26848,26861],{"className":26847},[775],[772,26849,26851],{"className":26850},[779],[781,26852,26853],{"xmlns":783},[785,26854,26855,26859],{},[788,26856,26857],{},[791,26858,22441],{},[795,26860,22441],{"encoding":797},[772,26862,26864],{"className":26863,"ariaHidden":803},[802],[772,26865,26867,26870],{"className":26866},[807],[772,26868],{"className":26869,"style":2758},[811],[772,26871,22441],{"className":26872,"style":22456},[816,817],[772,26874,26876,26889],{"className":26875},[775],[772,26877,26879],{"className":26878},[779],[781,26880,26881],{"xmlns":783},[785,26882,26883,26887],{},[788,26884,26885],{},[791,26886,22471],{},[795,26888,22471],{"encoding":797},[772,26890,26892],{"className":26891,"ariaHidden":803},[802],[772,26893,26895,26898],{"className":26894},[807],[772,26896],{"className":26897,"style":2758},[811],[772,26899,22471],{"className":26900,"style":22486},[816,817],". Verify ",[772,26903,26905,26933],{"className":26904},[775],[772,26906,26908],{"className":26907},[779],[781,26909,26910],{"xmlns":783},[785,26911,26912,26930],{},[788,26913,26914,26916,26918,26920,26922,26924,26926,26928],{},[791,26915,13735],{},[980,26917,1353],{},[791,26919,22441],{},[980,26921,1353],{},[791,26923,22471],{},[980,26925,982],{},[1084,26927,1086],{},[791,26929,907],{"mathvariant":906},[795,26931,26932],{"encoding":797},"A + B + C = 180°",[772,26934,26936,26954,26972,26990],{"className":26935,"ariaHidden":803},[802],[772,26937,26939,26942,26945,26948,26951],{"className":26938},[807],[772,26940],{"className":26941,"style":11398},[811],[772,26943,13735],{"className":26944},[816,817],[772,26946],{"className":26947,"style":1121},[1003],[772,26949,1353],{"className":26950},[1125],[772,26952],{"className":26953,"style":1121},[1003],[772,26955,26957,26960,26963,26966,26969],{"className":26956},[807],[772,26958],{"className":26959,"style":11398},[811],[772,26961,22441],{"className":26962,"style":22456},[816,817],[772,26964],{"className":26965,"style":1121},[1003],[772,26967,1353],{"className":26968},[1125],[772,26970],{"className":26971,"style":1121},[1003],[772,26973,26975,26978,26981,26984,26987],{"className":26974},[807],[772,26976],{"className":26977,"style":2758},[811],[772,26979,22471],{"className":26980,"style":22486},[816,817],[772,26982],{"className":26983,"style":1004},[1003],[772,26985,982],{"className":26986},[1008],[772,26988],{"className":26989,"style":1004},[1003],[772,26991,26993,26996],{"className":26992},[807],[772,26994],{"className":26995,"style":812},[811],[772,26997,1117],{"className":26998},[816],[706,27000,27002],{"id":27001},"step-2-barycentric-weights","Step 2: Barycentric weights",[11,27004,27005,27006,27034,27035,27063,27064,2896,27092,27063,27120,2896,27148,27063,27176,3802],{},"With observed angles at ",[772,27007,27009,27022],{"className":27008},[775],[772,27010,27012],{"className":27011},[779],[781,27013,27014],{"xmlns":783},[785,27015,27016,27020],{},[788,27017,27018],{},[791,27019,14943],{},[795,27021,14943],{"encoding":797},[772,27023,27025],{"className":27024,"ariaHidden":803},[802],[772,27026,27028,27031],{"className":27027},[807],[772,27029],{"className":27030,"style":2758},[811],[772,27032,14943],{"className":27033,"style":14964},[816,817]," (",[772,27036,27038,27051],{"className":27037},[775],[772,27039,27041],{"className":27040},[779],[781,27042,27043],{"xmlns":783},[785,27044,27045,27049],{},[788,27046,27047],{},[791,27048,5554],{},[795,27050,5557],{"encoding":797},[772,27052,27054],{"className":27053,"ariaHidden":803},[802],[772,27055,27057,27060],{"className":27056},[807],[772,27058],{"className":27059,"style":845},[811],[772,27061,5554],{"className":27062,"style":5570},[816,817]," opposite ",[772,27065,27067,27080],{"className":27066},[775],[772,27068,27070],{"className":27069},[779],[781,27071,27072],{"xmlns":783},[785,27073,27074,27078],{},[788,27075,27076],{},[791,27077,22471],{},[795,27079,22471],{"encoding":797},[772,27081,27083],{"className":27082,"ariaHidden":803},[802],[772,27084,27086,27089],{"className":27085},[807],[772,27087],{"className":27088,"style":2758},[811],[772,27090,22471],{"className":27091,"style":22486},[816,817],[772,27093,27095,27108],{"className":27094},[775],[772,27096,27098],{"className":27097},[779],[781,27099,27100],{"xmlns":783},[785,27101,27102,27106],{},[788,27103,27104],{},[791,27105,24666],{},[795,27107,24669],{"encoding":797},[772,27109,27111],{"className":27110,"ariaHidden":803},[802],[772,27112,27114,27117],{"className":27113},[807],[772,27115],{"className":27116,"style":22773},[811],[772,27118,24666],{"className":27119,"style":24682},[816,817],[772,27121,27123,27136],{"className":27122},[775],[772,27124,27126],{"className":27125},[779],[781,27127,27128],{"xmlns":783},[785,27129,27130,27134],{},[788,27131,27132],{},[791,27133,13735],{},[795,27135,13735],{"encoding":797},[772,27137,27139],{"className":27138,"ariaHidden":803},[802],[772,27140,27142,27145],{"className":27141},[807],[772,27143],{"className":27144,"style":2758},[811],[772,27146,13735],{"className":27147},[816,817],[772,27149,27151,27164],{"className":27150},[775],[772,27152,27154],{"className":27153},[779],[781,27155,27156],{"xmlns":783},[785,27157,27158,27162],{},[788,27159,27160],{},[791,27161,24799],{},[795,27163,24802],{"encoding":797},[772,27165,27167],{"className":27166,"ariaHidden":803},[802],[772,27168,27170,27173],{"className":27169},[807],[772,27171],{"className":27172,"style":11141},[811],[772,27174,24799],{"className":27175,"style":24815},[816,817],[772,27177,27179,27192],{"className":27178},[775],[772,27180,27182],{"className":27181},[779],[781,27183,27184],{"xmlns":783},[785,27185,27186,27190],{},[788,27187,27188],{},[791,27189,22441],{},[795,27191,22441],{"encoding":797},[772,27193,27195],{"className":27194,"ariaHidden":803},[802],[772,27196,27198,27201],{"className":27197},[807],[772,27199],{"className":27200,"style":2758},[811],[772,27202,22441],{"className":27203,"style":22456},[816,817],[11,27205,27206],{},[772,27207,27209,27249],{"className":27208},[775],[772,27210,27212],{"className":27211},[779],[781,27213,27214],{"xmlns":783},[785,27215,27216,27246],{},[788,27217,27218,27224,27226],{},[3525,27219,27220,27222],{},[791,27221,22758],{},[1084,27223,2170],{},[980,27225,982],{},[5599,27227,27228,27230],{},[1084,27229,2170],{},[788,27231,27232,27234,27236,27238,27240,27242,27244],{},[791,27233,25154],{},[980,27235,3252],{},[791,27237,13735],{},[980,27239,1091],{},[791,27241,25154],{},[980,27243,3252],{},[791,27245,24666],{},[795,27247,27248],{"encoding":797},"f_1 = \\frac{1}{\\cot A - \\cot \\beta}",[772,27250,27252,27307],{"className":27251,"ariaHidden":803},[802],[772,27253,27255,27258,27298,27301,27304],{"className":27254},[807],[772,27256],{"className":27257,"style":22773},[811],[772,27259,27261,27264],{"className":27260},[816],[772,27262,22758],{"className":27263,"style":22780},[816,817],[772,27265,27267],{"className":27266},[1752],[772,27268,27270,27290],{"className":27269},[1756,3556],[772,27271,27273,27287],{"className":27272},[1760],[772,27274,27276],{"className":27275,"style":3563},[1764],[772,27277,27278,27281],{"style":22795},[772,27279],{"className":27280,"style":1773},[1772],[772,27282,27284],{"className":27283},[1777,1778,1779,1780],[772,27285,2170],{"className":27286},[816,1780],[772,27288,3580],{"className":27289},[3579],[772,27291,27293],{"className":27292},[1760],[772,27294,27296],{"className":27295,"style":3587},[1764],[772,27297],{},[772,27299],{"className":27300,"style":1004},[1003],[772,27302,982],{"className":27303},[1008],[772,27305],{"className":27306,"style":1004},[1003],[772,27308,27310,27314],{"className":27309},[807],[772,27311],{"className":27312,"style":27313},[811],"height:1.3262em;vertical-align:-0.4811em;",[772,27315,27317,27320,27418],{"className":27316},[816],[772,27318],{"className":27319},[1113,5681],[772,27321,27323],{"className":27322},[5599],[772,27324,27326,27410],{"className":27325},[1756,3556],[772,27327,27329,27407],{"className":27328},[1760],[772,27330,27332,27385,27393],{"className":27331,"style":6558},[1764],[772,27333,27334,27337],{"style":5697},[772,27335],{"className":27336,"style":5422},[1772],[772,27338,27340],{"className":27339},[1777,1778,1779,1780],[772,27341,27343,27357,27361,27364,27367,27379,27382],{"className":27342},[816,1780],[772,27344,27346,27349,27353],{"className":27345},[3298,1780],[772,27347,26314],{"className":27348},[1780],[772,27350,27352],{"className":27351},[1780],"o",[772,27354,27356],{"className":27355},[1780],"t",[772,27358],{"className":27359,"style":27360},[1003,1780],"margin-right:0.1952em;",[772,27362,13735],{"className":27363},[816,817,1780],[772,27365,1091],{"className":27366},[1125,1780],[772,27368,27370,27373,27376],{"className":27369},[3298,1780],[772,27371,26314],{"className":27372},[1780],[772,27374,27352],{"className":27375},[1780],[772,27377,27356],{"className":27378},[1780],[772,27380],{"className":27381,"style":27360},[1003,1780],[772,27383,24666],{"className":27384,"style":24682},[816,817,1780],[772,27386,27387,27390],{"style":5719},[772,27388],{"className":27389,"style":5422},[1772],[772,27391],{"className":27392,"style":5727},[5726],[772,27394,27395,27398],{"style":6583},[772,27396],{"className":27397,"style":5422},[1772],[772,27399,27401],{"className":27400},[1777,1778,1779,1780],[772,27402,27404],{"className":27403},[816,1780],[772,27405,2170],{"className":27406},[816,1780],[772,27408,3580],{"className":27409},[3579],[772,27411,27413],{"className":27412},[1760],[772,27414,27416],{"className":27415,"style":23555},[1764],[772,27417],{},[772,27419],{"className":27420},[1141,5681],[11,27422,27423],{},[772,27424,27426,27466],{"className":27425},[775],[772,27427,27429],{"className":27428},[779],[781,27430,27431],{"xmlns":783},[785,27432,27433,27463],{},[788,27434,27435,27441,27443],{},[3525,27436,27437,27439],{},[791,27438,22758],{},[1084,27440,1872],{},[980,27442,982],{},[5599,27444,27445,27447],{},[1084,27446,2170],{},[788,27448,27449,27451,27453,27455,27457,27459,27461],{},[791,27450,25154],{},[980,27452,3252],{},[791,27454,22441],{},[980,27456,1091],{},[791,27458,25154],{},[980,27460,3252],{},[791,27462,24799],{},[795,27464,27465],{"encoding":797},"f_2 = \\frac{1}{\\cot B - \\cot \\gamma}",[772,27467,27469,27524],{"className":27468,"ariaHidden":803},[802],[772,27470,27472,27475,27515,27518,27521],{"className":27471},[807],[772,27473],{"className":27474,"style":22773},[811],[772,27476,27478,27481],{"className":27477},[816],[772,27479,22758],{"className":27480,"style":22780},[816,817],[772,27482,27484],{"className":27483},[1752],[772,27485,27487,27507],{"className":27486},[1756,3556],[772,27488,27490,27504],{"className":27489},[1760],[772,27491,27493],{"className":27492,"style":3563},[1764],[772,27494,27495,27498],{"style":22795},[772,27496],{"className":27497,"style":1773},[1772],[772,27499,27501],{"className":27500},[1777,1778,1779,1780],[772,27502,1872],{"className":27503},[816,1780],[772,27505,3580],{"className":27506},[3579],[772,27508,27510],{"className":27509},[1760],[772,27511,27513],{"className":27512,"style":3587},[1764],[772,27514],{},[772,27516],{"className":27517,"style":1004},[1003],[772,27519,982],{"className":27520},[1008],[772,27522],{"className":27523,"style":1004},[1003],[772,27525,27527,27530],{"className":27526},[807],[772,27528],{"className":27529,"style":27313},[811],[772,27531,27533,27536,27631],{"className":27532},[816],[772,27534],{"className":27535},[1113,5681],[772,27537,27539],{"className":27538},[5599],[772,27540,27542,27623],{"className":27541},[1756,3556],[772,27543,27545,27620],{"className":27544},[1760],[772,27546,27548,27598,27606],{"className":27547,"style":6558},[1764],[772,27549,27550,27553],{"style":5697},[772,27551],{"className":27552,"style":5422},[1772],[772,27554,27556],{"className":27555},[1777,1778,1779,1780],[772,27557,27559,27571,27574,27577,27580,27592,27595],{"className":27558},[816,1780],[772,27560,27562,27565,27568],{"className":27561},[3298,1780],[772,27563,26314],{"className":27564},[1780],[772,27566,27352],{"className":27567},[1780],[772,27569,27356],{"className":27570},[1780],[772,27572],{"className":27573,"style":27360},[1003,1780],[772,27575,22441],{"className":27576,"style":22456},[816,817,1780],[772,27578,1091],{"className":27579},[1125,1780],[772,27581,27583,27586,27589],{"className":27582},[3298,1780],[772,27584,26314],{"className":27585},[1780],[772,27587,27352],{"className":27588},[1780],[772,27590,27356],{"className":27591},[1780],[772,27593],{"className":27594,"style":27360},[1003,1780],[772,27596,24799],{"className":27597,"style":24815},[816,817,1780],[772,27599,27600,27603],{"style":5719},[772,27601],{"className":27602,"style":5422},[1772],[772,27604],{"className":27605,"style":5727},[5726],[772,27607,27608,27611],{"style":6583},[772,27609],{"className":27610,"style":5422},[1772],[772,27612,27614],{"className":27613},[1777,1778,1779,1780],[772,27615,27617],{"className":27616},[816,1780],[772,27618,2170],{"className":27619},[816,1780],[772,27621,3580],{"className":27622},[3579],[772,27624,27626],{"className":27625},[1760],[772,27627,27629],{"className":27628,"style":23555},[1764],[772,27630],{},[772,27632],{"className":27633},[1141,5681],[11,27635,27636],{},[772,27637,27639,27679],{"className":27638},[775],[772,27640,27642],{"className":27641},[779],[781,27643,27644],{"xmlns":783},[785,27645,27646,27676],{},[788,27647,27648,27654,27656],{},[3525,27649,27650,27652],{},[791,27651,22758],{},[1084,27653,1845],{},[980,27655,982],{},[5599,27657,27658,27660],{},[1084,27659,2170],{},[788,27661,27662,27664,27666,27668,27670,27672,27674],{},[791,27663,25154],{},[980,27665,3252],{},[791,27667,22471],{},[980,27669,1091],{},[791,27671,25154],{},[980,27673,3252],{},[791,27675,5554],{},[795,27677,27678],{"encoding":797},"f_3 = \\frac{1}{\\cot C - \\cot \\alpha}",[772,27680,27682,27737],{"className":27681,"ariaHidden":803},[802],[772,27683,27685,27688,27728,27731,27734],{"className":27684},[807],[772,27686],{"className":27687,"style":22773},[811],[772,27689,27691,27694],{"className":27690},[816],[772,27692,22758],{"className":27693,"style":22780},[816,817],[772,27695,27697],{"className":27696},[1752],[772,27698,27700,27720],{"className":27699},[1756,3556],[772,27701,27703,27717],{"className":27702},[1760],[772,27704,27706],{"className":27705,"style":3563},[1764],[772,27707,27708,27711],{"style":22795},[772,27709],{"className":27710,"style":1773},[1772],[772,27712,27714],{"className":27713},[1777,1778,1779,1780],[772,27715,1845],{"className":27716},[816,1780],[772,27718,3580],{"className":27719},[3579],[772,27721,27723],{"className":27722},[1760],[772,27724,27726],{"className":27725,"style":3587},[1764],[772,27727],{},[772,27729],{"className":27730,"style":1004},[1003],[772,27732,982],{"className":27733},[1008],[772,27735],{"className":27736,"style":1004},[1003],[772,27738,27740,27744],{"className":27739},[807],[772,27741],{"className":27742,"style":27743},[811],"height:1.2484em;vertical-align:-0.4033em;",[772,27745,27747,27750,27846],{"className":27746},[816],[772,27748],{"className":27749},[1113,5681],[772,27751,27753],{"className":27752},[5599],[772,27754,27756,27837],{"className":27755},[1756,3556],[772,27757,27759,27834],{"className":27758},[1760],[772,27760,27762,27812,27820],{"className":27761,"style":6558},[1764],[772,27763,27764,27767],{"style":5697},[772,27765],{"className":27766,"style":5422},[1772],[772,27768,27770],{"className":27769},[1777,1778,1779,1780],[772,27771,27773,27785,27788,27791,27794,27806,27809],{"className":27772},[816,1780],[772,27774,27776,27779,27782],{"className":27775},[3298,1780],[772,27777,26314],{"className":27778},[1780],[772,27780,27352],{"className":27781},[1780],[772,27783,27356],{"className":27784},[1780],[772,27786],{"className":27787,"style":27360},[1003,1780],[772,27789,22471],{"className":27790,"style":22486},[816,817,1780],[772,27792,1091],{"className":27793},[1125,1780],[772,27795,27797,27800,27803],{"className":27796},[3298,1780],[772,27798,26314],{"className":27799},[1780],[772,27801,27352],{"className":27802},[1780],[772,27804,27356],{"className":27805},[1780],[772,27807],{"className":27808,"style":27360},[1003,1780],[772,27810,5554],{"className":27811,"style":5570},[816,817,1780],[772,27813,27814,27817],{"style":5719},[772,27815],{"className":27816,"style":5422},[1772],[772,27818],{"className":27819,"style":5727},[5726],[772,27821,27822,27825],{"style":6583},[772,27823],{"className":27824,"style":5422},[1772],[772,27826,27828],{"className":27827},[1777,1778,1779,1780],[772,27829,27831],{"className":27830},[816,1780],[772,27832,2170],{"className":27833},[816,1780],[772,27835,3580],{"className":27836},[3579],[772,27838,27840],{"className":27839},[1760],[772,27841,27844],{"className":27842,"style":27843},[1764],"height:0.4033em;",[772,27845],{},[772,27847],{"className":27848},[1141,5681],[706,27850,27852,27853],{"id":27851},"step-3-coordinates-of-ppp","Step 3: Coordinates of ",[772,27854,27856,27869],{"className":27855},[775],[772,27857,27859],{"className":27858},[779],[781,27860,27861],{"xmlns":783},[785,27862,27863,27867],{},[788,27864,27865],{},[791,27866,14943],{},[795,27868,14943],{"encoding":797},[772,27870,27872],{"className":27871,"ariaHidden":803},[802],[772,27873,27875,27878],{"className":27874},[807],[772,27876],{"className":27877,"style":2758},[811],[772,27879,14943],{"className":27880,"style":14964},[816,817],[11,27882,27883],{},[772,27884,27886,28052],{"className":27885},[775],[772,27887,27889],{"className":27888},[779],[781,27890,27891],{"xmlns":783},[785,27892,27893,28049],{},[788,27894,27895,27901,27903,27971,27973,27979,27981],{},[3525,27896,27897,27899],{},[791,27898,2808],{},[791,27900,14943],{},[980,27902,982],{},[5599,27904,27905,27947],{},[788,27906,27907,27913,27919,27921,27927,27933,27935,27941],{},[3525,27908,27909,27911],{},[791,27910,22758],{},[1084,27912,2170],{},[3525,27914,27915,27917],{},[791,27916,2808],{},[791,27918,13735],{},[980,27920,1353],{},[3525,27922,27923,27925],{},[791,27924,22758],{},[1084,27926,1872],{},[3525,27928,27929,27931],{},[791,27930,2808],{},[791,27932,22441],{},[980,27934,1353],{},[3525,27936,27937,27939],{},[791,27938,22758],{},[1084,27940,1845],{},[3525,27942,27943,27945],{},[791,27944,2808],{},[791,27946,22471],{},[788,27948,27949,27955,27957,27963,27965],{},[3525,27950,27951,27953],{},[791,27952,22758],{},[1084,27954,2170],{},[980,27956,1353],{},[3525,27958,27959,27961],{},[791,27960,22758],{},[1084,27962,1872],{},[980,27964,1353],{},[3525,27966,27967,27969],{},[791,27968,22758],{},[1084,27970,1845],{},[1003,27972],{"width":12337},[3525,27974,27975,27977],{},[791,27976,2839],{},[791,27978,14943],{},[980,27980,982],{},[5599,27982,27983,28025],{},[788,27984,27985,27991,27997,27999,28005,28011,28013,28019],{},[3525,27986,27987,27989],{},[791,27988,22758],{},[1084,27990,2170],{},[3525,27992,27993,27995],{},[791,27994,2839],{},[791,27996,13735],{},[980,27998,1353],{},[3525,28000,28001,28003],{},[791,28002,22758],{},[1084,28004,1872],{},[3525,28006,28007,28009],{},[791,28008,2839],{},[791,28010,22441],{},[980,28012,1353],{},[3525,28014,28015,28017],{},[791,28016,22758],{},[1084,28018,1845],{},[3525,28020,28021,28023],{},[791,28022,2839],{},[791,28024,22471],{},[788,28026,28027,28033,28035,28041,28043],{},[3525,28028,28029,28031],{},[791,28030,22758],{},[1084,28032,2170],{},[980,28034,1353],{},[3525,28036,28037,28039],{},[791,28038,22758],{},[1084,28040,1872],{},[980,28042,1353],{},[3525,28044,28045,28047],{},[791,28046,22758],{},[1084,28048,1845],{},[795,28050,28051],{"encoding":797},"E_P = \\frac{f_1 E_A + f_2 E_B + f_3 E_C}{f_1 + f_2 + f_3} \\qquad N_P = \\frac{f_1 N_A + f_2 N_B + f_3 N_C}{f_1 + f_2 + f_3}",[772,28053,28055,28110,28602],{"className":28054,"ariaHidden":803},[802],[772,28056,28058,28061,28101,28104,28107],{"className":28057},[807],[772,28059],{"className":28060,"style":3543},[811],[772,28062,28064,28067],{"className":28063},[816],[772,28065,2808],{"className":28066,"style":2823},[816,817],[772,28068,28070],{"className":28069},[1752],[772,28071,28073,28093],{"className":28072},[1756,3556],[772,28074,28076,28090],{"className":28075},[1760],[772,28077,28079],{"className":28078,"style":23077},[1764],[772,28080,28081,28084],{"style":3566},[772,28082],{"className":28083,"style":1773},[1772],[772,28085,28087],{"className":28086},[1777,1778,1779,1780],[772,28088,14943],{"className":28089,"style":14964},[816,817,1780],[772,28091,3580],{"className":28092},[3579],[772,28094,28096],{"className":28095},[1760],[772,28097,28099],{"className":28098,"style":3587},[1764],[772,28100],{},[772,28102],{"className":28103,"style":1004},[1003],[772,28105,982],{"className":28106},[1008],[772,28108],{"className":28109,"style":1004},[1003],[772,28111,28113,28116,28550,28553,28593,28596,28599],{"className":28112},[807],[772,28114],{"className":28115,"style":23115},[811],[772,28117,28119,28122,28547],{"className":28118},[816],[772,28120],{"className":28121},[1113,5681],[772,28123,28125],{"className":28124},[5599],[772,28126,28128,28539],{"className":28127},[1756,3556],[772,28129,28131,28536],{"className":28130},[1760],[772,28132,28134,28271,28279],{"className":28133,"style":23134},[1764],[772,28135,28136,28139],{"style":5697},[772,28137],{"className":28138,"style":5422},[1772],[772,28140,28142],{"className":28141},[1777,1778,1779,1780],[772,28143,28145,28185,28188,28228,28231],{"className":28144},[816,1780],[772,28146,28148,28151],{"className":28147},[816,1780],[772,28149,22758],{"className":28150,"style":22780},[816,817,1780],[772,28152,28154],{"className":28153},[1752],[772,28155,28157,28177],{"className":28156},[1756,3556],[772,28158,28160,28174],{"className":28159},[1760],[772,28161,28163],{"className":28162,"style":23164},[1764],[772,28164,28165,28168],{"style":23167},[772,28166],{"className":28167,"style":23171},[1772],[772,28169,28171],{"className":28170},[1777,23175,6539,1780],[772,28172,2170],{"className":28173},[816,1780],[772,28175,3580],{"className":28176},[3579],[772,28178,28180],{"className":28179},[1760],[772,28181,28183],{"className":28182,"style":23188},[1764],[772,28184],{},[772,28186,1353],{"className":28187},[1125,1780],[772,28189,28191,28194],{"className":28190},[816,1780],[772,28192,22758],{"className":28193,"style":22780},[816,817,1780],[772,28195,28197],{"className":28196},[1752],[772,28198,28200,28220],{"className":28199},[1756,3556],[772,28201,28203,28217],{"className":28202},[1760],[772,28204,28206],{"className":28205,"style":23164},[1764],[772,28207,28208,28211],{"style":23167},[772,28209],{"className":28210,"style":23171},[1772],[772,28212,28214],{"className":28213},[1777,23175,6539,1780],[772,28215,1872],{"className":28216},[816,1780],[772,28218,3580],{"className":28219},[3579],[772,28221,28223],{"className":28222},[1760],[772,28224,28226],{"className":28225,"style":23188},[1764],[772,28227],{},[772,28229,1353],{"className":28230},[1125,1780],[772,28232,28234,28237],{"className":28233},[816,1780],[772,28235,22758],{"className":28236,"style":22780},[816,817,1780],[772,28238,28240],{"className":28239},[1752],[772,28241,28243,28263],{"className":28242},[1756,3556],[772,28244,28246,28260],{"className":28245},[1760],[772,28247,28249],{"className":28248,"style":23164},[1764],[772,28250,28251,28254],{"style":23167},[772,28252],{"className":28253,"style":23171},[1772],[772,28255,28257],{"className":28256},[1777,23175,6539,1780],[772,28258,1845],{"className":28259},[816,1780],[772,28261,3580],{"className":28262},[3579],[772,28264,28266],{"className":28265},[1760],[772,28267,28269],{"className":28268,"style":23188},[1764],[772,28270],{},[772,28272,28273,28276],{"style":5719},[772,28274],{"className":28275,"style":5422},[1772],[772,28277],{"className":28278,"style":5727},[5726],[772,28280,28281,28284],{"style":23287},[772,28282],{"className":28283,"style":5422},[1772],[772,28285,28287],{"className":28286},[1777,1778,1779,1780],[772,28288,28290,28330,28370,28373,28413,28453,28456,28496],{"className":28289},[816,1780],[772,28291,28293,28296],{"className":28292},[816,1780],[772,28294,22758],{"className":28295,"style":22780},[816,817,1780],[772,28297,28299],{"className":28298},[1752],[772,28300,28302,28322],{"className":28301},[1756,3556],[772,28303,28305,28319],{"className":28304},[1760],[772,28306,28308],{"className":28307,"style":23164},[1764],[772,28309,28310,28313],{"style":23167},[772,28311],{"className":28312,"style":23171},[1772],[772,28314,28316],{"className":28315},[1777,23175,6539,1780],[772,28317,2170],{"className":28318},[816,1780],[772,28320,3580],{"className":28321},[3579],[772,28323,28325],{"className":28324},[1760],[772,28326,28328],{"className":28327,"style":23188},[1764],[772,28329],{},[772,28331,28333,28336],{"className":28332},[816,1780],[772,28334,2808],{"className":28335,"style":2823},[816,817,1780],[772,28337,28339],{"className":28338},[1752],[772,28340,28342,28362],{"className":28341},[1756,3556],[772,28343,28345,28359],{"className":28344},[1760],[772,28346,28348],{"className":28347,"style":23355},[1764],[772,28349,28350,28353],{"style":23358},[772,28351],{"className":28352,"style":23171},[1772],[772,28354,28356],{"className":28355},[1777,23175,6539,1780],[772,28357,13735],{"className":28358},[816,817,1780],[772,28360,3580],{"className":28361},[3579],[772,28363,28365],{"className":28364},[1760],[772,28366,28368],{"className":28367,"style":23377},[1764],[772,28369],{},[772,28371,1353],{"className":28372},[1125,1780],[772,28374,28376,28379],{"className":28375},[816,1780],[772,28377,22758],{"className":28378,"style":22780},[816,817,1780],[772,28380,28382],{"className":28381},[1752],[772,28383,28385,28405],{"className":28384},[1756,3556],[772,28386,28388,28402],{"className":28387},[1760],[772,28389,28391],{"className":28390,"style":23164},[1764],[772,28392,28393,28396],{"style":23167},[772,28394],{"className":28395,"style":23171},[1772],[772,28397,28399],{"className":28398},[1777,23175,6539,1780],[772,28400,1872],{"className":28401},[816,1780],[772,28403,3580],{"className":28404},[3579],[772,28406,28408],{"className":28407},[1760],[772,28409,28411],{"className":28410,"style":23188},[1764],[772,28412],{},[772,28414,28416,28419],{"className":28415},[816,1780],[772,28417,2808],{"className":28418,"style":2823},[816,817,1780],[772,28420,28422],{"className":28421},[1752],[772,28423,28425,28445],{"className":28424},[1756,3556],[772,28426,28428,28442],{"className":28427},[1760],[772,28429,28431],{"className":28430,"style":23355},[1764],[772,28432,28433,28436],{"style":23358},[772,28434],{"className":28435,"style":23171},[1772],[772,28437,28439],{"className":28438},[1777,23175,6539,1780],[772,28440,22441],{"className":28441,"style":22456},[816,817,1780],[772,28443,3580],{"className":28444},[3579],[772,28446,28448],{"className":28447},[1760],[772,28449,28451],{"className":28450,"style":23377},[1764],[772,28452],{},[772,28454,1353],{"className":28455},[1125,1780],[772,28457,28459,28462],{"className":28458},[816,1780],[772,28460,22758],{"className":28461,"style":22780},[816,817,1780],[772,28463,28465],{"className":28464},[1752],[772,28466,28468,28488],{"className":28467},[1756,3556],[772,28469,28471,28485],{"className":28470},[1760],[772,28472,28474],{"className":28473,"style":23164},[1764],[772,28475,28476,28479],{"style":23167},[772,28477],{"className":28478,"style":23171},[1772],[772,28480,28482],{"className":28481},[1777,23175,6539,1780],[772,28483,1845],{"className":28484},[816,1780],[772,28486,3580],{"className":28487},[3579],[772,28489,28491],{"className":28490},[1760],[772,28492,28494],{"className":28493,"style":23188},[1764],[772,28495],{},[772,28497,28499,28502],{"className":28498},[816,1780],[772,28500,2808],{"className":28501,"style":2823},[816,817,1780],[772,28503,28505],{"className":28504},[1752],[772,28506,28508,28528],{"className":28507},[1756,3556],[772,28509,28511,28525],{"className":28510},[1760],[772,28512,28514],{"className":28513,"style":23355},[1764],[772,28515,28516,28519],{"style":23358},[772,28517],{"className":28518,"style":23171},[1772],[772,28520,28522],{"className":28521},[1777,23175,6539,1780],[772,28523,22471],{"className":28524,"style":22486},[816,817,1780],[772,28526,3580],{"className":28527},[3579],[772,28529,28531],{"className":28530},[1760],[772,28532,28534],{"className":28533,"style":23377},[1764],[772,28535],{},[772,28537,3580],{"className":28538},[3579],[772,28540,28542],{"className":28541},[1760],[772,28543,28545],{"className":28544,"style":23555},[1764],[772,28546],{},[772,28548],{"className":28549},[1141,5681],[772,28551],{"className":28552,"style":12489},[1003],[772,28554,28556,28559],{"className":28555},[816],[772,28557,2839],{"className":28558,"style":2854},[816,817],[772,28560,28562],{"className":28561},[1752],[772,28563,28565,28585],{"className":28564},[1756,3556],[772,28566,28568,28582],{"className":28567},[1760],[772,28569,28571],{"className":28570,"style":23077},[1764],[772,28572,28573,28576],{"style":3640},[772,28574],{"className":28575,"style":1773},[1772],[772,28577,28579],{"className":28578},[1777,1778,1779,1780],[772,28580,14943],{"className":28581,"style":14964},[816,817,1780],[772,28583,3580],{"className":28584},[3579],[772,28586,28588],{"className":28587},[1760],[772,28589,28591],{"className":28590,"style":3587},[1764],[772,28592],{},[772,28594],{"className":28595,"style":1004},[1003],[772,28597,982],{"className":28598},[1008],[772,28600],{"className":28601,"style":1004},[1003],[772,28603,28605,28608],{"className":28604},[807],[772,28606],{"className":28607,"style":23115},[811],[772,28609,28611,28614,29039],{"className":28610},[816],[772,28612],{"className":28613},[1113,5681],[772,28615,28617],{"className":28616},[5599],[772,28618,28620,29031],{"className":28619},[1756,3556],[772,28621,28623,29028],{"className":28622},[1760],[772,28624,28626,28763,28771],{"className":28625,"style":23134},[1764],[772,28627,28628,28631],{"style":5697},[772,28629],{"className":28630,"style":5422},[1772],[772,28632,28634],{"className":28633},[1777,1778,1779,1780],[772,28635,28637,28677,28680,28720,28723],{"className":28636},[816,1780],[772,28638,28640,28643],{"className":28639},[816,1780],[772,28641,22758],{"className":28642,"style":22780},[816,817,1780],[772,28644,28646],{"className":28645},[1752],[772,28647,28649,28669],{"className":28648},[1756,3556],[772,28650,28652,28666],{"className":28651},[1760],[772,28653,28655],{"className":28654,"style":23164},[1764],[772,28656,28657,28660],{"style":23167},[772,28658],{"className":28659,"style":23171},[1772],[772,28661,28663],{"className":28662},[1777,23175,6539,1780],[772,28664,2170],{"className":28665},[816,1780],[772,28667,3580],{"className":28668},[3579],[772,28670,28672],{"className":28671},[1760],[772,28673,28675],{"className":28674,"style":23188},[1764],[772,28676],{},[772,28678,1353],{"className":28679},[1125,1780],[772,28681,28683,28686],{"className":28682},[816,1780],[772,28684,22758],{"className":28685,"style":22780},[816,817,1780],[772,28687,28689],{"className":28688},[1752],[772,28690,28692,28712],{"className":28691},[1756,3556],[772,28693,28695,28709],{"className":28694},[1760],[772,28696,28698],{"className":28697,"style":23164},[1764],[772,28699,28700,28703],{"style":23167},[772,28701],{"className":28702,"style":23171},[1772],[772,28704,28706],{"className":28705},[1777,23175,6539,1780],[772,28707,1872],{"className":28708},[816,1780],[772,28710,3580],{"className":28711},[3579],[772,28713,28715],{"className":28714},[1760],[772,28716,28718],{"className":28717,"style":23188},[1764],[772,28719],{},[772,28721,1353],{"className":28722},[1125,1780],[772,28724,28726,28729],{"className":28725},[816,1780],[772,28727,22758],{"className":28728,"style":22780},[816,817,1780],[772,28730,28732],{"className":28731},[1752],[772,28733,28735,28755],{"className":28734},[1756,3556],[772,28736,28738,28752],{"className":28737},[1760],[772,28739,28741],{"className":28740,"style":23164},[1764],[772,28742,28743,28746],{"style":23167},[772,28744],{"className":28745,"style":23171},[1772],[772,28747,28749],{"className":28748},[1777,23175,6539,1780],[772,28750,1845],{"className":28751},[816,1780],[772,28753,3580],{"className":28754},[3579],[772,28756,28758],{"className":28757},[1760],[772,28759,28761],{"className":28760,"style":23188},[1764],[772,28762],{},[772,28764,28765,28768],{"style":5719},[772,28766],{"className":28767,"style":5422},[1772],[772,28769],{"className":28770,"style":5727},[5726],[772,28772,28773,28776],{"style":23287},[772,28774],{"className":28775,"style":5422},[1772],[772,28777,28779],{"className":28778},[1777,1778,1779,1780],[772,28780,28782,28822,28862,28865,28905,28945,28948,28988],{"className":28781},[816,1780],[772,28783,28785,28788],{"className":28784},[816,1780],[772,28786,22758],{"className":28787,"style":22780},[816,817,1780],[772,28789,28791],{"className":28790},[1752],[772,28792,28794,28814],{"className":28793},[1756,3556],[772,28795,28797,28811],{"className":28796},[1760],[772,28798,28800],{"className":28799,"style":23164},[1764],[772,28801,28802,28805],{"style":23167},[772,28803],{"className":28804,"style":23171},[1772],[772,28806,28808],{"className":28807},[1777,23175,6539,1780],[772,28809,2170],{"className":28810},[816,1780],[772,28812,3580],{"className":28813},[3579],[772,28815,28817],{"className":28816},[1760],[772,28818,28820],{"className":28819,"style":23188},[1764],[772,28821],{},[772,28823,28825,28828],{"className":28824},[816,1780],[772,28826,2839],{"className":28827,"style":2854},[816,817,1780],[772,28829,28831],{"className":28830},[1752],[772,28832,28834,28854],{"className":28833},[1756,3556],[772,28835,28837,28851],{"className":28836},[1760],[772,28838,28840],{"className":28839,"style":23355},[1764],[772,28841,28842,28845],{"style":23952},[772,28843],{"className":28844,"style":23171},[1772],[772,28846,28848],{"className":28847},[1777,23175,6539,1780],[772,28849,13735],{"className":28850},[816,817,1780],[772,28852,3580],{"className":28853},[3579],[772,28855,28857],{"className":28856},[1760],[772,28858,28860],{"className":28859,"style":23377},[1764],[772,28861],{},[772,28863,1353],{"className":28864},[1125,1780],[772,28866,28868,28871],{"className":28867},[816,1780],[772,28869,22758],{"className":28870,"style":22780},[816,817,1780],[772,28872,28874],{"className":28873},[1752],[772,28875,28877,28897],{"className":28876},[1756,3556],[772,28878,28880,28894],{"className":28879},[1760],[772,28881,28883],{"className":28882,"style":23164},[1764],[772,28884,28885,28888],{"style":23167},[772,28886],{"className":28887,"style":23171},[1772],[772,28889,28891],{"className":28890},[1777,23175,6539,1780],[772,28892,1872],{"className":28893},[816,1780],[772,28895,3580],{"className":28896},[3579],[772,28898,28900],{"className":28899},[1760],[772,28901,28903],{"className":28902,"style":23188},[1764],[772,28904],{},[772,28906,28908,28911],{"className":28907},[816,1780],[772,28909,2839],{"className":28910,"style":2854},[816,817,1780],[772,28912,28914],{"className":28913},[1752],[772,28915,28917,28937],{"className":28916},[1756,3556],[772,28918,28920,28934],{"className":28919},[1760],[772,28921,28923],{"className":28922,"style":23355},[1764],[772,28924,28925,28928],{"style":23952},[772,28926],{"className":28927,"style":23171},[1772],[772,28929,28931],{"className":28930},[1777,23175,6539,1780],[772,28932,22441],{"className":28933,"style":22456},[816,817,1780],[772,28935,3580],{"className":28936},[3579],[772,28938,28940],{"className":28939},[1760],[772,28941,28943],{"className":28942,"style":23377},[1764],[772,28944],{},[772,28946,1353],{"className":28947},[1125,1780],[772,28949,28951,28954],{"className":28950},[816,1780],[772,28952,22758],{"className":28953,"style":22780},[816,817,1780],[772,28955,28957],{"className":28956},[1752],[772,28958,28960,28980],{"className":28959},[1756,3556],[772,28961,28963,28977],{"className":28962},[1760],[772,28964,28966],{"className":28965,"style":23164},[1764],[772,28967,28968,28971],{"style":23167},[772,28969],{"className":28970,"style":23171},[1772],[772,28972,28974],{"className":28973},[1777,23175,6539,1780],[772,28975,1845],{"className":28976},[816,1780],[772,28978,3580],{"className":28979},[3579],[772,28981,28983],{"className":28982},[1760],[772,28984,28986],{"className":28985,"style":23188},[1764],[772,28987],{},[772,28989,28991,28994],{"className":28990},[816,1780],[772,28992,2839],{"className":28993,"style":2854},[816,817,1780],[772,28995,28997],{"className":28996},[1752],[772,28998,29000,29020],{"className":28999},[1756,3556],[772,29001,29003,29017],{"className":29002},[1760],[772,29004,29006],{"className":29005,"style":23355},[1764],[772,29007,29008,29011],{"style":23952},[772,29009],{"className":29010,"style":23171},[1772],[772,29012,29014],{"className":29013},[1777,23175,6539,1780],[772,29015,22471],{"className":29016,"style":22486},[816,817,1780],[772,29018,3580],{"className":29019},[3579],[772,29021,29023],{"className":29022},[1760],[772,29024,29026],{"className":29025,"style":23377},[1764],[772,29027],{},[772,29029,3580],{"className":29030},[3579],[772,29032,29034],{"className":29033},[1760],[772,29035,29037],{"className":29036,"style":23555},[1764],[772,29038],{},[772,29040],{"className":29041},[1141,5681],[323,29043,29045],{"id":29044},"worked-example","Worked Example",[706,29047,29049],{"id":29048},"given-data","Given data",[851,29051,29052,29065],{},[854,29053,29054],{},[857,29055,29056,29059,29062],{},[860,29057,29058],{},"Point",[860,29060,29061],{},"Easting (m)",[860,29063,29064],{},"Northing (m)",[876,29066,29067,29077,29088],{},[857,29068,29069,29073,29075],{},[881,29070,29071],{},[318,29072,13735],{},[881,29074,20364],{},[881,29076,20294],{},[857,29078,29079,29083,29086],{},[881,29080,29081],{},[318,29082,22441],{},[881,29084,29085],{},"3000.000",[881,29087,20294],{},[857,29089,29090,29094,29096],{},[881,29091,29092],{},[318,29093,22471],{},[881,29095,20294],{},[881,29097,29098],{},"3500.000",[11,29100,29101,29102,29130],{},"Observed angles at ",[772,29103,29105,29118],{"className":29104},[775],[772,29106,29108],{"className":29107},[779],[781,29109,29110],{"xmlns":783},[785,29111,29112,29116],{},[788,29113,29114],{},[791,29115,14943],{},[795,29117,14943],{"encoding":797},[772,29119,29121],{"className":29120,"ariaHidden":803},[802],[772,29122,29124,29127],{"className":29123},[807],[772,29125],{"className":29126,"style":2758},[811],[772,29128,14943],{"className":29129,"style":14964},[816,817]," (face-averaged, horizon closed):",[335,29132,29133,29262,29390],{},[338,29134,29135,27034,29190,29233,29234,1096],{},[772,29136,29138,29159],{"className":29137},[775],[772,29139,29141],{"className":29140},[779],[781,29142,29143],{"xmlns":783},[785,29144,29145,29156],{},[788,29146,29147,29149,29151,29154],{},[791,29148,5554],{},[980,29150,982],{},[1084,29152,29153],{},"118.4550",[791,29155,907],{"mathvariant":906},[795,29157,29158],{"encoding":797},"\\alpha = 118.4550°",[772,29160,29162,29180],{"className":29161,"ariaHidden":803},[802],[772,29163,29165,29168,29171,29174,29177],{"className":29164},[807],[772,29166],{"className":29167,"style":845},[811],[772,29169,5554],{"className":29170,"style":5570},[816,817],[772,29172],{"className":29173,"style":1004},[1003],[772,29175,982],{"className":29176},[1008],[772,29178],{"className":29179,"style":1004},[1003],[772,29181,29183,29186],{"className":29182},[807],[772,29184],{"className":29185,"style":812},[811],[772,29187,29189],{"className":29188},[816],"118.4550°",[772,29191,29193,29212],{"className":29192},[775],[772,29194,29196],{"className":29195},[779],[781,29197,29198],{"xmlns":783},[785,29199,29200,29210],{},[788,29201,29202,29204,29206,29208],{},[791,29203,24561],{"mathvariant":906},[791,29205,13735],{},[791,29207,14943],{},[791,29209,22441],{},[795,29211,24570],{"encoding":797},[772,29213,29215],{"className":29214,"ariaHidden":803},[802],[772,29216,29218,29221,29224,29227,29230],{"className":29217},[807],[772,29219],{"className":29220,"style":24580},[811],[772,29222,24561],{"className":29223},[816],[772,29225,13735],{"className":29226},[816,817],[772,29228,14943],{"className":29229,"style":14964},[816,817],[772,29231,22441],{"className":29232,"style":22456},[816,817],", opposite ",[772,29235,29237,29250],{"className":29236},[775],[772,29238,29240],{"className":29239},[779],[781,29241,29242],{"xmlns":783},[785,29243,29244,29248],{},[788,29245,29246],{},[791,29247,22471],{},[795,29249,22471],{"encoding":797},[772,29251,29253],{"className":29252,"ariaHidden":803},[802],[772,29254,29256,29259],{"className":29255},[807],[772,29257],{"className":29258,"style":2758},[811],[772,29260,22471],{"className":29261,"style":22486},[816,817],[338,29263,29264,27034,29319,29233,29362,1096],{},[772,29265,29267,29288],{"className":29266},[775],[772,29268,29270],{"className":29269},[779],[781,29271,29272],{"xmlns":783},[785,29273,29274,29285],{},[788,29275,29276,29278,29280,29283],{},[791,29277,24666],{},[980,29279,982],{},[1084,29281,29282],{},"104.3820",[791,29284,907],{"mathvariant":906},[795,29286,29287],{"encoding":797},"\\beta = 104.3820°",[772,29289,29291,29309],{"className":29290,"ariaHidden":803},[802],[772,29292,29294,29297,29300,29303,29306],{"className":29293},[807],[772,29295],{"className":29296,"style":22773},[811],[772,29298,24666],{"className":29299,"style":24682},[816,817],[772,29301],{"className":29302,"style":1004},[1003],[772,29304,982],{"className":29305},[1008],[772,29307],{"className":29308,"style":1004},[1003],[772,29310,29312,29315],{"className":29311},[807],[772,29313],{"className":29314,"style":812},[811],[772,29316,29318],{"className":29317},[816],"104.3820°",[772,29320,29322,29341],{"className":29321},[775],[772,29323,29325],{"className":29324},[779],[781,29326,29327],{"xmlns":783},[785,29328,29329,29339],{},[788,29330,29331,29333,29335,29337],{},[791,29332,24561],{"mathvariant":906},[791,29334,22441],{},[791,29336,14943],{},[791,29338,22471],{},[795,29340,24705],{"encoding":797},[772,29342,29344],{"className":29343,"ariaHidden":803},[802],[772,29345,29347,29350,29353,29356,29359],{"className":29346},[807],[772,29348],{"className":29349,"style":24580},[811],[772,29351,24561],{"className":29352},[816],[772,29354,22441],{"className":29355,"style":22456},[816,817],[772,29357,14943],{"className":29358,"style":14964},[816,817],[772,29360,22471],{"className":29361,"style":22486},[816,817],[772,29363,29365,29378],{"className":29364},[775],[772,29366,29368],{"className":29367},[779],[781,29369,29370],{"xmlns":783},[785,29371,29372,29376],{},[788,29373,29374],{},[791,29375,13735],{},[795,29377,13735],{"encoding":797},[772,29379,29381],{"className":29380,"ariaHidden":803},[802],[772,29382,29384,29387],{"className":29383},[807],[772,29385],{"className":29386,"style":2758},[811],[772,29388,13735],{"className":29389},[816,817],[338,29391,29392,27034,29447,29233,29490,1096],{},[772,29393,29395,29416],{"className":29394},[775],[772,29396,29398],{"className":29397},[779],[781,29399,29400],{"xmlns":783},[785,29401,29402,29413],{},[788,29403,29404,29406,29408,29411],{},[791,29405,24799],{},[980,29407,982],{},[1084,29409,29410],{},"137.1630",[791,29412,907],{"mathvariant":906},[795,29414,29415],{"encoding":797},"\\gamma = 137.1630°",[772,29417,29419,29437],{"className":29418,"ariaHidden":803},[802],[772,29420,29422,29425,29428,29431,29434],{"className":29421},[807],[772,29423],{"className":29424,"style":11141},[811],[772,29426,24799],{"className":29427,"style":24815},[816,817],[772,29429],{"className":29430,"style":1004},[1003],[772,29432,982],{"className":29433},[1008],[772,29435],{"className":29436,"style":1004},[1003],[772,29438,29440,29443],{"className":29439},[807],[772,29441],{"className":29442,"style":812},[811],[772,29444,29446],{"className":29445},[816],"137.1630°",[772,29448,29450,29469],{"className":29449},[775],[772,29451,29453],{"className":29452},[779],[781,29454,29455],{"xmlns":783},[785,29456,29457,29467],{},[788,29458,29459,29461,29463,29465],{},[791,29460,24561],{"mathvariant":906},[791,29462,22471],{},[791,29464,14943],{},[791,29466,13735],{},[795,29468,24838],{"encoding":797},[772,29470,29472],{"className":29471,"ariaHidden":803},[802],[772,29473,29475,29478,29481,29484,29487],{"className":29474},[807],[772,29476],{"className":29477,"style":24580},[811],[772,29479,24561],{"className":29480},[816],[772,29482,22471],{"className":29483,"style":22486},[816,817],[772,29485,14943],{"className":29486,"style":14964},[816,817],[772,29488,13735],{"className":29489},[816,817],[772,29491,29493,29506],{"className":29492},[775],[772,29494,29496],{"className":29495},[779],[781,29497,29498],{"xmlns":783},[785,29499,29500,29504],{},[788,29501,29502],{},[791,29503,22441],{},[795,29505,22441],{"encoding":797},[772,29507,29509],{"className":29508,"ariaHidden":803},[802],[772,29510,29512,29515],{"className":29511},[807],[772,29513],{"className":29514,"style":2758},[811],[772,29516,22441],{"className":29517,"style":22456},[816,817],[11,29519,29520,29521,5868],{},"Check: ",[772,29522,29524,29553],{"className":29523},[775],[772,29525,29527],{"className":29526},[779],[781,29528,29529],{"xmlns":783},[785,29530,29531,29550],{},[788,29532,29533,29535,29537,29539,29541,29543,29545,29548],{},[1084,29534,29153],{},[980,29536,1353],{},[1084,29538,29282],{},[980,29540,1353],{},[1084,29542,29410],{},[980,29544,982],{},[1084,29546,29547],{},"360.0000",[791,29549,907],{"mathvariant":906},[795,29551,29552],{"encoding":797},"118.4550 + 104.3820 + 137.1630 = 360.0000°",[772,29554,29556,29574,29592,29610],{"className":29555,"ariaHidden":803},[802],[772,29557,29559,29562,29565,29568,29571],{"className":29558},[807],[772,29560],{"className":29561,"style":7851},[811],[772,29563,29153],{"className":29564},[816],[772,29566],{"className":29567,"style":1121},[1003],[772,29569,1353],{"className":29570},[1125],[772,29572],{"className":29573,"style":1121},[1003],[772,29575,29577,29580,29583,29586,29589],{"className":29576},[807],[772,29578],{"className":29579,"style":7851},[811],[772,29581,29282],{"className":29582},[816],[772,29584],{"className":29585,"style":1121},[1003],[772,29587,1353],{"className":29588},[1125],[772,29590],{"className":29591,"style":1121},[1003],[772,29593,29595,29598,29601,29604,29607],{"className":29594},[807],[772,29596],{"className":29597,"style":5899},[811],[772,29599,29410],{"className":29600},[816],[772,29602],{"className":29603,"style":1004},[1003],[772,29605,982],{"className":29606},[1008],[772,29608],{"className":29609,"style":1004},[1003],[772,29611,29613,29616],{"className":29612},[807],[772,29614],{"className":29615,"style":812},[811],[772,29617,29619],{"className":29618},[816],"360.0000°",[706,29621,29623,29624],{"id":29622},"interior-angles-of-abctriangle-abcabc","Interior angles of ",[772,29625,29627,29646],{"className":29626},[775],[772,29628,29630],{"className":29629},[779],[781,29631,29632],{"xmlns":783},[785,29633,29634,29644],{},[788,29635,29636,29638,29640,29642],{},[791,29637,25074],{"mathvariant":906},[791,29639,13735],{},[791,29641,22441],{},[791,29643,22471],{},[795,29645,25083],{"encoding":797},[772,29647,29649],{"className":29648,"ariaHidden":803},[802],[772,29650,29652,29655,29658,29661,29664],{"className":29651},[807],[772,29653],{"className":29654,"style":22773},[811],[772,29656,25074],{"className":29657},[816],[772,29659,13735],{"className":29660},[816,817],[772,29662,22441],{"className":29663,"style":22456},[816,817],[772,29665,22471],{"className":29666,"style":22486},[816,817],[335,29668,29669,29939,30206],{},[338,29670,29671,2896,29777,29883,29884],{},[772,29672,29674,29703],{"className":29673},[775],[772,29675,29677],{"className":29676},[779],[781,29678,29679],{"xmlns":783},[785,29680,29681,29700],{},[788,29682,29683,29693,29695,29698],{},[3525,29684,29685,29687],{},[791,29686,793],{},[788,29688,29689,29691],{},[791,29690,13735],{},[791,29692,22441],{},[980,29694,982],{},[1084,29696,29697],{},"90.0000",[791,29699,907],{"mathvariant":906},[795,29701,29702],{"encoding":797},"\\theta_{AB} = 90.0000°",[772,29704,29706,29767],{"className":29705,"ariaHidden":803},[802],[772,29707,29709,29712,29758,29761,29764],{"className":29708},[807],[772,29710],{"className":29711,"style":5959},[811],[772,29713,29715,29718],{"className":29714},[816],[772,29716,793],{"className":29717,"style":818},[816,817],[772,29719,29721],{"className":29720},[1752],[772,29722,29724,29750],{"className":29723},[1756,3556],[772,29725,29727,29747],{"className":29726},[1760],[772,29728,29730],{"className":29729,"style":23077},[1764],[772,29731,29732,29735],{"style":5981},[772,29733],{"className":29734,"style":1773},[1772],[772,29736,29738],{"className":29737},[1777,1778,1779,1780],[772,29739,29741,29744],{"className":29740},[816,1780],[772,29742,13735],{"className":29743},[816,817,1780],[772,29745,22441],{"className":29746,"style":22456},[816,817,1780],[772,29748,3580],{"className":29749},[3579],[772,29751,29753],{"className":29752},[1760],[772,29754,29756],{"className":29755,"style":3587},[1764],[772,29757],{},[772,29759],{"className":29760,"style":1004},[1003],[772,29762,982],{"className":29763},[1008],[772,29765],{"className":29766,"style":1004},[1003],[772,29768,29770,29773],{"className":29769},[807],[772,29771],{"className":29772,"style":812},[811],[772,29774,29776],{"className":29775},[816],"90.0000°",[772,29778,29780,29809],{"className":29779},[775],[772,29781,29783],{"className":29782},[779],[781,29784,29785],{"xmlns":783},[785,29786,29787,29806],{},[788,29788,29789,29799,29801,29804],{},[3525,29790,29791,29793],{},[791,29792,793],{},[788,29794,29795,29797],{},[791,29796,13735],{},[791,29798,22471],{},[980,29800,982],{},[1084,29802,29803],{},"33.6901",[791,29805,907],{"mathvariant":906},[795,29807,29808],{"encoding":797},"\\theta_{AC} = 33.6901°",[772,29810,29812,29873],{"className":29811,"ariaHidden":803},[802],[772,29813,29815,29818,29864,29867,29870],{"className":29814},[807],[772,29816],{"className":29817,"style":5959},[811],[772,29819,29821,29824],{"className":29820},[816],[772,29822,793],{"className":29823,"style":818},[816,817],[772,29825,29827],{"className":29826},[1752],[772,29828,29830,29856],{"className":29829},[1756,3556],[772,29831,29833,29853],{"className":29832},[1760],[772,29834,29836],{"className":29835,"style":23077},[1764],[772,29837,29838,29841],{"style":5981},[772,29839],{"className":29840,"style":1773},[1772],[772,29842,29844],{"className":29843},[1777,1778,1779,1780],[772,29845,29847,29850],{"className":29846},[816,1780],[772,29848,13735],{"className":29849},[816,817,1780],[772,29851,22471],{"className":29852,"style":22486},[816,817,1780],[772,29854,3580],{"className":29855},[3579],[772,29857,29859],{"className":29858},[1760],[772,29860,29862],{"className":29861,"style":3587},[1764],[772,29863],{},[772,29865],{"className":29866,"style":1004},[1003],[772,29868,982],{"className":29869},[1008],[772,29871],{"className":29872,"style":1004},[1003],[772,29874,29876,29879],{"className":29875},[807],[772,29877],{"className":29878,"style":812},[811],[772,29880,29882],{"className":29881},[816],"33.6901°"," → ",[772,29885,29887,29908],{"className":29886},[775],[772,29888,29890],{"className":29889},[779],[781,29891,29892],{"xmlns":783},[785,29893,29894,29905],{},[788,29895,29896,29898,29900,29903],{},[791,29897,13735],{},[980,29899,982],{},[1084,29901,29902],{},"56.3099",[791,29904,907],{"mathvariant":906},[795,29906,29907],{"encoding":797},"A = 56.3099°",[772,29909,29911,29929],{"className":29910,"ariaHidden":803},[802],[772,29912,29914,29917,29920,29923,29926],{"className":29913},[807],[772,29915],{"className":29916,"style":2758},[811],[772,29918,13735],{"className":29919},[816,817],[772,29921],{"className":29922,"style":1004},[1003],[772,29924,982],{"className":29925},[1008],[772,29927],{"className":29928,"style":1004},[1003],[772,29930,29932,29935],{"className":29931},[807],[772,29933],{"className":29934,"style":812},[811],[772,29936,29938],{"className":29937},[816],"56.3099°",[338,29940,29941,2896,30047,29883,30153],{},[772,29942,29944,29973],{"className":29943},[775],[772,29945,29947],{"className":29946},[779],[781,29948,29949],{"xmlns":783},[785,29950,29951,29970],{},[788,29952,29953,29963,29965,29968],{},[3525,29954,29955,29957],{},[791,29956,793],{},[788,29958,29959,29961],{},[791,29960,22441],{},[791,29962,22471],{},[980,29964,982],{},[1084,29966,29967],{},"326.3099",[791,29969,907],{"mathvariant":906},[795,29971,29972],{"encoding":797},"\\theta_{BC} = 326.3099°",[772,29974,29976,30037],{"className":29975,"ariaHidden":803},[802],[772,29977,29979,29982,30028,30031,30034],{"className":29978},[807],[772,29980],{"className":29981,"style":5959},[811],[772,29983,29985,29988],{"className":29984},[816],[772,29986,793],{"className":29987,"style":818},[816,817],[772,29989,29991],{"className":29990},[1752],[772,29992,29994,30020],{"className":29993},[1756,3556],[772,29995,29997,30017],{"className":29996},[1760],[772,29998,30000],{"className":29999,"style":23077},[1764],[772,30001,30002,30005],{"style":5981},[772,30003],{"className":30004,"style":1773},[1772],[772,30006,30008],{"className":30007},[1777,1778,1779,1780],[772,30009,30011,30014],{"className":30010},[816,1780],[772,30012,22441],{"className":30013,"style":22456},[816,817,1780],[772,30015,22471],{"className":30016,"style":22486},[816,817,1780],[772,30018,3580],{"className":30019},[3579],[772,30021,30023],{"className":30022},[1760],[772,30024,30026],{"className":30025,"style":3587},[1764],[772,30027],{},[772,30029],{"className":30030,"style":1004},[1003],[772,30032,982],{"className":30033},[1008],[772,30035],{"className":30036,"style":1004},[1003],[772,30038,30040,30043],{"className":30039},[807],[772,30041],{"className":30042,"style":812},[811],[772,30044,30046],{"className":30045},[816],"326.3099°",[772,30048,30050,30079],{"className":30049},[775],[772,30051,30053],{"className":30052},[779],[781,30054,30055],{"xmlns":783},[785,30056,30057,30076],{},[788,30058,30059,30069,30071,30074],{},[3525,30060,30061,30063],{},[791,30062,793],{},[788,30064,30065,30067],{},[791,30066,22441],{},[791,30068,13735],{},[980,30070,982],{},[1084,30072,30073],{},"270.0000",[791,30075,907],{"mathvariant":906},[795,30077,30078],{"encoding":797},"\\theta_{BA} = 270.0000°",[772,30080,30082,30143],{"className":30081,"ariaHidden":803},[802],[772,30083,30085,30088,30134,30137,30140],{"className":30084},[807],[772,30086],{"className":30087,"style":5959},[811],[772,30089,30091,30094],{"className":30090},[816],[772,30092,793],{"className":30093,"style":818},[816,817],[772,30095,30097],{"className":30096},[1752],[772,30098,30100,30126],{"className":30099},[1756,3556],[772,30101,30103,30123],{"className":30102},[1760],[772,30104,30106],{"className":30105,"style":23077},[1764],[772,30107,30108,30111],{"style":5981},[772,30109],{"className":30110,"style":1773},[1772],[772,30112,30114],{"className":30113},[1777,1778,1779,1780],[772,30115,30117,30120],{"className":30116},[816,1780],[772,30118,22441],{"className":30119,"style":22456},[816,817,1780],[772,30121,13735],{"className":30122},[816,817,1780],[772,30124,3580],{"className":30125},[3579],[772,30127,30129],{"className":30128},[1760],[772,30130,30132],{"className":30131,"style":3587},[1764],[772,30133],{},[772,30135],{"className":30136,"style":1004},[1003],[772,30138,982],{"className":30139},[1008],[772,30141],{"className":30142,"style":1004},[1003],[772,30144,30146,30149],{"className":30145},[807],[772,30147],{"className":30148,"style":812},[811],[772,30150,30152],{"className":30151},[816],"270.0000°",[772,30154,30156,30176],{"className":30155},[775],[772,30157,30159],{"className":30158},[779],[781,30160,30161],{"xmlns":783},[785,30162,30163,30173],{},[788,30164,30165,30167,30169,30171],{},[791,30166,22441],{},[980,30168,982],{},[1084,30170,29902],{},[791,30172,907],{"mathvariant":906},[795,30174,30175],{"encoding":797},"B = 56.3099°",[772,30177,30179,30197],{"className":30178,"ariaHidden":803},[802],[772,30180,30182,30185,30188,30191,30194],{"className":30181},[807],[772,30183],{"className":30184,"style":2758},[811],[772,30186,22441],{"className":30187,"style":22456},[816,817],[772,30189],{"className":30190,"style":1004},[1003],[772,30192,982],{"className":30193},[1008],[772,30195],{"className":30196,"style":1004},[1003],[772,30198,30200,30203],{"className":30199},[807],[772,30201],{"className":30202,"style":812},[811],[772,30204,29938],{"className":30205},[816],[338,30207,30208,2896,30314,29883,30420],{},[772,30209,30211,30240],{"className":30210},[775],[772,30212,30214],{"className":30213},[779],[781,30215,30216],{"xmlns":783},[785,30217,30218,30237],{},[788,30219,30220,30230,30232,30235],{},[3525,30221,30222,30224],{},[791,30223,793],{},[788,30225,30226,30228],{},[791,30227,22471],{},[791,30229,13735],{},[980,30231,982],{},[1084,30233,30234],{},"213.6901",[791,30236,907],{"mathvariant":906},[795,30238,30239],{"encoding":797},"\\theta_{CA} = 213.6901°",[772,30241,30243,30304],{"className":30242,"ariaHidden":803},[802],[772,30244,30246,30249,30295,30298,30301],{"className":30245},[807],[772,30247],{"className":30248,"style":5959},[811],[772,30250,30252,30255],{"className":30251},[816],[772,30253,793],{"className":30254,"style":818},[816,817],[772,30256,30258],{"className":30257},[1752],[772,30259,30261,30287],{"className":30260},[1756,3556],[772,30262,30264,30284],{"className":30263},[1760],[772,30265,30267],{"className":30266,"style":23077},[1764],[772,30268,30269,30272],{"style":5981},[772,30270],{"className":30271,"style":1773},[1772],[772,30273,30275],{"className":30274},[1777,1778,1779,1780],[772,30276,30278,30281],{"className":30277},[816,1780],[772,30279,22471],{"className":30280,"style":22486},[816,817,1780],[772,30282,13735],{"className":30283},[816,817,1780],[772,30285,3580],{"className":30286},[3579],[772,30288,30290],{"className":30289},[1760],[772,30291,30293],{"className":30292,"style":3587},[1764],[772,30294],{},[772,30296],{"className":30297,"style":1004},[1003],[772,30299,982],{"className":30300},[1008],[772,30302],{"className":30303,"style":1004},[1003],[772,30305,30307,30310],{"className":30306},[807],[772,30308],{"className":30309,"style":812},[811],[772,30311,30313],{"className":30312},[816],"213.6901°",[772,30315,30317,30346],{"className":30316},[775],[772,30318,30320],{"className":30319},[779],[781,30321,30322],{"xmlns":783},[785,30323,30324,30343],{},[788,30325,30326,30336,30338,30341],{},[3525,30327,30328,30330],{},[791,30329,793],{},[788,30331,30332,30334],{},[791,30333,22471],{},[791,30335,22441],{},[980,30337,982],{},[1084,30339,30340],{},"146.3099",[791,30342,907],{"mathvariant":906},[795,30344,30345],{"encoding":797},"\\theta_{CB} = 146.3099°",[772,30347,30349,30410],{"className":30348,"ariaHidden":803},[802],[772,30350,30352,30355,30401,30404,30407],{"className":30351},[807],[772,30353],{"className":30354,"style":5959},[811],[772,30356,30358,30361],{"className":30357},[816],[772,30359,793],{"className":30360,"style":818},[816,817],[772,30362,30364],{"className":30363},[1752],[772,30365,30367,30393],{"className":30366},[1756,3556],[772,30368,30370,30390],{"className":30369},[1760],[772,30371,30373],{"className":30372,"style":23077},[1764],[772,30374,30375,30378],{"style":5981},[772,30376],{"className":30377,"style":1773},[1772],[772,30379,30381],{"className":30380},[1777,1778,1779,1780],[772,30382,30384,30387],{"className":30383},[816,1780],[772,30385,22471],{"className":30386,"style":22486},[816,817,1780],[772,30388,22441],{"className":30389,"style":22456},[816,817,1780],[772,30391,3580],{"className":30392},[3579],[772,30394,30396],{"className":30395},[1760],[772,30397,30399],{"className":30398,"style":3587},[1764],[772,30400],{},[772,30402],{"className":30403,"style":1004},[1003],[772,30405,982],{"className":30406},[1008],[772,30408],{"className":30409,"style":1004},[1003],[772,30411,30413,30416],{"className":30412},[807],[772,30414],{"className":30415,"style":812},[811],[772,30417,30419],{"className":30418},[816],"146.3099°",[772,30421,30423,30444],{"className":30422},[775],[772,30424,30426],{"className":30425},[779],[781,30427,30428],{"xmlns":783},[785,30429,30430,30441],{},[788,30431,30432,30434,30436,30439],{},[791,30433,22471],{},[980,30435,982],{},[1084,30437,30438],{},"67.3802",[791,30440,907],{"mathvariant":906},[795,30442,30443],{"encoding":797},"C = 67.3802°",[772,30445,30447,30465],{"className":30446,"ariaHidden":803},[802],[772,30448,30450,30453,30456,30459,30462],{"className":30449},[807],[772,30451],{"className":30452,"style":2758},[811],[772,30454,22471],{"className":30455,"style":22486},[816,817],[772,30457],{"className":30458,"style":1004},[1003],[772,30460,982],{"className":30461},[1008],[772,30463],{"className":30464,"style":1004},[1003],[772,30466,30468,30471],{"className":30467},[807],[772,30469],{"className":30470,"style":812},[811],[772,30472,30474],{"className":30473},[816],"67.3802°",[11,30476,30477,30478,5868],{},"Sum: ",[772,30479,30481,30498],{"className":30480},[775],[772,30482,30484],{"className":30483},[779],[781,30485,30486],{"xmlns":783},[785,30487,30488,30495],{},[788,30489,30490,30493],{},[1084,30491,30492],{},"180.0000",[791,30494,907],{"mathvariant":906},[795,30496,30497],{"encoding":797},"180.0000°",[772,30499,30501],{"className":30500,"ariaHidden":803},[802],[772,30502,30504,30507],{"className":30503},[807],[772,30505],{"className":30506,"style":812},[811],[772,30508,30497],{"className":30509},[816],[706,30511,30513],{"id":30512},"cotangents-and-weights","Cotangents and weights",[11,30515,30516],{},[772,30517,30519,30568],{"className":30518},[775],[772,30520,30522],{"className":30521},[779],[781,30523,30524],{"xmlns":783},[785,30525,30526,30565],{},[788,30527,30528,30530,30532,30534,30536,30539,30542,30544,30546,30548,30550,30552,30554,30556,30558,30560,30562],{},[791,30529,25154],{},[980,30531,3252],{},[791,30533,13735],{},[980,30535,7437],{},[1084,30537,30538],{},"0.666667",[1003,30540],{"width":30541},"1em",[791,30543,25154],{},[980,30545,3252],{},[791,30547,22441],{},[980,30549,7437],{},[1084,30551,30538],{},[1003,30553],{"width":30541},[791,30555,25154],{},[980,30557,3252],{},[791,30559,22471],{},[980,30561,7437],{},[1084,30563,30564],{},"0.416667",[795,30566,30567],{"encoding":797},"\\cot A \\approx 0.666667 \\quad \\cot B \\approx 0.666667 \\quad \\cot C \\approx 0.416667",[772,30569,30571,30595,30629,30662],{"className":30570,"ariaHidden":803},[802],[772,30572,30574,30577,30580,30583,30586,30589,30592],{"className":30573},[807],[772,30575],{"className":30576,"style":2758},[811],[772,30578,25154],{"className":30579},[3298],[772,30581],{"className":30582,"style":3294},[1003],[772,30584,13735],{"className":30585},[816,817],[772,30587],{"className":30588,"style":1004},[1003],[772,30590,7437],{"className":30591},[1008],[772,30593],{"className":30594,"style":1004},[1003],[772,30596,30598,30601,30604,30608,30611,30614,30617,30620,30623,30626],{"className":30597},[807],[772,30599],{"className":30600,"style":2758},[811],[772,30602,30538],{"className":30603},[816],[772,30605],{"className":30606,"style":30607},[1003],"margin-right:1em;",[772,30609],{"className":30610,"style":3294},[1003],[772,30612,25154],{"className":30613},[3298],[772,30615],{"className":30616,"style":3294},[1003],[772,30618,22441],{"className":30619,"style":22456},[816,817],[772,30621],{"className":30622,"style":1004},[1003],[772,30624,7437],{"className":30625},[1008],[772,30627],{"className":30628,"style":1004},[1003],[772,30630,30632,30635,30638,30641,30644,30647,30650,30653,30656,30659],{"className":30631},[807],[772,30633],{"className":30634,"style":2758},[811],[772,30636,30538],{"className":30637},[816],[772,30639],{"className":30640,"style":30607},[1003],[772,30642],{"className":30643,"style":3294},[1003],[772,30645,25154],{"className":30646},[3298],[772,30648],{"className":30649,"style":3294},[1003],[772,30651,22471],{"className":30652,"style":22486},[816,817],[772,30654],{"className":30655,"style":1004},[1003],[772,30657,7437],{"className":30658},[1008],[772,30660],{"className":30661,"style":1004},[1003],[772,30663,30665,30668],{"className":30664},[807],[772,30666],{"className":30667,"style":5899},[811],[772,30669,30564],{"className":30670},[816],[11,30672,30673],{},[772,30674,30676,30731],{"className":30675},[775],[772,30677,30679],{"className":30678},[779],[781,30680,30681],{"xmlns":783},[785,30682,30683,30728],{},[788,30684,30685,30687,30689,30691,30693,30695,30698,30700,30702,30704,30706,30708,30710,30713,30715,30717,30719,30721,30723,30725],{},[791,30686,25154],{},[980,30688,3252],{},[791,30690,5554],{},[980,30692,7437],{},[980,30694,1091],{},[1084,30696,30697],{},"0.541951",[1003,30699],{"width":30541},[791,30701,25154],{},[980,30703,3252],{},[791,30705,24666],{},[980,30707,7437],{},[980,30709,1091],{},[1084,30711,30712],{},"0.256428",[1003,30714],{"width":30541},[791,30716,25154],{},[980,30718,3252],{},[791,30720,24799],{},[980,30722,7437],{},[980,30724,1091],{},[1084,30726,30727],{},"1.078441",[795,30729,30730],{"encoding":797},"\\cot\\alpha \\approx -0.541951 \\quad \\cot\\beta \\approx -0.256428 \\quad \\cot\\gamma \\approx -1.078441",[772,30732,30734,30758,30794,30830],{"className":30733,"ariaHidden":803},[802],[772,30735,30737,30740,30743,30746,30749,30752,30755],{"className":30736},[807],[772,30738],{"className":30739,"style":25362},[811],[772,30741,25154],{"className":30742},[3298],[772,30744],{"className":30745,"style":3294},[1003],[772,30747,5554],{"className":30748,"style":5570},[816,817],[772,30750],{"className":30751,"style":1004},[1003],[772,30753,7437],{"className":30754},[1008],[772,30756],{"className":30757,"style":1004},[1003],[772,30759,30761,30764,30767,30770,30773,30776,30779,30782,30785,30788,30791],{"className":30760},[807],[772,30762],{"className":30763,"style":22773},[811],[772,30765,1091],{"className":30766},[816],[772,30768,30697],{"className":30769},[816],[772,30771],{"className":30772,"style":30607},[1003],[772,30774],{"className":30775,"style":3294},[1003],[772,30777,25154],{"className":30778},[3298],[772,30780],{"className":30781,"style":3294},[1003],[772,30783,24666],{"className":30784,"style":24682},[816,817],[772,30786],{"className":30787,"style":1004},[1003],[772,30789,7437],{"className":30790},[1008],[772,30792],{"className":30793,"style":1004},[1003],[772,30795,30797,30800,30803,30806,30809,30812,30815,30818,30821,30824,30827],{"className":30796},[807],[772,30798],{"className":30799,"style":11620},[811],[772,30801,1091],{"className":30802},[816],[772,30804,30712],{"className":30805},[816],[772,30807],{"className":30808,"style":30607},[1003],[772,30810],{"className":30811,"style":3294},[1003],[772,30813,25154],{"className":30814},[3298],[772,30816],{"className":30817,"style":3294},[1003],[772,30819,24799],{"className":30820,"style":24815},[816,817],[772,30822],{"className":30823,"style":1004},[1003],[772,30825,7437],{"className":30826},[1008],[772,30828],{"className":30829,"style":1004},[1003],[772,30831,30833,30836,30839],{"className":30832},[807],[772,30834],{"className":30835,"style":7851},[811],[772,30837,1091],{"className":30838},[816],[772,30840,30727],{"className":30841},[816],[11,30843,30844],{},[772,30845,30847,30916],{"className":30846},[775],[772,30848,30850],{"className":30849},[779],[781,30851,30852],{"xmlns":783},[785,30853,30854,30913],{},[788,30855,30856,30862,30864,30882,30884,30887,30889,30895,30897,30900,30902,30908,30910],{},[3525,30857,30858,30860],{},[791,30859,22758],{},[1084,30861,2170],{},[980,30863,982],{},[5599,30865,30866,30868],{},[1084,30867,2170],{},[788,30869,30870,30872,30874,30876,30878,30880],{},[1084,30871,30538],{},[980,30873,1091],{},[980,30875,1082],{"stretchy":1081},[980,30877,1091],{},[1084,30879,30712],{},[980,30881,1096],{"stretchy":1081},[980,30883,7437],{},[1084,30885,30886],{},"1.083",[1003,30888],{"width":12337},[3525,30890,30891,30893],{},[791,30892,22758],{},[1084,30894,1872],{},[980,30896,7437],{},[1084,30898,30899],{},"0.573",[1003,30901],{"width":12337},[3525,30903,30904,30906],{},[791,30905,22758],{},[1084,30907,1845],{},[980,30909,7437],{},[1084,30911,30912],{},"1.043",[795,30914,30915],{"encoding":797},"f_1 = \\frac{1}{0.666667 - (-0.256428)} \\approx 1.083 \\qquad f_2 \\approx 0.573 \\qquad f_3 \\approx 1.043",[772,30917,30919,30974,31073,31134,31195],{"className":30918,"ariaHidden":803},[802],[772,30920,30922,30925,30965,30968,30971],{"className":30921},[807],[772,30923],{"className":30924,"style":22773},[811],[772,30926,30928,30931],{"className":30927},[816],[772,30929,22758],{"className":30930,"style":22780},[816,817],[772,30932,30934],{"className":30933},[1752],[772,30935,30937,30957],{"className":30936},[1756,3556],[772,30938,30940,30954],{"className":30939},[1760],[772,30941,30943],{"className":30942,"style":3563},[1764],[772,30944,30945,30948],{"style":22795},[772,30946],{"className":30947,"style":1773},[1772],[772,30949,30951],{"className":30950},[1777,1778,1779,1780],[772,30952,2170],{"className":30953},[816,1780],[772,30955,3580],{"className":30956},[3579],[772,30958,30960],{"className":30959},[1760],[772,30961,30963],{"className":30962,"style":3587},[1764],[772,30964],{},[772,30966],{"className":30967,"style":1004},[1003],[772,30969,982],{"className":30970},[1008],[772,30972],{"className":30973,"style":1004},[1003],[772,30975,30977,30981,31064,31067,31070],{"className":30976},[807],[772,30978],{"className":30979,"style":30980},[811],"height:1.3651em;vertical-align:-0.52em;",[772,30982,30984,30987,31061],{"className":30983},[816],[772,30985],{"className":30986},[1113,5681],[772,30988,30990],{"className":30989},[5599],[772,30991,30993,31053],{"className":30992},[1756,3556],[772,30994,30996,31050],{"className":30995},[1760],[772,30997,30999,31028,31036],{"className":30998,"style":6558},[1764],[772,31000,31001,31004],{"style":5697},[772,31002],{"className":31003,"style":5422},[1772],[772,31005,31007],{"className":31006},[1777,1778,1779,1780],[772,31008,31010,31013,31016,31019,31022,31025],{"className":31009},[816,1780],[772,31011,30538],{"className":31012},[816,1780],[772,31014,1091],{"className":31015},[1125,1780],[772,31017,1082],{"className":31018},[1113,1780],[772,31020,1091],{"className":31021},[816,1780],[772,31023,30712],{"className":31024},[816,1780],[772,31026,1096],{"className":31027},[1141,1780],[772,31029,31030,31033],{"style":5719},[772,31031],{"className":31032,"style":5422},[1772],[772,31034],{"className":31035,"style":5727},[5726],[772,31037,31038,31041],{"style":6583},[772,31039],{"className":31040,"style":5422},[1772],[772,31042,31044],{"className":31043},[1777,1778,1779,1780],[772,31045,31047],{"className":31046},[816,1780],[772,31048,2170],{"className":31049},[816,1780],[772,31051,3580],{"className":31052},[3579],[772,31054,31056],{"className":31055},[1760],[772,31057,31059],{"className":31058,"style":5758},[1764],[772,31060],{},[772,31062],{"className":31063},[1141,5681],[772,31065],{"className":31066,"style":1004},[1003],[772,31068,7437],{"className":31069},[1008],[772,31071],{"className":31072,"style":1004},[1003],[772,31074,31076,31079,31082,31085,31125,31128,31131],{"className":31075},[807],[772,31077],{"className":31078,"style":22773},[811],[772,31080,30886],{"className":31081},[816],[772,31083],{"className":31084,"style":12489},[1003],[772,31086,31088,31091],{"className":31087},[816],[772,31089,22758],{"className":31090,"style":22780},[816,817],[772,31092,31094],{"className":31093},[1752],[772,31095,31097,31117],{"className":31096},[1756,3556],[772,31098,31100,31114],{"className":31099},[1760],[772,31101,31103],{"className":31102,"style":3563},[1764],[772,31104,31105,31108],{"style":22795},[772,31106],{"className":31107,"style":1773},[1772],[772,31109,31111],{"className":31110},[1777,1778,1779,1780],[772,31112,1872],{"className":31113},[816,1780],[772,31115,3580],{"className":31116},[3579],[772,31118,31120],{"className":31119},[1760],[772,31121,31123],{"className":31122,"style":3587},[1764],[772,31124],{},[772,31126],{"className":31127,"style":1004},[1003],[772,31129,7437],{"className":31130},[1008],[772,31132],{"className":31133,"style":1004},[1003],[772,31135,31137,31140,31143,31146,31186,31189,31192],{"className":31136},[807],[772,31138],{"className":31139,"style":22773},[811],[772,31141,30899],{"className":31142},[816],[772,31144],{"className":31145,"style":12489},[1003],[772,31147,31149,31152],{"className":31148},[816],[772,31150,22758],{"className":31151,"style":22780},[816,817],[772,31153,31155],{"className":31154},[1752],[772,31156,31158,31178],{"className":31157},[1756,3556],[772,31159,31161,31175],{"className":31160},[1760],[772,31162,31164],{"className":31163,"style":3563},[1764],[772,31165,31166,31169],{"style":22795},[772,31167],{"className":31168,"style":1773},[1772],[772,31170,31172],{"className":31171},[1777,1778,1779,1780],[772,31173,1845],{"className":31174},[816,1780],[772,31176,3580],{"className":31177},[3579],[772,31179,31181],{"className":31180},[1760],[772,31182,31184],{"className":31183,"style":3587},[1764],[772,31185],{},[772,31187],{"className":31188,"style":1004},[1003],[772,31190,7437],{"className":31191},[1008],[772,31193],{"className":31194,"style":1004},[1003],[772,31196,31198,31201],{"className":31197},[807],[772,31199],{"className":31200,"style":5899},[811],[772,31202,30912],{"className":31203},[816],[11,31205,31206],{},[772,31207,31209,31230],{"className":31208},[775],[772,31210,31212],{"className":31211},[779],[781,31213,31214],{"xmlns":783},[785,31215,31216,31227],{},[788,31217,31218,31220,31222,31224],{},[980,31219,13882],{},[791,31221,22758],{},[980,31223,7437],{},[1084,31225,31226],{},"2.699",[795,31228,31229],{"encoding":797},"\\sum f \\approx 2.699",[772,31231,31233,31257],{"className":31232,"ariaHidden":803},[802],[772,31234,31236,31239,31242,31245,31248,31251,31254],{"className":31235},[807],[772,31237],{"className":31238,"style":1109},[811],[772,31240,13882],{"className":31241,"style":14054},[3298,14052,14053],[772,31243],{"className":31244,"style":3294},[1003],[772,31246,22758],{"className":31247,"style":22780},[816,817],[772,31249],{"className":31250,"style":1004},[1003],[772,31252,7437],{"className":31253},[1008],[772,31255],{"className":31256,"style":1004},[1003],[772,31258,31260,31263],{"className":31259},[807],[772,31261],{"className":31262,"style":5899},[811],[772,31264,31226],{"className":31265},[816],[706,31267,31269],{"id":31268},"result","Result",[11,31271,31272],{},[772,31273,31275,31315],{"className":31274},[775],[772,31276,31278],{"className":31277},[779],[781,31279,31280],{"xmlns":783},[785,31281,31282,31312],{},[788,31283,31284,31290,31292,31295,31297,31299,31305,31307,31310],{},[3525,31285,31286,31288],{},[791,31287,2808],{},[791,31289,14943],{},[980,31291,7437],{},[1084,31293,31294],{},"1810.962",[1625,31296,20954],{},[1003,31298],{"width":12337},[3525,31300,31301,31303],{},[791,31302,2839],{},[791,31304,14943],{},[980,31306,7437],{},[1084,31308,31309],{},"2579.650",[1625,31311,20954],{},[795,31313,31314],{"encoding":797},"E_P \\approx 1810.962 \\text{ m} \\qquad N_P \\approx 2579.650 \\text{ m}",[772,31316,31318,31373,31440],{"className":31317,"ariaHidden":803},[802],[772,31319,31321,31324,31364,31367,31370],{"className":31320},[807],[772,31322],{"className":31323,"style":3543},[811],[772,31325,31327,31330],{"className":31326},[816],[772,31328,2808],{"className":31329,"style":2823},[816,817],[772,31331,31333],{"className":31332},[1752],[772,31334,31336,31356],{"className":31335},[1756,3556],[772,31337,31339,31353],{"className":31338},[1760],[772,31340,31342],{"className":31341,"style":23077},[1764],[772,31343,31344,31347],{"style":3566},[772,31345],{"className":31346,"style":1773},[1772],[772,31348,31350],{"className":31349},[1777,1778,1779,1780],[772,31351,14943],{"className":31352,"style":14964},[816,817,1780],[772,31354,3580],{"className":31355},[3579],[772,31357,31359],{"className":31358},[1760],[772,31360,31362],{"className":31361,"style":3587},[1764],[772,31363],{},[772,31365],{"className":31366,"style":1004},[1003],[772,31368,7437],{"className":31369},[1008],[772,31371],{"className":31372,"style":1004},[1003],[772,31374,31376,31379,31382,31388,31391,31431,31434,31437],{"className":31375},[807],[772,31377],{"className":31378,"style":3543},[811],[772,31380,31294],{"className":31381},[816],[772,31383,31385],{"className":31384},[816,1709],[772,31386,20954],{"className":31387},[816],[772,31389],{"className":31390,"style":12489},[1003],[772,31392,31394,31397],{"className":31393},[816],[772,31395,2839],{"className":31396,"style":2854},[816,817],[772,31398,31400],{"className":31399},[1752],[772,31401,31403,31423],{"className":31402},[1756,3556],[772,31404,31406,31420],{"className":31405},[1760],[772,31407,31409],{"className":31408,"style":23077},[1764],[772,31410,31411,31414],{"style":3640},[772,31412],{"className":31413,"style":1773},[1772],[772,31415,31417],{"className":31416},[1777,1778,1779,1780],[772,31418,14943],{"className":31419,"style":14964},[816,817,1780],[772,31421,3580],{"className":31422},[3579],[772,31424,31426],{"className":31425},[1760],[772,31427,31429],{"className":31428,"style":3587},[1764],[772,31430],{},[772,31432],{"className":31433,"style":1004},[1003],[772,31435,7437],{"className":31436},[1008],[772,31438],{"className":31439,"style":1004},[1003],[772,31441,31443,31446,31449],{"className":31442},[807],[772,31444],{"className":31445,"style":5899},[811],[772,31447,31309],{"className":31448},[816],[772,31450,31452],{"className":31451},[816,1709],[772,31453,20954],{"className":31454},[816],[11,31456,31457,31458,31460],{},"Enter the same values in the ",[693,31459,22343],{"href":22342}," to verify.",[323,31462,31464],{"id":31463},"checking-your-result","Checking Your Result",[706,31466,31468],{"id":31467},"back-bearing-check","Back-bearing check",[11,31470,31471,31472,31500,31501,31570,31571,2896,31599,2896,31627,31655],{},"From computed ",[772,31473,31475,31488],{"className":31474},[775],[772,31476,31478],{"className":31477},[779],[781,31479,31480],{"xmlns":783},[785,31481,31482,31486],{},[788,31483,31484],{},[791,31485,14943],{},[795,31487,14943],{"encoding":797},[772,31489,31491],{"className":31490,"ariaHidden":803},[802],[772,31492,31494,31497],{"className":31493},[807],[772,31495],{"className":31496,"style":2758},[811],[772,31498,14943],{"className":31499,"style":14964},[816,817],", calculate grid bearings to each control point using ",[772,31502,31504,31531],{"className":31503},[775],[772,31505,31507],{"className":31506},[779],[781,31508,31509],{"xmlns":783},[785,31510,31511,31529],{},[788,31512,31513,31515,31517,31519,31521,31523,31525,31527],{},[1625,31514,5886],{},[980,31516,1082],{"stretchy":1081},[791,31518,2963],{"mathvariant":906},[791,31520,2808],{},[980,31522,5940],{"separator":803},[791,31524,2963],{"mathvariant":906},[791,31526,2839],{},[980,31528,1096],{"stretchy":1081},[795,31530,6972],{"encoding":797},[772,31532,31534],{"className":31533,"ariaHidden":803},[802],[772,31535,31537,31540,31546,31549,31552,31555,31558,31561,31564,31567],{"className":31536},[807],[772,31538],{"className":31539,"style":1109},[811],[772,31541,31543],{"className":31542},[816,1709],[772,31544,5886],{"className":31545},[816],[772,31547,1082],{"className":31548},[1113],[772,31550,2963],{"className":31551},[816],[772,31553,2808],{"className":31554,"style":2823},[816,817],[772,31556,5940],{"className":31557},[6041],[772,31559],{"className":31560,"style":3294},[1003],[772,31562,2963],{"className":31563},[816],[772,31565,2839],{"className":31566,"style":2854},[816,817],[772,31568,1096],{"className":31569},[1141],", then reconstruct ",[772,31572,31574,31587],{"className":31573},[775],[772,31575,31577],{"className":31576},[779],[781,31578,31579],{"xmlns":783},[785,31580,31581,31585],{},[788,31582,31583],{},[791,31584,5554],{},[795,31586,5557],{"encoding":797},[772,31588,31590],{"className":31589,"ariaHidden":803},[802],[772,31591,31593,31596],{"className":31592},[807],[772,31594],{"className":31595,"style":845},[811],[772,31597,5554],{"className":31598,"style":5570},[816,817],[772,31600,31602,31615],{"className":31601},[775],[772,31603,31605],{"className":31604},[779],[781,31606,31607],{"xmlns":783},[785,31608,31609,31613],{},[788,31610,31611],{},[791,31612,24666],{},[795,31614,24669],{"encoding":797},[772,31616,31618],{"className":31617,"ariaHidden":803},[802],[772,31619,31621,31624],{"className":31620},[807],[772,31622],{"className":31623,"style":22773},[811],[772,31625,24666],{"className":31626,"style":24682},[816,817],[772,31628,31630,31643],{"className":31629},[775],[772,31631,31633],{"className":31632},[779],[781,31634,31635],{"xmlns":783},[785,31636,31637,31641],{},[788,31638,31639],{},[791,31640,24799],{},[795,31642,24802],{"encoding":797},[772,31644,31646],{"className":31645,"ariaHidden":803},[802],[772,31647,31649,31652],{"className":31648},[807],[772,31650],{"className":31651,"style":11141},[811],[772,31653,24799],{"className":31654,"style":24815},[816,817],". Reconstructed angles should match field values within a few seconds of arc.",[706,31657,31659],{"id":31658},"alternative-methods","Alternative methods",[335,31661,31662,31668,31784],{},[338,31663,31664,31667],{},[318,31665,31666],{},"Collins point"," — auxiliary construction on the circumcircle",[338,31669,31670,31673,31674,3474,31729],{},[318,31671,31672],{},"Cassini"," — intersecting auxiliary circles through ",[772,31675,31677,31699],{"className":31676},[775],[772,31678,31680],{"className":31679},[779],[781,31681,31682],{"xmlns":783},[785,31683,31684,31696],{},[788,31685,31686,31688,31690,31692,31694],{},[791,31687,13735],{},[980,31689,5940],{"separator":803},[791,31691,22441],{},[980,31693,5940],{"separator":803},[791,31695,14943],{},[795,31697,31698],{"encoding":797},"A,B,P",[772,31700,31702],{"className":31701,"ariaHidden":803},[802],[772,31703,31705,31708,31711,31714,31717,31720,31723,31726],{"className":31704},[807],[772,31706],{"className":31707,"style":1705},[811],[772,31709,13735],{"className":31710},[816,817],[772,31712,5940],{"className":31713},[6041],[772,31715],{"className":31716,"style":3294},[1003],[772,31718,22441],{"className":31719,"style":22456},[816,817],[772,31721,5940],{"className":31722},[6041],[772,31724],{"className":31725,"style":3294},[1003],[772,31727,14943],{"className":31728,"style":14964},[816,817],[772,31730,31732,31754],{"className":31731},[775],[772,31733,31735],{"className":31734},[779],[781,31736,31737],{"xmlns":783},[785,31738,31739,31751],{},[788,31740,31741,31743,31745,31747,31749],{},[791,31742,22441],{},[980,31744,5940],{"separator":803},[791,31746,22471],{},[980,31748,5940],{"separator":803},[791,31750,14943],{},[795,31752,31753],{"encoding":797},"B,C,P",[772,31755,31757],{"className":31756,"ariaHidden":803},[802],[772,31758,31760,31763,31766,31769,31772,31775,31778,31781],{"className":31759},[807],[772,31761],{"className":31762,"style":1705},[811],[772,31764,22441],{"className":31765,"style":22456},[816,817],[772,31767,5940],{"className":31768},[6041],[772,31770],{"className":31771,"style":3294},[1003],[772,31773,22471],{"className":31774,"style":22486},[816,817],[772,31776,5940],{"className":31777},[6041],[772,31779],{"className":31780,"style":3294},[1003],[772,31782,14943],{"className":31783,"style":14964},[816,817],[338,31785,31786,31789],{},[318,31787,31788],{},"Least-squares resection"," — when distances and redundant observations are available",[323,31791,31793],{"id":31792},"limitations-and-practical-warnings","Limitations and Practical Warnings",[706,31795,31797],{"id":31796},"sensitivity-to-angular-errors","Sensitivity to angular errors",[11,31799,31800,31801,3474,31831,31861,31862,31890],{},"Cotangents blow up near ",[772,31802,31804,31819],{"className":31803},[775],[772,31805,31807],{"className":31806},[779],[781,31808,31809],{"xmlns":783},[785,31810,31811,31817],{},[788,31812,31813,31815],{},[1084,31814,1748],{},[791,31816,907],{"mathvariant":906},[795,31818,12955],{"encoding":797},[772,31820,31822],{"className":31821,"ariaHidden":803},[802],[772,31823,31825,31828],{"className":31824},[807],[772,31826],{"className":31827,"style":812},[811],[772,31829,12955],{"className":31830},[816],[772,31832,31834,31849],{"className":31833},[775],[772,31835,31837],{"className":31836},[779],[781,31838,31839],{"xmlns":783},[785,31840,31841,31847],{},[788,31842,31843,31845],{},[1084,31844,1086],{},[791,31846,907],{"mathvariant":906},[795,31848,1117],{"encoding":797},[772,31850,31852],{"className":31851,"ariaHidden":803},[802],[772,31853,31855,31858],{"className":31854},[807],[772,31856],{"className":31857,"style":812},[811],[772,31859,1117],{"className":31860},[816],". Small pointing errors can shift ",[772,31863,31865,31878],{"className":31864},[775],[772,31866,31868],{"className":31867},[779],[781,31869,31870],{"xmlns":783},[785,31871,31872,31876],{},[788,31873,31874],{},[791,31875,14943],{},[795,31877,14943],{"encoding":797},[772,31879,31881],{"className":31880,"ariaHidden":803},[802],[772,31882,31884,31887],{"className":31883},[807],[772,31885],{"className":31886,"style":2758},[811],[772,31888,14943],{"className":31889,"style":14964},[816,817]," by metres when angles are ill-conditioned.",[706,31892,31894],{"id":31893},"collinear-control","Collinear control",[11,31896,31897,31898,2896,31926,2896,31954,31982],{},"Nearly collinear ",[772,31899,31901,31914],{"className":31900},[775],[772,31902,31904],{"className":31903},[779],[781,31905,31906],{"xmlns":783},[785,31907,31908,31912],{},[788,31909,31910],{},[791,31911,13735],{},[795,31913,13735],{"encoding":797},[772,31915,31917],{"className":31916,"ariaHidden":803},[802],[772,31918,31920,31923],{"className":31919},[807],[772,31921],{"className":31922,"style":2758},[811],[772,31924,13735],{"className":31925},[816,817],[772,31927,31929,31942],{"className":31928},[775],[772,31930,31932],{"className":31931},[779],[781,31933,31934],{"xmlns":783},[785,31935,31936,31940],{},[788,31937,31938],{},[791,31939,22441],{},[795,31941,22441],{"encoding":797},[772,31943,31945],{"className":31944,"ariaHidden":803},[802],[772,31946,31948,31951],{"className":31947},[807],[772,31949],{"className":31950,"style":2758},[811],[772,31952,22441],{"className":31953,"style":22456},[816,817],[772,31955,31957,31970],{"className":31956},[775],[772,31958,31960],{"className":31959},[779],[781,31961,31962],{"xmlns":783},[785,31963,31964,31968],{},[788,31965,31966],{},[791,31967,22471],{},[795,31969,22471],{"encoding":797},[772,31971,31973],{"className":31972,"ariaHidden":803},[802],[772,31974,31976,31979],{"className":31975},[807],[772,31977],{"className":31978,"style":2758},[811],[772,31980,22471],{"className":31981,"style":22486},[816,817]," collapses triangle area and makes side-of-line ambiguity severe.",[706,31984,31986],{"id":31985},"no-redundancy","No redundancy",[11,31988,31989],{},"Three angles solve two unknowns exactly — no redundancy to catch misidentified targets or disturbed control. Observe a fourth point for least-squares adjustment when possible.",[323,31991,11205],{"id":11204},[11,31993,31994,31995,5868],{},"Tienstra resection establishes 2D control from angles alone — elegant and field-practical when geometry is strong and the danger circle is avoided. For rapid computation without manual cotangent steps, use the ",[693,31996,22343],{"href":22342},{"title":15,"searchDepth":16,"depth":16,"links":31998},[31999,32000,32004,32005,32010,32015,32021,32028,32032,32037],{"id":22299,"depth":16,"text":22300},{"id":22347,"depth":16,"text":22348,"children":32001},[32002,32003],{"id":22354,"depth":73,"text":22355},{"id":22361,"depth":73,"text":22362},{"id":22673,"depth":16,"text":22674},{"id":24256,"depth":16,"text":24257,"children":32006},[32007,32008,32009],{"id":24263,"depth":73,"text":24264},{"id":25051,"depth":73,"text":25052},{"id":25392,"depth":73,"text":25393},{"id":25657,"depth":16,"text":25658,"children":32011},[32012,32013,32014],{"id":25661,"depth":73,"text":25662},{"id":25676,"depth":73,"text":25677},{"id":26472,"depth":73,"text":26473},{"id":26479,"depth":16,"text":26480,"children":32016},[32017,32018,32019],{"id":26483,"depth":73,"text":26484},{"id":27001,"depth":73,"text":27002},{"id":27851,"depth":73,"text":32020},"Step 3: Coordinates of PPP",{"id":29044,"depth":16,"text":29045,"children":32022},[32023,32024,32026,32027],{"id":29048,"depth":73,"text":29049},{"id":29622,"depth":73,"text":32025},"Interior angles of △ABC\\triangle ABC△ABC",{"id":30512,"depth":73,"text":30513},{"id":31268,"depth":73,"text":31269},{"id":31463,"depth":16,"text":31464,"children":32029},[32030,32031],{"id":31467,"depth":73,"text":31468},{"id":31658,"depth":73,"text":31659},{"id":31792,"depth":16,"text":31793,"children":32033},[32034,32035,32036],{"id":31796,"depth":73,"text":31797},{"id":31893,"depth":73,"text":31894},{"id":31985,"depth":73,"text":31986},{"id":11204,"depth":16,"text":11205},"Three-point angular resection — Tienstra barycentric method, danger circle, field practice, worked example, and verification.",{},"\u002Flearn\u002Ftools\u002Ftienstra-resection",{"title":22294,"description":32038},{"loc":32040},"learn\u002Ftools\u002Ftienstra-resection","KKNmMPhJCVWQct9uWXqsB7Mr5U3vo23rUHK8hTAyCzA",{"id":32046,"title":32047,"body":32048,"category":11245,"description":39397,"extension":20,"meta":39398,"navigation":22,"order":16,"path":39399,"seo":39400,"sitemap":39401,"stem":39402,"__hash__":39403},"learn\u002Flearn\u002Ftools\u002Ftraverse-and-bowditch.md","Traverse Closure & Bowditch Adjustment",{"type":8,"value":32049,"toc":39371},[32050,32052,32055,32058,32107,32111,32114,32124,32138,32141,32145,32205,32208,32339,32345,32459,32523,32872,32875,33140,33143,33376,33379,33586,33647,33651,33683,33993,34057,34089,34093,34096,34168,34204,34318,34455,34459,34462,34465,34469,34482,34614,35175,35178,35677,35884,36335,36338,36681,36952,36957,36961,37154,37158,37219,37223,37341,37345,37689,37693,37766,37959,37963,38025,38029,38085,38089,38743,38747,38754,38924,38927,38931,38935,39233,39237,39307,39311,39314,39318,39321,39323],[323,32051,685],{"id":684},[11,32053,32054],{},"Establishing a reliable horizontal control network is fundamental in civil engineering, boundary determination, and construction monitoring. Horizontal traversing remains the most versatile field method for extending coordinates and checking spatial consistency. Traversing yields connected lines whose directions and lengths are measured successively.",[11,32056,32057],{},"Professional traverse resolution splits into two phases: analyze raw measurements for geometric closure, then apply a systematic adjustment to distribute random measurement error. Surveying Core provides:",[335,32059,32060,32095,32102],{},[338,32061,11033,32062,32066,32067,2925],{},[693,32063,32065],{"href":32064},"\u002Ftools\u002Ftraverse-closure","Traverse closure calculator"," — departures and latitudes, linear misclosure, and relative precision (1:",[772,32068,32070,32083],{"className":32069},[775],[772,32071,32073],{"className":32072},[779],[781,32074,32075],{"xmlns":783},[785,32076,32077,32081],{},[788,32078,32079],{},[791,32080,2839],{},[795,32082,2839],{"encoding":797},[772,32084,32086],{"className":32085,"ariaHidden":803},[802],[772,32087,32089,32092],{"className":32088},[807],[772,32090],{"className":32091,"style":2758},[811],[772,32093,2839],{"className":32094,"style":2854},[816,817],[338,32096,11033,32097,32101],{},[693,32098,32100],{"href":32099},"\u002Ftools\u002Fbowditch-traverse","Bowditch adjustment calculator"," — compass-rule corrections, closed coordinates, adjusted bearings and distances.",[338,32103,11033,32104,32106],{},[693,32105,696],{"href":695}," — auxiliary coordinate and orientation conversions.",[323,32108,32110],{"id":32109},"closed-traverses-in-engineering-surveying","Closed Traverses in Engineering Surveying",[11,32112,32113],{},"A horizontal traverse is a sequence of legs between control stations. The goal is plane rectangular coordinates (eastings and northings) relative to a project grid or geodetic datum.",[11,32115,32116,32117,32120,32121,4135],{},"Traverses are ",[318,32118,32119],{},"open"," (start known, end unknown — no redundancy) or ",[318,32122,32123],{},"closed",[1602,32125,32126,32132],{},[338,32127,32128,32131],{},[318,32129,32130],{},"Closed-loop (polygon)"," — starts and ends at the same station; interior-angle and coordinate sums check against theory.",[338,32133,32134,32137],{},[318,32135,32136],{},"Closed-link (connecting)"," — starts at known control, runs through new stations, closes on a different known point — common for corridors anchored to geodetic control.",[11,32139,32140],{},"Field data per station: clockwise horizontal angles (backsight to foresight) and horizontal distances (EDM or tape). Slope reduction and curvature\u002Frefraction are pre-processing steps before the horizontal model discussed here.",[323,32142,32144],{"id":32143},"computing-traverse-closure","Computing Traverse Closure",[11,32146,32147,32148,32176,32177,5868],{},"Project each leg onto the horizontal plane using grid azimuth ",[772,32149,32151,32164],{"className":32150},[775],[772,32152,32154],{"className":32153},[779],[781,32155,32156],{"xmlns":783},[785,32157,32158,32162],{},[788,32159,32160],{},[791,32161,793],{},[795,32163,798],{"encoding":797},[772,32165,32167],{"className":32166,"ariaHidden":803},[802],[772,32168,32170,32173],{"className":32169},[807],[772,32171],{"className":32172,"style":812},[811],[772,32174,793],{"className":32175,"style":818},[816,817]," (clockwise from north, decimal degrees) and horizontal distance ",[772,32178,32180,32193],{"className":32179},[775],[772,32181,32183],{"className":32182},[779],[781,32184,32185],{"xmlns":783},[785,32186,32187,32191],{},[788,32188,32189],{},[791,32190,3180],{},[795,32192,3180],{"encoding":797},[772,32194,32196],{"className":32195,"ariaHidden":803},[802],[772,32197,32199,32202],{"className":32198},[807],[772,32200],{"className":32201,"style":2758},[811],[772,32203,3180],{"className":32204,"style":818},[816,817],[11,32206,32207],{},"Departures and latitudes (coordinate increments):",[11,32209,32210],{},[772,32211,32213,32255],{"className":32212},[775],[772,32214,32216],{"className":32215},[779],[781,32217,32218],{"xmlns":783},[785,32219,32220,32252],{},[788,32221,32222,32224,32226,32228,32230,32232,32234,32236,32238,32240,32242,32244,32246,32248,32250],{},[791,32223,2963],{"mathvariant":906},[791,32225,2808],{},[980,32227,982],{},[791,32229,3180],{},[791,32231,3249],{},[980,32233,3252],{},[791,32235,793],{},[1003,32237],{"width":12337},[791,32239,2963],{"mathvariant":906},[791,32241,2839],{},[980,32243,982],{},[791,32245,3180],{},[791,32247,3327],{},[980,32249,3252],{},[791,32251,793],{},[795,32253,32254],{"encoding":797},"\\Delta E = D \\sin\\theta \\qquad \\Delta N = D \\cos\\theta",[772,32256,32258,32279,32318],{"className":32257,"ariaHidden":803},[802],[772,32259,32261,32264,32267,32270,32273,32276],{"className":32260},[807],[772,32262],{"className":32263,"style":2758},[811],[772,32265,2963],{"className":32266},[816],[772,32268,2808],{"className":32269,"style":2823},[816,817],[772,32271],{"className":32272,"style":1004},[1003],[772,32274,982],{"className":32275},[1008],[772,32277],{"className":32278,"style":1004},[1003],[772,32280,32282,32285,32288,32291,32294,32297,32300,32303,32306,32309,32312,32315],{"className":32281},[807],[772,32283],{"className":32284,"style":812},[811],[772,32286,3180],{"className":32287,"style":818},[816,817],[772,32289],{"className":32290,"style":3294},[1003],[772,32292,3249],{"className":32293},[3298],[772,32295],{"className":32296,"style":3294},[1003],[772,32298,793],{"className":32299,"style":818},[816,817],[772,32301],{"className":32302,"style":12489},[1003],[772,32304,2963],{"className":32305},[816],[772,32307,2839],{"className":32308,"style":2854},[816,817],[772,32310],{"className":32311,"style":1004},[1003],[772,32313,982],{"className":32314},[1008],[772,32316],{"className":32317,"style":1004},[1003],[772,32319,32321,32324,32327,32330,32333,32336],{"className":32320},[807],[772,32322],{"className":32323,"style":812},[811],[772,32325,3180],{"className":32326,"style":818},[816,817],[772,32328],{"className":32329,"style":3294},[1003],[772,32331,3327],{"className":32332},[3298],[772,32334],{"className":32335,"style":3294},[1003],[772,32337,793],{"className":32338,"style":818},[816,817],[11,32340,32341,32342,4135],{},"For a perfect ",[318,32343,32344],{},"closed loop",[11,32346,32347],{},[772,32348,32350,32384],{"className":32349},[775],[772,32351,32353],{"className":32352},[779],[781,32354,32355],{"xmlns":783},[785,32356,32357,32381],{},[788,32358,32359,32361,32363,32365,32367,32369,32371,32373,32375,32377,32379],{},[980,32360,13882],{},[791,32362,2963],{"mathvariant":906},[791,32364,2808],{},[980,32366,982],{},[1084,32368,1748],{},[1003,32370],{"width":12337},[980,32372,13882],{},[791,32374,2963],{"mathvariant":906},[791,32376,2839],{},[980,32378,982],{},[1084,32380,1748],{},[795,32382,32383],{"encoding":797},"\\sum \\Delta E = 0 \\qquad \\sum \\Delta N = 0",[772,32385,32387,32414,32450],{"className":32386,"ariaHidden":803},[802],[772,32388,32390,32393,32396,32399,32402,32405,32408,32411],{"className":32389},[807],[772,32391],{"className":32392,"style":1109},[811],[772,32394,13882],{"className":32395,"style":14054},[3298,14052,14053],[772,32397],{"className":32398,"style":3294},[1003],[772,32400,2963],{"className":32401},[816],[772,32403,2808],{"className":32404,"style":2823},[816,817],[772,32406],{"className":32407,"style":1004},[1003],[772,32409,982],{"className":32410},[1008],[772,32412],{"className":32413,"style":1004},[1003],[772,32415,32417,32420,32423,32426,32429,32432,32435,32438,32441,32444,32447],{"className":32416},[807],[772,32418],{"className":32419,"style":1109},[811],[772,32421,1748],{"className":32422},[816],[772,32424],{"className":32425,"style":12489},[1003],[772,32427],{"className":32428,"style":3294},[1003],[772,32430,13882],{"className":32431,"style":14054},[3298,14052,14053],[772,32433],{"className":32434,"style":3294},[1003],[772,32436,2963],{"className":32437},[816],[772,32439,2839],{"className":32440,"style":2854},[816,817],[772,32442],{"className":32443,"style":1004},[1003],[772,32445,982],{"className":32446},[1008],[772,32448],{"className":32449,"style":1004},[1003],[772,32451,32453,32456],{"className":32452},[807],[772,32454],{"className":32455,"style":5899},[811],[772,32457,1748],{"className":32458},[816],[11,32460,32461,32462,32465,32466,6126,32495,4135],{},"For a ",[318,32463,32464],{},"closed link"," from station ",[772,32467,32469,32483],{"className":32468},[775],[772,32470,32472],{"className":32471},[779],[781,32473,32474],{"xmlns":783},[785,32475,32476,32481],{},[788,32477,32478],{},[791,32479,32480],{},"S",[795,32482,32480],{"encoding":797},[772,32484,32486],{"className":32485,"ariaHidden":803},[802],[772,32487,32489,32492],{"className":32488},[807],[772,32490],{"className":32491,"style":2758},[811],[772,32493,32480],{"className":32494,"style":2823},[816,817],[772,32496,32498,32511],{"className":32497},[775],[772,32499,32501],{"className":32500},[779],[781,32502,32503],{"xmlns":783},[785,32504,32505,32509],{},[788,32506,32507],{},[791,32508,2808],{},[795,32510,2808],{"encoding":797},[772,32512,32514],{"className":32513,"ariaHidden":803},[802],[772,32515,32517,32520],{"className":32516},[807],[772,32518],{"className":32519,"style":2758},[811],[772,32521,2808],{"className":32522,"style":2823},[816,817],[11,32524,32525],{},[772,32526,32528,32588],{"className":32527},[775],[772,32529,32531],{"className":32530},[779],[781,32532,32533],{"xmlns":783},[785,32534,32535,32585],{},[788,32536,32537,32539,32541,32543,32545,32552,32554,32561,32563,32565,32567,32569,32571,32577,32579],{},[980,32538,13882],{},[791,32540,2963],{"mathvariant":906},[791,32542,2808],{},[980,32544,982],{},[3525,32546,32547,32549],{},[791,32548,2808],{},[1625,32550,32551],{},"end",[980,32553,1091],{},[3525,32555,32556,32558],{},[791,32557,2808],{},[1625,32559,32560],{},"start",[1003,32562],{"width":12337},[980,32564,13882],{},[791,32566,2963],{"mathvariant":906},[791,32568,2839],{},[980,32570,982],{},[3525,32572,32573,32575],{},[791,32574,2839],{},[1625,32576,32551],{},[980,32578,1091],{},[3525,32580,32581,32583],{},[791,32582,2839],{},[1625,32584,32560],{},[795,32586,32587],{"encoding":797},"\\sum \\Delta E = E_{\\text{end}} - E_{\\text{start}} \\qquad \\sum \\Delta N = N_{\\text{end}} - N_{\\text{start}}",[772,32589,32591,32618,32679,32759,32820],{"className":32590,"ariaHidden":803},[802],[772,32592,32594,32597,32600,32603,32606,32609,32612,32615],{"className":32593},[807],[772,32595],{"className":32596,"style":1109},[811],[772,32598,13882],{"className":32599,"style":14054},[3298,14052,14053],[772,32601],{"className":32602,"style":3294},[1003],[772,32604,2963],{"className":32605},[816],[772,32607,2808],{"className":32608,"style":2823},[816,817],[772,32610],{"className":32611,"style":1004},[1003],[772,32613,982],{"className":32614},[1008],[772,32616],{"className":32617,"style":1004},[1003],[772,32619,32621,32624,32670,32673,32676],{"className":32620},[807],[772,32622],{"className":32623,"style":3543},[811],[772,32625,32627,32630],{"className":32626},[816],[772,32628,2808],{"className":32629,"style":2823},[816,817],[772,32631,32633],{"className":32632},[1752],[772,32634,32636,32662],{"className":32635},[1756,3556],[772,32637,32639,32659],{"className":32638},[1760],[772,32640,32642],{"className":32641,"style":5978},[1764],[772,32643,32644,32647],{"style":3566},[772,32645],{"className":32646,"style":1773},[1772],[772,32648,32650],{"className":32649},[1777,1778,1779,1780],[772,32651,32653],{"className":32652},[816,1780],[772,32654,32656],{"className":32655},[816,1709,1780],[772,32657,32551],{"className":32658},[816,1780],[772,32660,3580],{"className":32661},[3579],[772,32663,32665],{"className":32664},[1760],[772,32666,32668],{"className":32667,"style":3587},[1764],[772,32669],{},[772,32671],{"className":32672,"style":1121},[1003],[772,32674,1091],{"className":32675},[1125],[772,32677],{"className":32678,"style":1121},[1003],[772,32680,32682,32685,32732,32735,32738,32741,32744,32747,32750,32753,32756],{"className":32681},[807],[772,32683],{"className":32684,"style":1109},[811],[772,32686,32688,32691],{"className":32687},[816],[772,32689,2808],{"className":32690,"style":2823},[816,817],[772,32692,32694],{"className":32693},[1752],[772,32695,32697,32724],{"className":32696},[1756,3556],[772,32698,32700,32721],{"className":32699},[1760],[772,32701,32704],{"className":32702,"style":32703},[1764],"height:0.2806em;",[772,32705,32706,32709],{"style":3566},[772,32707],{"className":32708,"style":1773},[1772],[772,32710,32712],{"className":32711},[1777,1778,1779,1780],[772,32713,32715],{"className":32714},[816,1780],[772,32716,32718],{"className":32717},[816,1709,1780],[772,32719,32560],{"className":32720},[816,1780],[772,32722,3580],{"className":32723},[3579],[772,32725,32727],{"className":32726},[1760],[772,32728,32730],{"className":32729,"style":3587},[1764],[772,32731],{},[772,32733],{"className":32734,"style":12489},[1003],[772,32736],{"className":32737,"style":3294},[1003],[772,32739,13882],{"className":32740,"style":14054},[3298,14052,14053],[772,32742],{"className":32743,"style":3294},[1003],[772,32745,2963],{"className":32746},[816],[772,32748,2839],{"className":32749,"style":2854},[816,817],[772,32751],{"className":32752,"style":1004},[1003],[772,32754,982],{"className":32755},[1008],[772,32757],{"className":32758,"style":1004},[1003],[772,32760,32762,32765,32811,32814,32817],{"className":32761},[807],[772,32763],{"className":32764,"style":3543},[811],[772,32766,32768,32771],{"className":32767},[816],[772,32769,2839],{"className":32770,"style":2854},[816,817],[772,32772,32774],{"className":32773},[1752],[772,32775,32777,32803],{"className":32776},[1756,3556],[772,32778,32780,32800],{"className":32779},[1760],[772,32781,32783],{"className":32782,"style":5978},[1764],[772,32784,32785,32788],{"style":3640},[772,32786],{"className":32787,"style":1773},[1772],[772,32789,32791],{"className":32790},[1777,1778,1779,1780],[772,32792,32794],{"className":32793},[816,1780],[772,32795,32797],{"className":32796},[816,1709,1780],[772,32798,32551],{"className":32799},[816,1780],[772,32801,3580],{"className":32802},[3579],[772,32804,32806],{"className":32805},[1760],[772,32807,32809],{"className":32808,"style":3587},[1764],[772,32810],{},[772,32812],{"className":32813,"style":1121},[1003],[772,32815,1091],{"className":32816},[1125],[772,32818],{"className":32819,"style":1121},[1003],[772,32821,32823,32826],{"className":32822},[807],[772,32824],{"className":32825,"style":3543},[811],[772,32827,32829,32832],{"className":32828},[816],[772,32830,2839],{"className":32831,"style":2854},[816,817],[772,32833,32835],{"className":32834},[1752],[772,32836,32838,32864],{"className":32837},[1756,3556],[772,32839,32841,32861],{"className":32840},[1760],[772,32842,32844],{"className":32843,"style":32703},[1764],[772,32845,32846,32849],{"style":3640},[772,32847],{"className":32848,"style":1773},[1772],[772,32850,32852],{"className":32851},[1777,1778,1779,1780],[772,32853,32855],{"className":32854},[816,1780],[772,32856,32858],{"className":32857},[816,1709,1780],[772,32859,32560],{"className":32860},[816,1780],[772,32862,3580],{"className":32863},[3579],[772,32865,32867],{"className":32866},[1760],[772,32868,32870],{"className":32869,"style":3587},[1764],[772,32871],{},[11,32873,32874],{},"Raw field sums rarely hit these targets. Misclosure components:",[11,32876,32877],{},[772,32878,32880,32940],{"className":32879},[775],[772,32881,32883],{"className":32882},[779],[781,32884,32885],{"xmlns":783},[785,32886,32887,32937],{},[788,32888,32889,32896,32898,32900,32902,32904,32906,32909,32911,32913,32915,32921,32923,32925,32927,32929,32931,32933,32935],{},[3525,32890,32891,32894],{},[791,32892,32893],{},"e",[791,32895,2808],{},[980,32897,982],{},[980,32899,13882],{},[791,32901,2963],{"mathvariant":906},[791,32903,2808],{},[980,32905,1091],{},[1625,32907,32908],{},"target ",[791,32910,2963],{"mathvariant":906},[791,32912,2808],{},[1003,32914],{"width":12337},[3525,32916,32917,32919],{},[791,32918,32893],{},[791,32920,2839],{},[980,32922,982],{},[980,32924,13882],{},[791,32926,2963],{"mathvariant":906},[791,32928,2839],{},[980,32930,1091],{},[1625,32932,32908],{},[791,32934,2963],{"mathvariant":906},[791,32936,2839],{},[795,32938,32939],{"encoding":797},"e_E = \\sum \\Delta E - \\text{target } \\Delta E \\qquad e_N = \\sum \\Delta N - \\text{target } \\Delta N",[772,32941,32943,32998,33025,33095,33122],{"className":32942,"ariaHidden":803},[802],[772,32944,32946,32949,32989,32992,32995],{"className":32945},[807],[772,32947],{"className":32948,"style":26076},[811],[772,32950,32952,32955],{"className":32951},[816],[772,32953,32893],{"className":32954},[816,817],[772,32956,32958],{"className":32957},[1752],[772,32959,32961,32981],{"className":32960},[1756,3556],[772,32962,32964,32978],{"className":32963},[1760],[772,32965,32967],{"className":32966,"style":23077},[1764],[772,32968,32969,32972],{"style":16341},[772,32970],{"className":32971,"style":1773},[1772],[772,32973,32975],{"className":32974},[1777,1778,1779,1780],[772,32976,2808],{"className":32977,"style":2823},[816,817,1780],[772,32979,3580],{"className":32980},[3579],[772,32982,32984],{"className":32983},[1760],[772,32985,32987],{"className":32986,"style":3587},[1764],[772,32988],{},[772,32990],{"className":32991,"style":1004},[1003],[772,32993,982],{"className":32994},[1008],[772,32996],{"className":32997,"style":1004},[1003],[772,32999,33001,33004,33007,33010,33013,33016,33019,33022],{"className":33000},[807],[772,33002],{"className":33003,"style":1109},[811],[772,33005,13882],{"className":33006,"style":14054},[3298,14052,14053],[772,33008],{"className":33009,"style":3294},[1003],[772,33011,2963],{"className":33012},[816],[772,33014,2808],{"className":33015,"style":2823},[816,817],[772,33017],{"className":33018,"style":1121},[1003],[772,33020,1091],{"className":33021},[1125],[772,33023],{"className":33024,"style":1121},[1003],[772,33026,33028,33031,33037,33040,33043,33046,33086,33089,33092],{"className":33027},[807],[772,33029],{"className":33030,"style":1705},[811],[772,33032,33034],{"className":33033},[816,1709],[772,33035,32908],{"className":33036},[816],[772,33038,2963],{"className":33039},[816],[772,33041,2808],{"className":33042,"style":2823},[816,817],[772,33044],{"className":33045,"style":12489},[1003],[772,33047,33049,33052],{"className":33048},[816],[772,33050,32893],{"className":33051},[816,817],[772,33053,33055],{"className":33054},[1752],[772,33056,33058,33078],{"className":33057},[1756,3556],[772,33059,33061,33075],{"className":33060},[1760],[772,33062,33064],{"className":33063,"style":23077},[1764],[772,33065,33066,33069],{"style":16341},[772,33067],{"className":33068,"style":1773},[1772],[772,33070,33072],{"className":33071},[1777,1778,1779,1780],[772,33073,2839],{"className":33074,"style":2854},[816,817,1780],[772,33076,3580],{"className":33077},[3579],[772,33079,33081],{"className":33080},[1760],[772,33082,33084],{"className":33083,"style":3587},[1764],[772,33085],{},[772,33087],{"className":33088,"style":1004},[1003],[772,33090,982],{"className":33091},[1008],[772,33093],{"className":33094,"style":1004},[1003],[772,33096,33098,33101,33104,33107,33110,33113,33116,33119],{"className":33097},[807],[772,33099],{"className":33100,"style":1109},[811],[772,33102,13882],{"className":33103,"style":14054},[3298,14052,14053],[772,33105],{"className":33106,"style":3294},[1003],[772,33108,2963],{"className":33109},[816],[772,33111,2839],{"className":33112,"style":2854},[816,817],[772,33114],{"className":33115,"style":1121},[1003],[772,33117,1091],{"className":33118},[1125],[772,33120],{"className":33121,"style":1121},[1003],[772,33123,33125,33128,33134,33137],{"className":33124},[807],[772,33126],{"className":33127,"style":1705},[811],[772,33129,33131],{"className":33130},[816,1709],[772,33132,32908],{"className":33133},[816],[772,33135,2963],{"className":33136},[816],[772,33138,2839],{"className":33139,"style":2854},[816,817],[11,33141,33142],{},"Linear misclosure:",[11,33144,33145],{},[772,33146,33148,33186],{"className":33147},[775],[772,33149,33151],{"className":33150},[779],[781,33152,33153],{"xmlns":783},[785,33154,33155,33183],{},[788,33156,33157,33159,33161],{},[791,33158,32893],{},[980,33160,982],{},[5347,33162,33163],{},[788,33164,33165,33173,33175],{},[13878,33166,33167,33169,33171],{},[791,33168,32893],{},[791,33170,2808],{},[1084,33172,1872],{},[980,33174,1353],{},[13878,33176,33177,33179,33181],{},[791,33178,32893],{},[791,33180,2839],{},[1084,33182,1872],{},[795,33184,33185],{"encoding":797},"e = \\sqrt{e_E^2 + e_N^2}",[772,33187,33189,33207],{"className":33188,"ariaHidden":803},[802],[772,33190,33192,33195,33198,33201,33204],{"className":33191},[807],[772,33193],{"className":33194,"style":845},[811],[772,33196,32893],{"className":33197},[816,817],[772,33199],{"className":33200,"style":1004},[1003],[772,33202,982],{"className":33203},[1008],[772,33205],{"className":33206,"style":1004},[1003],[772,33208,33210,33214],{"className":33209},[807],[772,33211],{"className":33212,"style":33213},[811],"height:1.24em;vertical-align:-0.3038em;",[772,33215,33217],{"className":33216},[816,5403],[772,33218,33220,33367],{"className":33219},[1756,3556],[772,33221,33223,33364],{"className":33222},[1760],[772,33224,33227,33351],{"className":33225,"style":33226},[1764],"height:0.9362em;",[772,33228,33230,33233],{"className":33229,"style":9455},[5417],[772,33231],{"className":33232,"style":9459},[1772],[772,33234,33236,33291,33294,33297,33300],{"className":33235,"style":9463},[816],[772,33237,33239,33242],{"className":33238},[816],[772,33240,32893],{"className":33241},[816,817],[772,33243,33245],{"className":33244},[1752],[772,33246,33248,33282],{"className":33247},[1756,3556],[772,33249,33251,33279],{"className":33250},[1760],[772,33252,33255,33267],{"className":33253,"style":33254},[1764],"height:0.7959em;",[772,33256,33258,33261],{"style":33257},"top:-2.4065em;margin-left:0em;margin-right:0.05em;",[772,33259],{"className":33260,"style":1773},[1772],[772,33262,33264],{"className":33263},[1777,1778,1779,1780],[772,33265,2808],{"className":33266,"style":2823},[816,817,1780],[772,33268,33270,33273],{"style":33269},"top:-3.0448em;margin-right:0.05em;",[772,33271],{"className":33272,"style":1773},[1772],[772,33274,33276],{"className":33275},[1777,1778,1779,1780],[772,33277,1872],{"className":33278},[816,1780],[772,33280,3580],{"className":33281},[3579],[772,33283,33285],{"className":33284},[1760],[772,33286,33289],{"className":33287,"style":33288},[1764],"height:0.2935em;",[772,33290],{},[772,33292],{"className":33293,"style":1121},[1003],[772,33295,1353],{"className":33296},[1125],[772,33298],{"className":33299,"style":1121},[1003],[772,33301,33303,33306],{"className":33302},[816],[772,33304,32893],{"className":33305},[816,817],[772,33307,33309],{"className":33308},[1752],[772,33310,33312,33343],{"className":33311},[1756,3556],[772,33313,33315,33340],{"className":33314},[1760],[772,33316,33318,33329],{"className":33317,"style":33254},[1764],[772,33319,33320,33323],{"style":33257},[772,33321],{"className":33322,"style":1773},[1772],[772,33324,33326],{"className":33325},[1777,1778,1779,1780],[772,33327,2839],{"className":33328,"style":2854},[816,817,1780],[772,33330,33331,33334],{"style":33269},[772,33332],{"className":33333,"style":1773},[1772],[772,33335,33337],{"className":33336},[1777,1778,1779,1780],[772,33338,1872],{"className":33339},[816,1780],[772,33341,3580],{"className":33342},[3579],[772,33344,33346],{"className":33345},[1760],[772,33347,33349],{"className":33348,"style":33288},[1764],[772,33350],{},[772,33352,33354,33357],{"style":33353},"top:-2.8962em;",[772,33355],{"className":33356,"style":9459},[1772],[772,33358,33360],{"className":33359,"style":9555},[5511],[5514,33361,33362],{"xmlns":5516,"width":5517,"height":9558,"viewBox":9559,"preserveAspectRatio":5520},[5522,33363],{"d":9562},[772,33365,3580],{"className":33366},[3579],[772,33368,33370],{"className":33369},[1760],[772,33371,33374],{"className":33372,"style":33373},[1764],"height:0.3038em;",[772,33375],{},[11,33377,33378],{},"Misclosure azimuth:",[11,33380,33381],{},[772,33382,33384,33424],{"className":33383},[775],[772,33385,33387],{"className":33386},[779],[781,33388,33389],{"xmlns":783},[785,33390,33391,33421],{},[788,33392,33393,33399,33401,33403,33405,33411,33413,33419],{},[3525,33394,33395,33397],{},[791,33396,793],{},[791,33398,32893],{},[980,33400,982],{},[1625,33402,5886],{},[980,33404,1082],{"stretchy":1081},[3525,33406,33407,33409],{},[791,33408,32893],{},[791,33410,2808],{},[980,33412,5940],{"separator":803},[3525,33414,33415,33417],{},[791,33416,32893],{},[791,33418,2839],{},[980,33420,1096],{"stretchy":1081},[795,33422,33423],{"encoding":797},"\\theta_e = \\text{atan2}(e_E, e_N)",[772,33425,33427,33482],{"className":33426,"ariaHidden":803},[802],[772,33428,33430,33433,33473,33476,33479],{"className":33429},[807],[772,33431],{"className":33432,"style":5959},[811],[772,33434,33436,33439],{"className":33435},[816],[772,33437,793],{"className":33438,"style":818},[816,817],[772,33440,33442],{"className":33441},[1752],[772,33443,33445,33465],{"className":33444},[1756,3556],[772,33446,33448,33462],{"className":33447},[1760],[772,33449,33451],{"className":33450,"style":15612},[1764],[772,33452,33453,33456],{"style":5981},[772,33454],{"className":33455,"style":1773},[1772],[772,33457,33459],{"className":33458},[1777,1778,1779,1780],[772,33460,32893],{"className":33461},[816,817,1780],[772,33463,3580],{"className":33464},[3579],[772,33466,33468],{"className":33467},[1760],[772,33469,33471],{"className":33470,"style":3587},[1764],[772,33472],{},[772,33474],{"className":33475,"style":1004},[1003],[772,33477,982],{"className":33478},[1008],[772,33480],{"className":33481,"style":1004},[1003],[772,33483,33485,33488,33494,33497,33537,33540,33543,33583],{"className":33484},[807],[772,33486],{"className":33487,"style":1109},[811],[772,33489,33491],{"className":33490},[816,1709],[772,33492,5886],{"className":33493},[816],[772,33495,1082],{"className":33496},[1113],[772,33498,33500,33503],{"className":33499},[816],[772,33501,32893],{"className":33502},[816,817],[772,33504,33506],{"className":33505},[1752],[772,33507,33509,33529],{"className":33508},[1756,3556],[772,33510,33512,33526],{"className":33511},[1760],[772,33513,33515],{"className":33514,"style":23077},[1764],[772,33516,33517,33520],{"style":16341},[772,33518],{"className":33519,"style":1773},[1772],[772,33521,33523],{"className":33522},[1777,1778,1779,1780],[772,33524,2808],{"className":33525,"style":2823},[816,817,1780],[772,33527,3580],{"className":33528},[3579],[772,33530,33532],{"className":33531},[1760],[772,33533,33535],{"className":33534,"style":3587},[1764],[772,33536],{},[772,33538,5940],{"className":33539},[6041],[772,33541],{"className":33542,"style":3294},[1003],[772,33544,33546,33549],{"className":33545},[816],[772,33547,32893],{"className":33548},[816,817],[772,33550,33552],{"className":33551},[1752],[772,33553,33555,33575],{"className":33554},[1756,3556],[772,33556,33558,33572],{"className":33557},[1760],[772,33559,33561],{"className":33560,"style":23077},[1764],[772,33562,33563,33566],{"style":16341},[772,33564],{"className":33565,"style":1773},[1772],[772,33567,33569],{"className":33568},[1777,1778,1779,1780],[772,33570,2839],{"className":33571,"style":2854},[816,817,1780],[772,33573,3580],{"className":33574},[3579],[772,33576,33578],{"className":33577},[1760],[772,33579,33581],{"className":33580,"style":3587},[1764],[772,33582],{},[772,33584,1096],{"className":33585},[1141],[11,33587,11033,33588,33590,33591,2896,33619,2925],{},[693,33589,32065],{"href":32064}," assumes a closed loop (target misclosure ",[772,33592,33594,33607],{"className":33593},[775],[772,33595,33597],{"className":33596},[779],[781,33598,33599],{"xmlns":783},[785,33600,33601,33605],{},[788,33602,33603],{},[1084,33604,1748],{},[795,33606,1748],{"encoding":797},[772,33608,33610],{"className":33609,"ariaHidden":803},[802],[772,33611,33613,33616],{"className":33612},[807],[772,33614],{"className":33615,"style":5899},[811],[772,33617,1748],{"className":33618},[816],[772,33620,33622,33635],{"className":33621},[775],[772,33623,33625],{"className":33624},[779],[781,33626,33627],{"xmlns":783},[785,33628,33629,33633],{},[788,33630,33631],{},[1084,33632,1748],{},[795,33634,1748],{"encoding":797},[772,33636,33638],{"className":33637,"ariaHidden":803},[802],[772,33639,33641,33644],{"className":33640},[807],[772,33642],{"className":33643,"style":5899},[811],[772,33645,1748],{"className":33646},[816],[323,33648,33650],{"id":33649},"misclosure-and-relative-precision","Misclosure and Relative Precision",[11,33652,33653,33654,33682],{},"Absolute misclosure ",[772,33655,33657,33670],{"className":33656},[775],[772,33658,33660],{"className":33659},[779],[781,33661,33662],{"xmlns":783},[785,33663,33664,33668],{},[788,33665,33666],{},[791,33667,32893],{},[795,33669,32893],{"encoding":797},[772,33671,33673],{"className":33672,"ariaHidden":803},[802],[772,33674,33676,33679],{"className":33675},[807],[772,33677],{"className":33678,"style":845},[811],[772,33680,32893],{"className":33681},[816,817]," must be judged against traverse size. Relative precision normalizes quality:",[11,33684,33685],{},[772,33686,33688,33737],{"className":33687},[775],[772,33689,33691],{"className":33690},[779],[781,33692,33693],{"xmlns":783},[785,33694,33695,33734],{},[788,33696,33697,33699,33701,33703,33709,33711,33714,33716,33722,33724,33726,33728],{},[791,33698,14943],{},[980,33700,982],{},[980,33702,13882],{},[3525,33704,33705,33707],{},[791,33706,3180],{},[791,33708,13887],{},[1003,33710],{"width":12337},[1625,33712,33713],{},"Relative precision",[980,33715,982],{},[5599,33717,33718,33720],{},[791,33719,32893],{},[791,33721,14943],{},[1003,33723],{"width":12337},[791,33725,2839],{},[980,33727,982],{},[5599,33729,33730,33732],{},[791,33731,14943],{},[791,33733,32893],{},[795,33735,33736],{"encoding":797},"P = \\sum D_i \\qquad \\text{Relative precision} = \\frac{e}{P} \\qquad N = \\frac{P}{e}",[772,33738,33740,33758,33828,33917],{"className":33739,"ariaHidden":803},[802],[772,33741,33743,33746,33749,33752,33755],{"className":33742},[807],[772,33744],{"className":33745,"style":2758},[811],[772,33747,14943],{"className":33748,"style":14964},[816,817],[772,33750],{"className":33751,"style":1004},[1003],[772,33753,982],{"className":33754},[1008],[772,33756],{"className":33757,"style":1004},[1003],[772,33759,33761,33764,33767,33770,33810,33813,33819,33822,33825],{"className":33760},[807],[772,33762],{"className":33763,"style":1109},[811],[772,33765,13882],{"className":33766,"style":14054},[3298,14052,14053],[772,33768],{"className":33769,"style":3294},[1003],[772,33771,33773,33776],{"className":33772},[816],[772,33774,3180],{"className":33775,"style":818},[816,817],[772,33777,33779],{"className":33778},[1752],[772,33780,33782,33802],{"className":33781},[1756,3556],[772,33783,33785,33799],{"className":33784},[1760],[772,33786,33788],{"className":33787,"style":14137},[1764],[772,33789,33790,33793],{"style":5981},[772,33791],{"className":33792,"style":1773},[1772],[772,33794,33796],{"className":33795},[1777,1778,1779,1780],[772,33797,13887],{"className":33798},[816,817,1780],[772,33800,3580],{"className":33801},[3579],[772,33803,33805],{"className":33804},[1760],[772,33806,33808],{"className":33807,"style":3587},[1764],[772,33809],{},[772,33811],{"className":33812,"style":12489},[1003],[772,33814,33816],{"className":33815},[816,1709],[772,33817,33713],{"className":33818},[816],[772,33820],{"className":33821,"style":1004},[1003],[772,33823,982],{"className":33824},[1008],[772,33826],{"className":33827,"style":1004},[1003],[772,33829,33831,33834,33902,33905,33908,33911,33914],{"className":33830},[807],[772,33832],{"className":33833,"style":10957},[811],[772,33835,33837,33840,33899],{"className":33836},[816],[772,33838],{"className":33839},[1113,5681],[772,33841,33843],{"className":33842},[5599],[772,33844,33846,33891],{"className":33845},[1756,3556],[772,33847,33849,33888],{"className":33848},[1760],[772,33850,33852,33866,33874],{"className":33851,"style":7589},[1764],[772,33853,33854,33857],{"style":5697},[772,33855],{"className":33856,"style":5422},[1772],[772,33858,33860],{"className":33859},[1777,1778,1779,1780],[772,33861,33863],{"className":33862},[816,1780],[772,33864,14943],{"className":33865,"style":14964},[816,817,1780],[772,33867,33868,33871],{"style":5719},[772,33869],{"className":33870,"style":5422},[1772],[772,33872],{"className":33873,"style":5727},[5726],[772,33875,33876,33879],{"style":6583},[772,33877],{"className":33878,"style":5422},[1772],[772,33880,33882],{"className":33881},[1777,1778,1779,1780],[772,33883,33885],{"className":33884},[816,1780],[772,33886,32893],{"className":33887},[816,817,1780],[772,33889,3580],{"className":33890},[3579],[772,33892,33894],{"className":33893},[1760],[772,33895,33897],{"className":33896,"style":6605},[1764],[772,33898],{},[772,33900],{"className":33901},[1141,5681],[772,33903],{"className":33904,"style":12489},[1003],[772,33906,2839],{"className":33907,"style":2854},[816,817],[772,33909],{"className":33910,"style":1004},[1003],[772,33912,982],{"className":33913},[1008],[772,33915],{"className":33916,"style":1004},[1003],[772,33918,33920,33924],{"className":33919},[807],[772,33921],{"className":33922,"style":33923},[811],"height:1.2173em;vertical-align:-0.345em;",[772,33925,33927,33930,33990],{"className":33926},[816],[772,33928],{"className":33929},[1113,5681],[772,33931,33933],{"className":33932},[5599],[772,33934,33936,33982],{"className":33935},[1756,3556],[772,33937,33939,33979],{"className":33938},[1760],[772,33940,33943,33957,33965],{"className":33941,"style":33942},[1764],"height:0.8723em;",[772,33944,33945,33948],{"style":5697},[772,33946],{"className":33947,"style":5422},[1772],[772,33949,33951],{"className":33950},[1777,1778,1779,1780],[772,33952,33954],{"className":33953},[816,1780],[772,33955,32893],{"className":33956},[816,817,1780],[772,33958,33959,33962],{"style":5719},[772,33960],{"className":33961,"style":5422},[1772],[772,33963],{"className":33964,"style":5727},[5726],[772,33966,33967,33970],{"style":6583},[772,33968],{"className":33969,"style":5422},[1772],[772,33971,33973],{"className":33972},[1777,1778,1779,1780],[772,33974,33976],{"className":33975},[816,1780],[772,33977,14943],{"className":33978,"style":14964},[816,817,1780],[772,33980,3580],{"className":33981},[3579],[772,33983,33985],{"className":33984},[1760],[772,33986,33988],{"className":33987,"style":6605},[1764],[772,33989],{},[772,33991],{"className":33992},[1141,5681],[11,33994,33995,33996,34027,34028,34056],{},"Formatted as ",[318,33997,33998,33999],{},"1 : ",[772,34000,34002,34015],{"className":34001},[775],[772,34003,34005],{"className":34004},[779],[781,34006,34007],{"xmlns":783},[785,34008,34009,34013],{},[788,34010,34011],{},[791,34012,2839],{},[795,34014,2839],{"encoding":797},[772,34016,34018],{"className":34017,"ariaHidden":803},[802],[772,34019,34021,34024],{"className":34020},[807],[772,34022],{"className":34023,"style":2758},[811],[772,34025,2839],{"className":34026,"style":2854},[816,817]," — one unit of error per ",[772,34029,34031,34044],{"className":34030},[775],[772,34032,34034],{"className":34033},[779],[781,34035,34036],{"xmlns":783},[785,34037,34038,34042],{},[788,34039,34040],{},[791,34041,2839],{},[795,34043,2839],{"encoding":797},[772,34045,34047],{"className":34046,"ariaHidden":803},[802],[772,34048,34050,34053],{"className":34049},[807],[772,34051],{"className":34052,"style":2758},[811],[772,34054,2839],{"className":34055,"style":2854},[816,817]," units measured. Example: 1 : 10,000 means ~1 m error per 10 km surveyed.",[11,34058,34059,34060,34088],{},"Relative precision indicates observation consistency, not absolute positional accuracy at every point. High 1 : ",[772,34061,34063,34076],{"className":34062},[775],[772,34064,34066],{"className":34065},[779],[781,34067,34068],{"xmlns":783},[785,34069,34070,34074],{},[788,34071,34072],{},[791,34073,2839],{},[795,34075,2839],{"encoding":797},[772,34077,34079],{"className":34078,"ariaHidden":803},[802],[772,34080,34082,34085],{"className":34081},[807],[772,34083],{"className":34084,"style":2758},[811],[772,34086,2839],{"className":34087,"style":2854},[816,817]," can still hide systematic orientation or scale errors (uncalibrated EDM, wrong starting azimuth).",[323,34090,34092],{"id":34091},"is-the-traverse-acceptable","Is the Traverse Acceptable?",[11,34094,34095],{},"Before adjustment, compare relative precision to project specifications. Adjusting a traverse that fails tolerance only spreads bad data through the network.",[851,34097,34098,34111],{},[854,34099,34100],{},[857,34101,34102,34105,34108],{},[860,34103,34104],{},"Survey order \u002F class",[860,34106,34107],{},"Minimum relative precision",[860,34109,34110],{},"Typical applications",[876,34112,34113,34124,34135,34146,34157],{},[857,34114,34115,34118,34121],{},[881,34116,34117],{},"First-order",[881,34119,34120],{},"1 : 100,000",[881,34122,34123],{},"National geodetic control, deformation monitoring",[857,34125,34126,34129,34132],{},[881,34127,34128],{},"Second-order, Class I",[881,34130,34131],{},"1 : 50,000",[881,34133,34134],{},"Major infrastructure, municipal control",[857,34136,34137,34140,34143],{},[881,34138,34139],{},"Second-order, Class II",[881,34141,34142],{},"1 : 20,000",[881,34144,34145],{},"Engineering control, construction baselines",[857,34147,34148,34151,34154],{},[881,34149,34150],{},"Third-order, Class I",[881,34152,34153],{},"1 : 10,000",[881,34155,34156],{},"Local layout, suburban boundaries, topo control",[857,34158,34159,34162,34165],{},[881,34160,34161],{},"Third-order, Class II",[881,34163,34164],{},"1 : 5,000",[881,34166,34167],{},"Rural boundaries, preliminary route surveys",[11,34169,34170,34171,34174,34175,34203],{},"Closed loops also require ",[318,34172,34173],{},"angular closure",". For ",[772,34176,34178,34191],{"className":34177},[775],[772,34179,34181],{"className":34180},[779],[781,34182,34183],{"xmlns":783},[785,34184,34185,34189],{},[788,34186,34187],{},[791,34188,13833],{},[795,34190,13833],{"encoding":797},[772,34192,34194],{"className":34193,"ariaHidden":803},[802],[772,34195,34197,34200],{"className":34196},[807],[772,34198],{"className":34199,"style":845},[811],[772,34201,13833],{"className":34202},[816,817]," sides:",[11,34205,34206],{},[772,34207,34209,34243],{"className":34208},[775],[772,34210,34212],{"className":34211},[779],[781,34213,34214],{"xmlns":783},[785,34215,34216,34240],{},[788,34217,34218,34221,34223,34225,34227,34229,34231,34233,34236,34238],{},[1625,34219,34220],{},"Theoretical sum",[980,34222,982],{},[980,34224,1082],{"stretchy":1081},[791,34226,13833],{},[980,34228,1091],{},[1084,34230,1872],{},[980,34232,1096],{"stretchy":1081},[980,34234,34235],{},"⋅",[1084,34237,1086],{},[791,34239,907],{"mathvariant":906},[795,34241,34242],{"encoding":797},"\\text{Theoretical sum} = (n - 2) \\cdot 180°",[772,34244,34246,34267,34288,34309],{"className":34245,"ariaHidden":803},[802],[772,34247,34249,34252,34258,34261,34264],{"className":34248},[807],[772,34250],{"className":34251,"style":812},[811],[772,34253,34255],{"className":34254},[816,1709],[772,34256,34220],{"className":34257},[816],[772,34259],{"className":34260,"style":1004},[1003],[772,34262,982],{"className":34263},[1008],[772,34265],{"className":34266,"style":1004},[1003],[772,34268,34270,34273,34276,34279,34282,34285],{"className":34269},[807],[772,34271],{"className":34272,"style":1109},[811],[772,34274,1082],{"className":34275},[1113],[772,34277,13833],{"className":34278},[816,817],[772,34280],{"className":34281,"style":1121},[1003],[772,34283,1091],{"className":34284},[1125],[772,34286],{"className":34287,"style":1121},[1003],[772,34289,34291,34294,34297,34300,34303,34306],{"className":34290},[807],[772,34292],{"className":34293,"style":1109},[811],[772,34295,1872],{"className":34296},[816],[772,34298,1096],{"className":34299},[1141],[772,34301],{"className":34302,"style":1121},[1003],[772,34304,34235],{"className":34305},[1125],[772,34307],{"className":34308,"style":1121},[1003],[772,34310,34312,34315],{"className":34311},[807],[772,34313],{"className":34314,"style":812},[811],[772,34316,1117],{"className":34317},[816],[11,34319,34320,34321,34425,34426,34454],{},"Angular misclosure must not exceed ",[772,34322,34324,34347],{"className":34323},[775],[772,34325,34327],{"className":34326},[779],[781,34328,34329],{"xmlns":783},[785,34330,34331,34344],{},[788,34332,34333,34335,34337,34340],{},[791,34334,26314],{},[980,34336,982],{},[791,34338,34339],{},"k",[5347,34341,34342],{},[791,34343,13833],{},[795,34345,34346],{"encoding":797},"c = k\\sqrt{n}",[772,34348,34350,34368],{"className":34349,"ariaHidden":803},[802],[772,34351,34353,34356,34359,34362,34365],{"className":34352},[807],[772,34354],{"className":34355,"style":845},[811],[772,34357,26314],{"className":34358},[816,817],[772,34360],{"className":34361,"style":1004},[1003],[772,34363,982],{"className":34364},[1008],[772,34366],{"className":34367,"style":1004},[1003],[772,34369,34371,34374,34378],{"className":34370},[807],[772,34372],{"className":34373,"style":26331},[811],[772,34375,34339],{"className":34376,"style":34377},[816,817],"margin-right:0.0315em;",[772,34379,34381],{"className":34380},[816,5403],[772,34382,34384,34417],{"className":34383},[1756,3556],[772,34385,34387,34414],{"className":34386},[1760],[772,34388,34390,34402],{"className":34389,"style":26350},[1764],[772,34391,34393,34396],{"className":34392,"style":5418},[5417],[772,34394],{"className":34395,"style":5422},[1772],[772,34397,34399],{"className":34398,"style":5426},[816],[772,34400,13833],{"className":34401},[816,817],[772,34403,34404,34407],{"style":26365},[772,34405],{"className":34406,"style":5422},[1772],[772,34408,34410],{"className":34409,"style":5512},[5511],[5514,34411,34412],{"xmlns":5516,"width":5517,"height":5518,"viewBox":5519,"preserveAspectRatio":5520},[5522,34413],{"d":5524},[772,34415,3580],{"className":34416},[3579],[772,34418,34420],{"className":34419},[1760],[772,34421,34423],{"className":34422,"style":26385},[1764],[772,34424],{}," ( ",[772,34427,34429,34442],{"className":34428},[775],[772,34430,34432],{"className":34431},[779],[781,34433,34434],{"xmlns":783},[785,34435,34436,34440],{},[788,34437,34438],{},[791,34439,34339],{},[795,34441,34339],{"encoding":797},[772,34443,34445],{"className":34444,"ariaHidden":803},[802],[772,34446,34448,34451],{"className":34447},[807],[772,34449],{"className":34450,"style":812},[811],[772,34452,34339],{"className":34453,"style":34377},[816,817]," in seconds, instrument-dependent). Balance within tolerance before computing preliminary azimuths; reject and re-observe if over limit.",[323,34456,34458],{"id":34457},"why-adjustment-is-mathematically-necessary","Why Adjustment is Mathematically Necessary",[11,34460,34461],{},"Even at 1 : 25,000, unadjusted coordinates are inconsistent — the closing station gets two positions. That causes CAD\u002FGIS gaps, stakeout disagreements between legs, and boundary geometry that does not close legally.",[11,34463,34464],{},"Adjustment distributes random error across legs so one consistent coordinate set exists per station. Systematic errors should be removed by calibration and field procedure first.",[323,34466,34468],{"id":34467},"bowditch-compass-rule-adjustment","Bowditch (Compass Rule) Adjustment",[11,34470,11033,34471,11764,34474,34477,34478,34481],{},[318,34472,34473],{},"Bowditch",[318,34475,34476],{},"compass rule"," (Bowditch \u002F Adrain, c. 1808) suits traverses where angles and distances have comparable relative precision. Corrections to each leg are ",[318,34479,34480],{},"proportional to leg length"," relative to total perimeter.",[11,34483,34484,34485,34514,34515,34585,34586,4135],{},"For leg ",[772,34486,34488,34501],{"className":34487},[775],[772,34489,34491],{"className":34490},[779],[781,34492,34493],{"xmlns":783},[785,34494,34495,34499],{},[788,34496,34497],{},[791,34498,13887],{},[795,34500,13887],{"encoding":797},[772,34502,34504],{"className":34503,"ariaHidden":803},[802],[772,34505,34507,34511],{"className":34506},[807],[772,34508],{"className":34509,"style":34510},[811],"height:0.6595em;",[772,34512,13887],{"className":34513},[816,817]," with distance ",[772,34516,34518,34536],{"className":34517},[775],[772,34519,34521],{"className":34520},[779],[781,34522,34523],{"xmlns":783},[785,34524,34525,34533],{},[788,34526,34527],{},[3525,34528,34529,34531],{},[791,34530,3180],{},[791,34532,13887],{},[795,34534,34535],{"encoding":797},"D_i",[772,34537,34539],{"className":34538,"ariaHidden":803},[802],[772,34540,34542,34545],{"className":34541},[807],[772,34543],{"className":34544,"style":3543},[811],[772,34546,34548,34551],{"className":34547},[816],[772,34549,3180],{"className":34550,"style":818},[816,817],[772,34552,34554],{"className":34553},[1752],[772,34555,34557,34577],{"className":34556},[1756,3556],[772,34558,34560,34574],{"className":34559},[1760],[772,34561,34563],{"className":34562,"style":14137},[1764],[772,34564,34565,34568],{"style":5981},[772,34566],{"className":34567,"style":1773},[1772],[772,34569,34571],{"className":34570},[1777,1778,1779,1780],[772,34572,13887],{"className":34573},[816,817,1780],[772,34575,3580],{"className":34576},[3579],[772,34578,34580],{"className":34579},[1760],[772,34581,34583],{"className":34582,"style":3587},[1764],[772,34584],{}," and perimeter ",[772,34587,34589,34602],{"className":34588},[775],[772,34590,34592],{"className":34591},[779],[781,34593,34594],{"xmlns":783},[785,34595,34596,34600],{},[788,34597,34598],{},[791,34599,14943],{},[795,34601,14943],{"encoding":797},[772,34603,34605],{"className":34604,"ariaHidden":803},[802],[772,34606,34608,34611],{"className":34607},[807],[772,34609],{"className":34610,"style":2758},[811],[772,34612,14943],{"className":34613,"style":14964},[816,817],[11,34615,34616],{},[772,34617,34619,34701],{"className":34618},[775],[772,34620,34622],{"className":34621},[779],[781,34623,34624],{"xmlns":783},[785,34625,34626,34698],{},[788,34627,34628,34640,34642,34644,34650,34652,34662,34664,34676,34678,34680,34686,34688],{},[3525,34629,34630,34632],{},[791,34631,22471],{},[788,34633,34634,34636,34638],{},[791,34635,2808],{},[980,34637,5940],{"separator":803},[791,34639,13887],{},[980,34641,982],{},[980,34643,1091],{},[3525,34645,34646,34648],{},[791,34647,32893],{},[791,34649,2808],{},[980,34651,34235],{},[5599,34653,34654,34660],{},[3525,34655,34656,34658],{},[791,34657,3180],{},[791,34659,13887],{},[791,34661,14943],{},[1003,34663],{"width":12337},[3525,34665,34666,34668],{},[791,34667,22471],{},[788,34669,34670,34672,34674],{},[791,34671,2839],{},[980,34673,5940],{"separator":803},[791,34675,13887],{},[980,34677,982],{},[980,34679,1091],{},[3525,34681,34682,34684],{},[791,34683,32893],{},[791,34685,2839],{},[980,34687,34235],{},[5599,34689,34690,34696],{},[3525,34691,34692,34694],{},[791,34693,3180],{},[791,34695,13887],{},[791,34697,14943],{},[795,34699,34700],{"encoding":797},"C_{E,i} = -e_E \\cdot \\frac{D_i}{P} \\qquad C_{N,i} = -e_N \\cdot \\frac{D_i}{P}",[772,34702,34704,34770,34829,35006,35064],{"className":34703,"ariaHidden":803},[802],[772,34705,34707,34711,34761,34764,34767],{"className":34706},[807],[772,34708],{"className":34709,"style":34710},[811],"height:0.9694em;vertical-align:-0.2861em;",[772,34712,34714,34717],{"className":34713},[816],[772,34715,22471],{"className":34716,"style":22486},[816,817],[772,34718,34720],{"className":34719},[1752],[772,34721,34723,34753],{"className":34722},[1756,3556],[772,34724,34726,34750],{"className":34725},[1760],[772,34727,34729],{"className":34728,"style":23077},[1764],[772,34730,34732,34735],{"style":34731},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[772,34733],{"className":34734,"style":1773},[1772],[772,34736,34738],{"className":34737},[1777,1778,1779,1780],[772,34739,34741,34744,34747],{"className":34740},[816,1780],[772,34742,2808],{"className":34743,"style":2823},[816,817,1780],[772,34745,5940],{"className":34746},[6041,1780],[772,34748,13887],{"className":34749},[816,817,1780],[772,34751,3580],{"className":34752},[3579],[772,34754,34756],{"className":34755},[1760],[772,34757,34759],{"className":34758,"style":6301},[1764],[772,34760],{},[772,34762],{"className":34763,"style":1004},[1003],[772,34765,982],{"className":34766},[1008],[772,34768],{"className":34769,"style":1004},[1003],[772,34771,34773,34777,34780,34820,34823,34826],{"className":34772},[807],[772,34774],{"className":34775,"style":34776},[811],"height:0.7333em;vertical-align:-0.15em;",[772,34778,1091],{"className":34779},[816],[772,34781,34783,34786],{"className":34782},[816],[772,34784,32893],{"className":34785},[816,817],[772,34787,34789],{"className":34788},[1752],[772,34790,34792,34812],{"className":34791},[1756,3556],[772,34793,34795,34809],{"className":34794},[1760],[772,34796,34798],{"className":34797,"style":23077},[1764],[772,34799,34800,34803],{"style":16341},[772,34801],{"className":34802,"style":1773},[1772],[772,34804,34806],{"className":34805},[1777,1778,1779,1780],[772,34807,2808],{"className":34808,"style":2823},[816,817,1780],[772,34810,3580],{"className":34811},[3579],[772,34813,34815],{"className":34814},[1760],[772,34816,34818],{"className":34817,"style":3587},[1764],[772,34819],{},[772,34821],{"className":34822,"style":1121},[1003],[772,34824,34235],{"className":34825},[1125],[772,34827],{"className":34828,"style":1121},[1003],[772,34830,34832,34836,34945,34948,34997,35000,35003],{"className":34831},[807],[772,34833],{"className":34834,"style":34835},[811],"height:1.2334em;vertical-align:-0.345em;",[772,34837,34839,34842,34942],{"className":34838},[816],[772,34840],{"className":34841},[1113,5681],[772,34843,34845],{"className":34844},[5599],[772,34846,34848,34934],{"className":34847},[1756,3556],[772,34849,34851,34931],{"className":34850},[1760],[772,34852,34855,34869,34877],{"className":34853,"style":34854},[1764],"height:0.8884em;",[772,34856,34857,34860],{"style":5697},[772,34858],{"className":34859,"style":5422},[1772],[772,34861,34863],{"className":34862},[1777,1778,1779,1780],[772,34864,34866],{"className":34865},[816,1780],[772,34867,14943],{"className":34868,"style":14964},[816,817,1780],[772,34870,34871,34874],{"style":5719},[772,34872],{"className":34873,"style":5422},[1772],[772,34875],{"className":34876,"style":5727},[5726],[772,34878,34880,34883],{"style":34879},"top:-3.4101em;",[772,34881],{"className":34882,"style":5422},[1772],[772,34884,34886],{"className":34885},[1777,1778,1779,1780],[772,34887,34889],{"className":34888},[816,1780],[772,34890,34892,34895],{"className":34891},[816,1780],[772,34893,3180],{"className":34894,"style":818},[816,817,1780],[772,34896,34898],{"className":34897},[1752],[772,34899,34901,34923],{"className":34900},[1756,3556],[772,34902,34904,34920],{"className":34903},[1760],[772,34905,34908],{"className":34906,"style":34907},[1764],"height:0.3281em;",[772,34909,34911,34914],{"style":34910},"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;",[772,34912],{"className":34913,"style":23171},[1772],[772,34915,34917],{"className":34916},[1777,23175,6539,1780],[772,34918,13887],{"className":34919},[816,817,1780],[772,34921,3580],{"className":34922},[3579],[772,34924,34926],{"className":34925},[1760],[772,34927,34929],{"className":34928,"style":23188},[1764],[772,34930],{},[772,34932,3580],{"className":34933},[3579],[772,34935,34937],{"className":34936},[1760],[772,34938,34940],{"className":34939,"style":6605},[1764],[772,34941],{},[772,34943],{"className":34944},[1141,5681],[772,34946],{"className":34947,"style":12489},[1003],[772,34949,34951,34954],{"className":34950},[816],[772,34952,22471],{"className":34953,"style":22486},[816,817],[772,34955,34957],{"className":34956},[1752],[772,34958,34960,34989],{"className":34959},[1756,3556],[772,34961,34963,34986],{"className":34962},[1760],[772,34964,34966],{"className":34965,"style":23077},[1764],[772,34967,34968,34971],{"style":34731},[772,34969],{"className":34970,"style":1773},[1772],[772,34972,34974],{"className":34973},[1777,1778,1779,1780],[772,34975,34977,34980,34983],{"className":34976},[816,1780],[772,34978,2839],{"className":34979,"style":2854},[816,817,1780],[772,34981,5940],{"className":34982},[6041,1780],[772,34984,13887],{"className":34985},[816,817,1780],[772,34987,3580],{"className":34988},[3579],[772,34990,34992],{"className":34991},[1760],[772,34993,34995],{"className":34994,"style":6301},[1764],[772,34996],{},[772,34998],{"className":34999,"style":1004},[1003],[772,35001,982],{"className":35002},[1008],[772,35004],{"className":35005,"style":1004},[1003],[772,35007,35009,35012,35015,35055,35058,35061],{"className":35008},[807],[772,35010],{"className":35011,"style":34776},[811],[772,35013,1091],{"className":35014},[816],[772,35016,35018,35021],{"className":35017},[816],[772,35019,32893],{"className":35020},[816,817],[772,35022,35024],{"className":35023},[1752],[772,35025,35027,35047],{"className":35026},[1756,3556],[772,35028,35030,35044],{"className":35029},[1760],[772,35031,35033],{"className":35032,"style":23077},[1764],[772,35034,35035,35038],{"style":16341},[772,35036],{"className":35037,"style":1773},[1772],[772,35039,35041],{"className":35040},[1777,1778,1779,1780],[772,35042,2839],{"className":35043,"style":2854},[816,817,1780],[772,35045,3580],{"className":35046},[3579],[772,35048,35050],{"className":35049},[1760],[772,35051,35053],{"className":35052,"style":3587},[1764],[772,35054],{},[772,35056],{"className":35057,"style":1121},[1003],[772,35059,34235],{"className":35060},[1125],[772,35062],{"className":35063,"style":1121},[1003],[772,35065,35067,35070],{"className":35066},[807],[772,35068],{"className":35069,"style":34835},[811],[772,35071,35073,35076,35172],{"className":35072},[816],[772,35074],{"className":35075},[1113,5681],[772,35077,35079],{"className":35078},[5599],[772,35080,35082,35164],{"className":35081},[1756,3556],[772,35083,35085,35161],{"className":35084},[1760],[772,35086,35088,35102,35110],{"className":35087,"style":34854},[1764],[772,35089,35090,35093],{"style":5697},[772,35091],{"className":35092,"style":5422},[1772],[772,35094,35096],{"className":35095},[1777,1778,1779,1780],[772,35097,35099],{"className":35098},[816,1780],[772,35100,14943],{"className":35101,"style":14964},[816,817,1780],[772,35103,35104,35107],{"style":5719},[772,35105],{"className":35106,"style":5422},[1772],[772,35108],{"className":35109,"style":5727},[5726],[772,35111,35112,35115],{"style":34879},[772,35113],{"className":35114,"style":5422},[1772],[772,35116,35118],{"className":35117},[1777,1778,1779,1780],[772,35119,35121],{"className":35120},[816,1780],[772,35122,35124,35127],{"className":35123},[816,1780],[772,35125,3180],{"className":35126,"style":818},[816,817,1780],[772,35128,35130],{"className":35129},[1752],[772,35131,35133,35153],{"className":35132},[1756,3556],[772,35134,35136,35150],{"className":35135},[1760],[772,35137,35139],{"className":35138,"style":34907},[1764],[772,35140,35141,35144],{"style":34910},[772,35142],{"className":35143,"style":23171},[1772],[772,35145,35147],{"className":35146},[1777,23175,6539,1780],[772,35148,13887],{"className":35149},[816,817,1780],[772,35151,3580],{"className":35152},[3579],[772,35154,35156],{"className":35155},[1760],[772,35157,35159],{"className":35158,"style":23188},[1764],[772,35160],{},[772,35162,3580],{"className":35163},[3579],[772,35165,35167],{"className":35166},[1760],[772,35168,35170],{"className":35169,"style":6605},[1764],[772,35171],{},[772,35173],{"className":35174},[1141,5681],[11,35176,35177],{},"Adjusted components:",[11,35179,35180],{},[772,35181,35183,35287],{"className":35182},[775],[772,35184,35186],{"className":35185},[779],[781,35187,35188],{"xmlns":783},[785,35189,35190,35284],{},[788,35191,35192,35194,35207,35209,35211,35224,35226,35238,35240,35242,35254,35256,35258,35270,35272],{},[791,35193,2963],{"mathvariant":906},[3525,35195,35196,35198],{},[791,35197,2808],{},[788,35199,35200,35202,35204],{},[791,35201,13887],{},[980,35203,5940],{"separator":803},[1625,35205,35206],{},"adj",[980,35208,982],{},[791,35210,2963],{"mathvariant":906},[3525,35212,35213,35215],{},[791,35214,2808],{},[788,35216,35217,35219,35221],{},[791,35218,13887],{},[980,35220,5940],{"separator":803},[1625,35222,35223],{},"raw",[980,35225,1353],{},[3525,35227,35228,35230],{},[791,35229,22471],{},[788,35231,35232,35234,35236],{},[791,35233,2808],{},[980,35235,5940],{"separator":803},[791,35237,13887],{},[1003,35239],{"width":12337},[791,35241,2963],{"mathvariant":906},[3525,35243,35244,35246],{},[791,35245,2839],{},[788,35247,35248,35250,35252],{},[791,35249,13887],{},[980,35251,5940],{"separator":803},[1625,35253,35206],{},[980,35255,982],{},[791,35257,2963],{"mathvariant":906},[3525,35259,35260,35262],{},[791,35261,2839],{},[788,35263,35264,35266,35268],{},[791,35265,13887],{},[980,35267,5940],{"separator":803},[1625,35269,35223],{},[980,35271,1353],{},[3525,35273,35274,35276],{},[791,35275,22471],{},[788,35277,35278,35280,35282],{},[791,35279,2839],{},[980,35281,5940],{"separator":803},[791,35283,13887],{},[795,35285,35286],{"encoding":797},"\\Delta E_{i,\\text{adj}} = \\Delta E_{i,\\text{raw}} + C_{E,i} \\qquad \\Delta N_{i,\\text{adj}} = \\Delta N_{i,\\text{raw}} + C_{N,i}",[772,35288,35290,35360,35430,35552,35622],{"className":35289,"ariaHidden":803},[802],[772,35291,35293,35296,35299,35351,35354,35357],{"className":35292},[807],[772,35294],{"className":35295,"style":34710},[811],[772,35297,2963],{"className":35298},[816],[772,35300,35302,35305],{"className":35301},[816],[772,35303,2808],{"className":35304,"style":2823},[816,817],[772,35306,35308],{"className":35307},[1752],[772,35309,35311,35343],{"className":35310},[1756,3556],[772,35312,35314,35340],{"className":35313},[1760],[772,35315,35317],{"className":35316,"style":5978},[1764],[772,35318,35319,35322],{"style":3566},[772,35320],{"className":35321,"style":1773},[1772],[772,35323,35325],{"className":35324},[1777,1778,1779,1780],[772,35326,35328,35331,35334],{"className":35327},[816,1780],[772,35329,13887],{"className":35330},[816,817,1780],[772,35332,5940],{"className":35333},[6041,1780],[772,35335,35337],{"className":35336},[816,1709,1780],[772,35338,35206],{"className":35339},[816,1780],[772,35341,3580],{"className":35342},[3579],[772,35344,35346],{"className":35345},[1760],[772,35347,35349],{"className":35348,"style":6301},[1764],[772,35350],{},[772,35352],{"className":35353,"style":1004},[1003],[772,35355,982],{"className":35356},[1008],[772,35358],{"className":35359,"style":1004},[1003],[772,35361,35363,35366,35369,35421,35424,35427],{"className":35362},[807],[772,35364],{"className":35365,"style":34710},[811],[772,35367,2963],{"className":35368},[816],[772,35370,35372,35375],{"className":35371},[816],[772,35373,2808],{"className":35374,"style":2823},[816,817],[772,35376,35378],{"className":35377},[1752],[772,35379,35381,35413],{"className":35380},[1756,3556],[772,35382,35384,35410],{"className":35383},[1760],[772,35385,35387],{"className":35386,"style":14137},[1764],[772,35388,35389,35392],{"style":3566},[772,35390],{"className":35391,"style":1773},[1772],[772,35393,35395],{"className":35394},[1777,1778,1779,1780],[772,35396,35398,35401,35404],{"className":35397},[816,1780],[772,35399,13887],{"className":35400},[816,817,1780],[772,35402,5940],{"className":35403},[6041,1780],[772,35405,35407],{"className":35406},[816,1709,1780],[772,35408,35223],{"className":35409},[816,1780],[772,35411,3580],{"className":35412},[3579],[772,35414,35416],{"className":35415},[1760],[772,35417,35419],{"className":35418,"style":6301},[1764],[772,35420],{},[772,35422],{"className":35423,"style":1121},[1003],[772,35425,1353],{"className":35426},[1125],[772,35428],{"className":35429,"style":1121},[1003],[772,35431,35433,35436,35485,35488,35491,35543,35546,35549],{"className":35432},[807],[772,35434],{"className":35435,"style":34710},[811],[772,35437,35439,35442],{"className":35438},[816],[772,35440,22471],{"className":35441,"style":22486},[816,817],[772,35443,35445],{"className":35444},[1752],[772,35446,35448,35477],{"className":35447},[1756,3556],[772,35449,35451,35474],{"className":35450},[1760],[772,35452,35454],{"className":35453,"style":23077},[1764],[772,35455,35456,35459],{"style":34731},[772,35457],{"className":35458,"style":1773},[1772],[772,35460,35462],{"className":35461},[1777,1778,1779,1780],[772,35463,35465,35468,35471],{"className":35464},[816,1780],[772,35466,2808],{"className":35467,"style":2823},[816,817,1780],[772,35469,5940],{"className":35470},[6041,1780],[772,35472,13887],{"className":35473},[816,817,1780],[772,35475,3580],{"className":35476},[3579],[772,35478,35480],{"className":35479},[1760],[772,35481,35483],{"className":35482,"style":6301},[1764],[772,35484],{},[772,35486],{"className":35487,"style":12489},[1003],[772,35489,2963],{"className":35490},[816],[772,35492,35494,35497],{"className":35493},[816],[772,35495,2839],{"className":35496,"style":2854},[816,817],[772,35498,35500],{"className":35499},[1752],[772,35501,35503,35535],{"className":35502},[1756,3556],[772,35504,35506,35532],{"className":35505},[1760],[772,35507,35509],{"className":35508,"style":5978},[1764],[772,35510,35511,35514],{"style":3640},[772,35512],{"className":35513,"style":1773},[1772],[772,35515,35517],{"className":35516},[1777,1778,1779,1780],[772,35518,35520,35523,35526],{"className":35519},[816,1780],[772,35521,13887],{"className":35522},[816,817,1780],[772,35524,5940],{"className":35525},[6041,1780],[772,35527,35529],{"className":35528},[816,1709,1780],[772,35530,35206],{"className":35531},[816,1780],[772,35533,3580],{"className":35534},[3579],[772,35536,35538],{"className":35537},[1760],[772,35539,35541],{"className":35540,"style":6301},[1764],[772,35542],{},[772,35544],{"className":35545,"style":1004},[1003],[772,35547,982],{"className":35548},[1008],[772,35550],{"className":35551,"style":1004},[1003],[772,35553,35555,35558,35561,35613,35616,35619],{"className":35554},[807],[772,35556],{"className":35557,"style":34710},[811],[772,35559,2963],{"className":35560},[816],[772,35562,35564,35567],{"className":35563},[816],[772,35565,2839],{"className":35566,"style":2854},[816,817],[772,35568,35570],{"className":35569},[1752],[772,35571,35573,35605],{"className":35572},[1756,3556],[772,35574,35576,35602],{"className":35575},[1760],[772,35577,35579],{"className":35578,"style":14137},[1764],[772,35580,35581,35584],{"style":3640},[772,35582],{"className":35583,"style":1773},[1772],[772,35585,35587],{"className":35586},[1777,1778,1779,1780],[772,35588,35590,35593,35596],{"className":35589},[816,1780],[772,35591,13887],{"className":35592},[816,817,1780],[772,35594,5940],{"className":35595},[6041,1780],[772,35597,35599],{"className":35598},[816,1709,1780],[772,35600,35223],{"className":35601},[816,1780],[772,35603,3580],{"className":35604},[3579],[772,35606,35608],{"className":35607},[1760],[772,35609,35611],{"className":35610,"style":6301},[1764],[772,35612],{},[772,35614],{"className":35615,"style":1121},[1003],[772,35617,1353],{"className":35618},[1125],[772,35620],{"className":35621,"style":1121},[1003],[772,35623,35625,35628],{"className":35624},[807],[772,35626],{"className":35627,"style":34710},[811],[772,35629,35631,35634],{"className":35630},[816],[772,35632,22471],{"className":35633,"style":22486},[816,817],[772,35635,35637],{"className":35636},[1752],[772,35638,35640,35669],{"className":35639},[1756,3556],[772,35641,35643,35666],{"className":35642},[1760],[772,35644,35646],{"className":35645,"style":23077},[1764],[772,35647,35648,35651],{"style":34731},[772,35649],{"className":35650,"style":1773},[1772],[772,35652,35654],{"className":35653},[1777,1778,1779,1780],[772,35655,35657,35660,35663],{"className":35656},[816,1780],[772,35658,2839],{"className":35659,"style":2854},[816,817,1780],[772,35661,5940],{"className":35662},[6041,1780],[772,35664,13887],{"className":35665},[816,817,1780],[772,35667,3580],{"className":35668},[3579],[772,35670,35672],{"className":35671},[1760],[772,35673,35675],{"className":35674,"style":6301},[1764],[772,35676],{},[11,35678,35679,35680,3474,35782,35883],{},"Then ",[772,35681,35683,35711],{"className":35682},[775],[772,35684,35686],{"className":35685},[779],[781,35687,35688],{"xmlns":783},[785,35689,35690,35708],{},[788,35691,35692,35694,35696],{},[980,35693,13882],{},[791,35695,2963],{"mathvariant":906},[3525,35697,35698,35700],{},[791,35699,2808],{},[788,35701,35702,35704,35706],{},[791,35703,13887],{},[980,35705,5940],{"separator":803},[1625,35707,35206],{},[795,35709,35710],{"encoding":797},"\\sum \\Delta E_{i,\\text{adj}}",[772,35712,35714],{"className":35713,"ariaHidden":803},[802],[772,35715,35717,35721,35724,35727,35730],{"className":35716},[807],[772,35718],{"className":35719,"style":35720},[811],"height:1.0361em;vertical-align:-0.2861em;",[772,35722,13882],{"className":35723,"style":14054},[3298,14052,14053],[772,35725],{"className":35726,"style":3294},[1003],[772,35728,2963],{"className":35729},[816],[772,35731,35733,35736],{"className":35732},[816],[772,35734,2808],{"className":35735,"style":2823},[816,817],[772,35737,35739],{"className":35738},[1752],[772,35740,35742,35774],{"className":35741},[1756,3556],[772,35743,35745,35771],{"className":35744},[1760],[772,35746,35748],{"className":35747,"style":5978},[1764],[772,35749,35750,35753],{"style":3566},[772,35751],{"className":35752,"style":1773},[1772],[772,35754,35756],{"className":35755},[1777,1778,1779,1780],[772,35757,35759,35762,35765],{"className":35758},[816,1780],[772,35760,13887],{"className":35761},[816,817,1780],[772,35763,5940],{"className":35764},[6041,1780],[772,35766,35768],{"className":35767},[816,1709,1780],[772,35769,35206],{"className":35770},[816,1780],[772,35772,3580],{"className":35773},[3579],[772,35775,35777],{"className":35776},[1760],[772,35778,35780],{"className":35779,"style":6301},[1764],[772,35781],{},[772,35783,35785,35813],{"className":35784},[775],[772,35786,35788],{"className":35787},[779],[781,35789,35790],{"xmlns":783},[785,35791,35792,35810],{},[788,35793,35794,35796,35798],{},[980,35795,13882],{},[791,35797,2963],{"mathvariant":906},[3525,35799,35800,35802],{},[791,35801,2839],{},[788,35803,35804,35806,35808],{},[791,35805,13887],{},[980,35807,5940],{"separator":803},[1625,35809,35206],{},[795,35811,35812],{"encoding":797},"\\sum \\Delta N_{i,\\text{adj}}",[772,35814,35816],{"className":35815,"ariaHidden":803},[802],[772,35817,35819,35822,35825,35828,35831],{"className":35818},[807],[772,35820],{"className":35821,"style":35720},[811],[772,35823,13882],{"className":35824,"style":14054},[3298,14052,14053],[772,35826],{"className":35827,"style":3294},[1003],[772,35829,2963],{"className":35830},[816],[772,35832,35834,35837],{"className":35833},[816],[772,35835,2839],{"className":35836,"style":2854},[816,817],[772,35838,35840],{"className":35839},[1752],[772,35841,35843,35875],{"className":35842},[1756,3556],[772,35844,35846,35872],{"className":35845},[1760],[772,35847,35849],{"className":35848,"style":5978},[1764],[772,35850,35851,35854],{"style":3640},[772,35852],{"className":35853,"style":1773},[1772],[772,35855,35857],{"className":35856},[1777,1778,1779,1780],[772,35858,35860,35863,35866],{"className":35859},[816,1780],[772,35861,13887],{"className":35862},[816,817,1780],[772,35864,5940],{"className":35865},[6041,1780],[772,35867,35869],{"className":35868},[816,1709,1780],[772,35870,35206],{"className":35871},[816,1780],[772,35873,3580],{"className":35874},[3579],[772,35876,35878],{"className":35877},[1760],[772,35879,35881],{"className":35880,"style":6301},[1764],[772,35882],{}," match targets. Progressive coordinates:",[11,35885,35886],{},[772,35887,35889,35975],{"className":35888},[775],[772,35890,35892],{"className":35891},[779],[781,35893,35894],{"xmlns":783},[785,35895,35896,35972],{},[788,35897,35898,35910,35912,35918,35920,35922,35934,35936,35948,35950,35956,35958,35960],{},[3525,35899,35900,35902],{},[791,35901,2808],{},[788,35903,35904,35906,35908],{},[791,35905,13887],{},[980,35907,1353],{},[1084,35909,2170],{},[980,35911,982],{},[3525,35913,35914,35916],{},[791,35915,2808],{},[791,35917,13887],{},[980,35919,1353],{},[791,35921,2963],{"mathvariant":906},[3525,35923,35924,35926],{},[791,35925,2808],{},[788,35927,35928,35930,35932],{},[791,35929,13887],{},[980,35931,5940],{"separator":803},[1625,35933,35206],{},[1003,35935],{"width":12337},[3525,35937,35938,35940],{},[791,35939,2839],{},[788,35941,35942,35944,35946],{},[791,35943,13887],{},[980,35945,1353],{},[1084,35947,2170],{},[980,35949,982],{},[3525,35951,35952,35954],{},[791,35953,2839],{},[791,35955,13887],{},[980,35957,1353],{},[791,35959,2963],{"mathvariant":906},[3525,35961,35962,35964],{},[791,35963,2839],{},[788,35965,35966,35968,35970],{},[791,35967,13887],{},[980,35969,5940],{"separator":803},[1625,35971,35206],{},[795,35973,35974],{"encoding":797},"E_{i+1} = E_i + \\Delta E_{i,\\text{adj}} \\qquad N_{i+1} = N_i + \\Delta N_{i,\\text{adj}}",[772,35976,35978,36042,36097,36219,36274],{"className":35977,"ariaHidden":803},[802],[772,35979,35981,35984,36033,36036,36039],{"className":35980},[807],[772,35982],{"className":35983,"style":14673},[811],[772,35985,35987,35990],{"className":35986},[816],[772,35988,2808],{"className":35989,"style":2823},[816,817],[772,35991,35993],{"className":35992},[1752],[772,35994,35996,36025],{"className":35995},[1756,3556],[772,35997,35999,36022],{"className":35998},[1760],[772,36000,36002],{"className":36001,"style":14137},[1764],[772,36003,36004,36007],{"style":3566},[772,36005],{"className":36006,"style":1773},[1772],[772,36008,36010],{"className":36009},[1777,1778,1779,1780],[772,36011,36013,36016,36019],{"className":36012},[816,1780],[772,36014,13887],{"className":36015},[816,817,1780],[772,36017,1353],{"className":36018},[1125,1780],[772,36020,2170],{"className":36021},[816,1780],[772,36023,3580],{"className":36024},[3579],[772,36026,36028],{"className":36027},[1760],[772,36029,36031],{"className":36030,"style":14207},[1764],[772,36032],{},[772,36034],{"className":36035,"style":1004},[1003],[772,36037,982],{"className":36038},[1008],[772,36040],{"className":36041,"style":1004},[1003],[772,36043,36045,36048,36088,36091,36094],{"className":36044},[807],[772,36046],{"className":36047,"style":3543},[811],[772,36049,36051,36054],{"className":36050},[816],[772,36052,2808],{"className":36053,"style":2823},[816,817],[772,36055,36057],{"className":36056},[1752],[772,36058,36060,36080],{"className":36059},[1756,3556],[772,36061,36063,36077],{"className":36062},[1760],[772,36064,36066],{"className":36065,"style":14137},[1764],[772,36067,36068,36071],{"style":3566},[772,36069],{"className":36070,"style":1773},[1772],[772,36072,36074],{"className":36073},[1777,1778,1779,1780],[772,36075,13887],{"className":36076},[816,817,1780],[772,36078,3580],{"className":36079},[3579],[772,36081,36083],{"className":36082},[1760],[772,36084,36086],{"className":36085,"style":3587},[1764],[772,36087],{},[772,36089],{"className":36090,"style":1121},[1003],[772,36092,1353],{"className":36093},[1125],[772,36095],{"className":36096,"style":1121},[1003],[772,36098,36100,36103,36106,36158,36161,36210,36213,36216],{"className":36099},[807],[772,36101],{"className":36102,"style":34710},[811],[772,36104,2963],{"className":36105},[816],[772,36107,36109,36112],{"className":36108},[816],[772,36110,2808],{"className":36111,"style":2823},[816,817],[772,36113,36115],{"className":36114},[1752],[772,36116,36118,36150],{"className":36117},[1756,3556],[772,36119,36121,36147],{"className":36120},[1760],[772,36122,36124],{"className":36123,"style":5978},[1764],[772,36125,36126,36129],{"style":3566},[772,36127],{"className":36128,"style":1773},[1772],[772,36130,36132],{"className":36131},[1777,1778,1779,1780],[772,36133,36135,36138,36141],{"className":36134},[816,1780],[772,36136,13887],{"className":36137},[816,817,1780],[772,36139,5940],{"className":36140},[6041,1780],[772,36142,36144],{"className":36143},[816,1709,1780],[772,36145,35206],{"className":36146},[816,1780],[772,36148,3580],{"className":36149},[3579],[772,36151,36153],{"className":36152},[1760],[772,36154,36156],{"className":36155,"style":6301},[1764],[772,36157],{},[772,36159],{"className":36160,"style":12489},[1003],[772,36162,36164,36167],{"className":36163},[816],[772,36165,2839],{"className":36166,"style":2854},[816,817],[772,36168,36170],{"className":36169},[1752],[772,36171,36173,36202],{"className":36172},[1756,3556],[772,36174,36176,36199],{"className":36175},[1760],[772,36177,36179],{"className":36178,"style":14137},[1764],[772,36180,36181,36184],{"style":3640},[772,36182],{"className":36183,"style":1773},[1772],[772,36185,36187],{"className":36186},[1777,1778,1779,1780],[772,36188,36190,36193,36196],{"className":36189},[816,1780],[772,36191,13887],{"className":36192},[816,817,1780],[772,36194,1353],{"className":36195},[1125,1780],[772,36197,2170],{"className":36198},[816,1780],[772,36200,3580],{"className":36201},[3579],[772,36203,36205],{"className":36204},[1760],[772,36206,36208],{"className":36207,"style":14207},[1764],[772,36209],{},[772,36211],{"className":36212,"style":1004},[1003],[772,36214,982],{"className":36215},[1008],[772,36217],{"className":36218,"style":1004},[1003],[772,36220,36222,36225,36265,36268,36271],{"className":36221},[807],[772,36223],{"className":36224,"style":3543},[811],[772,36226,36228,36231],{"className":36227},[816],[772,36229,2839],{"className":36230,"style":2854},[816,817],[772,36232,36234],{"className":36233},[1752],[772,36235,36237,36257],{"className":36236},[1756,3556],[772,36238,36240,36254],{"className":36239},[1760],[772,36241,36243],{"className":36242,"style":14137},[1764],[772,36244,36245,36248],{"style":3640},[772,36246],{"className":36247,"style":1773},[1772],[772,36249,36251],{"className":36250},[1777,1778,1779,1780],[772,36252,13887],{"className":36253},[816,817,1780],[772,36255,3580],{"className":36256},[3579],[772,36258,36260],{"className":36259},[1760],[772,36261,36263],{"className":36262,"style":3587},[1764],[772,36264],{},[772,36266],{"className":36267,"style":1121},[1003],[772,36269,1353],{"className":36270},[1125],[772,36272],{"className":36273,"style":1121},[1003],[772,36275,36277,36280,36283],{"className":36276},[807],[772,36278],{"className":36279,"style":34710},[811],[772,36281,2963],{"className":36282},[816],[772,36284,36286,36289],{"className":36285},[816],[772,36287,2839],{"className":36288,"style":2854},[816,817],[772,36290,36292],{"className":36291},[1752],[772,36293,36295,36327],{"className":36294},[1756,3556],[772,36296,36298,36324],{"className":36297},[1760],[772,36299,36301],{"className":36300,"style":5978},[1764],[772,36302,36303,36306],{"style":3640},[772,36304],{"className":36305,"style":1773},[1772],[772,36307,36309],{"className":36308},[1777,1778,1779,1780],[772,36310,36312,36315,36318],{"className":36311},[816,1780],[772,36313,13887],{"className":36314},[816,817,1780],[772,36316,5940],{"className":36317},[6041,1780],[772,36319,36321],{"className":36320},[816,1709,1780],[772,36322,35206],{"className":36323},[816,1780],[772,36325,3580],{"className":36326},[3579],[772,36328,36330],{"className":36329},[1760],[772,36331,36333],{"className":36332,"style":6301},[1764],[772,36334],{},[11,36336,36337],{},"Back-calculate adjusted distance and azimuth:",[11,36339,36340],{},[772,36341,36343,36407],{"className":36342},[775],[772,36344,36346],{"className":36345},[779],[781,36347,36348],{"xmlns":783},[785,36349,36350,36404],{},[788,36351,36352,36364,36366],{},[3525,36353,36354,36356],{},[791,36355,3180],{},[788,36357,36358,36360,36362],{},[791,36359,13887],{},[980,36361,5940],{"separator":803},[1625,36363,35206],{},[980,36365,982],{},[5347,36367,36368],{},[788,36369,36370,36372,36386,36388,36390],{},[791,36371,2963],{"mathvariant":906},[13878,36373,36374,36376,36384],{},[791,36375,2808],{},[788,36377,36378,36380,36382],{},[791,36379,13887],{},[980,36381,5940],{"separator":803},[1625,36383,35206],{},[1084,36385,1872],{},[980,36387,1353],{},[791,36389,2963],{"mathvariant":906},[13878,36391,36392,36394,36402],{},[791,36393,2839],{},[788,36395,36396,36398,36400],{},[791,36397,13887],{},[980,36399,5940],{"separator":803},[1625,36401,35206],{},[1084,36403,1872],{},[795,36405,36406],{"encoding":797},"D_{i,\\text{adj}} = \\sqrt{\\Delta E_{i,\\text{adj}}^2 + \\Delta N_{i,\\text{adj}}^2}",[772,36408,36410,36477],{"className":36409,"ariaHidden":803},[802],[772,36411,36413,36416,36468,36471,36474],{"className":36412},[807],[772,36414],{"className":36415,"style":34710},[811],[772,36417,36419,36422],{"className":36418},[816],[772,36420,3180],{"className":36421,"style":818},[816,817],[772,36423,36425],{"className":36424},[1752],[772,36426,36428,36460],{"className":36427},[1756,3556],[772,36429,36431,36457],{"className":36430},[1760],[772,36432,36434],{"className":36433,"style":5978},[1764],[772,36435,36436,36439],{"style":5981},[772,36437],{"className":36438,"style":1773},[1772],[772,36440,36442],{"className":36441},[1777,1778,1779,1780],[772,36443,36445,36448,36451],{"className":36444},[816,1780],[772,36446,13887],{"className":36447},[816,817,1780],[772,36449,5940],{"className":36450},[6041,1780],[772,36452,36454],{"className":36453},[816,1709,1780],[772,36455,35206],{"className":36456},[816,1780],[772,36458,3580],{"className":36459},[3579],[772,36461,36463],{"className":36462},[1760],[772,36464,36466],{"className":36465,"style":6301},[1764],[772,36467],{},[772,36469],{"className":36470,"style":1004},[1003],[772,36472,982],{"className":36473},[1008],[772,36475],{"className":36476,"style":1004},[1003],[772,36478,36480,36484],{"className":36479},[807],[772,36481],{"className":36482,"style":36483},[811],"height:1.84em;vertical-align:-0.6758em;",[772,36485,36487],{"className":36486},[816,5403],[772,36488,36490,36672],{"className":36489},[1756,3556],[772,36491,36493,36669],{"className":36492},[1760],[772,36494,36497,36652],{"className":36495,"style":36496},[1764],"height:1.1642em;",[772,36498,36501,36505],{"className":36499,"style":36500},[5417],"top:-3.8em;",[772,36502],{"className":36503,"style":36504},[1772],"height:3.8em;",[772,36506,36508,36511,36576,36579,36582,36585,36588],{"className":36507,"style":9463},[816],[772,36509,2963],{"className":36510},[816],[772,36512,36514,36517],{"className":36513},[816],[772,36515,2808],{"className":36516,"style":2823},[816,817],[772,36518,36520],{"className":36519},[1752],[772,36521,36523,36567],{"className":36522},[1756,3556],[772,36524,36526,36564],{"className":36525},[1760],[772,36527,36529,36553],{"className":36528,"style":33254},[1764],[772,36530,36532,36535],{"style":36531},"top:-2.3987em;margin-left:-0.0576em;margin-right:0.05em;",[772,36533],{"className":36534,"style":1773},[1772],[772,36536,36538],{"className":36537},[1777,1778,1779,1780],[772,36539,36541,36544,36547],{"className":36540},[816,1780],[772,36542,13887],{"className":36543},[816,817,1780],[772,36545,5940],{"className":36546},[6041,1780],[772,36548,36550],{"className":36549},[816,1709,1780],[772,36551,35206],{"className":36552},[816,1780],[772,36554,36555,36558],{"style":33269},[772,36556],{"className":36557,"style":1773},[1772],[772,36559,36561],{"className":36560},[1777,1778,1779,1780],[772,36562,1872],{"className":36563},[816,1780],[772,36565,3580],{"className":36566},[3579],[772,36568,36570],{"className":36569},[1760],[772,36571,36574],{"className":36572,"style":36573},[1764],"height:0.4374em;",[772,36575],{},[772,36577],{"className":36578,"style":1121},[1003],[772,36580,1353],{"className":36581},[1125],[772,36583],{"className":36584,"style":1121},[1003],[772,36586,2963],{"className":36587},[816],[772,36589,36591,36594],{"className":36590},[816],[772,36592,2839],{"className":36593,"style":2854},[816,817],[772,36595,36597],{"className":36596},[1752],[772,36598,36600,36644],{"className":36599},[1756,3556],[772,36601,36603,36641],{"className":36602},[1760],[772,36604,36606,36630],{"className":36605,"style":33254},[1764],[772,36607,36609,36612],{"style":36608},"top:-2.3987em;margin-left:-0.109em;margin-right:0.05em;",[772,36610],{"className":36611,"style":1773},[1772],[772,36613,36615],{"className":36614},[1777,1778,1779,1780],[772,36616,36618,36621,36624],{"className":36617},[816,1780],[772,36619,13887],{"className":36620},[816,817,1780],[772,36622,5940],{"className":36623},[6041,1780],[772,36625,36627],{"className":36626},[816,1709,1780],[772,36628,35206],{"className":36629},[816,1780],[772,36631,36632,36635],{"style":33269},[772,36633],{"className":36634,"style":1773},[1772],[772,36636,36638],{"className":36637},[1777,1778,1779,1780],[772,36639,1872],{"className":36640},[816,1780],[772,36642,3580],{"className":36643},[3579],[772,36645,36647],{"className":36646},[1760],[772,36648,36650],{"className":36649,"style":36573},[1764],[772,36651],{},[772,36653,36655,36658],{"style":36654},"top:-3.1242em;",[772,36656],{"className":36657,"style":36504},[1772],[772,36659,36662],{"className":36660,"style":36661},[5511],"min-width:1.02em;height:1.88em;",[5514,36663,36666],{"xmlns":5516,"width":5517,"height":36664,"viewBox":36665,"preserveAspectRatio":5520},"1.88em","0 0 400000 1944",[5522,36667],{"d":36668},"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z",[772,36670,3580],{"className":36671},[3579],[772,36673,36675],{"className":36674},[1760],[772,36676,36679],{"className":36677,"style":36678},[1764],"height:0.6758em;",[772,36680],{},[11,36682,36683],{},[772,36684,36686,36748],{"className":36685},[775],[772,36687,36689],{"className":36688},[779],[781,36690,36691],{"xmlns":783},[785,36692,36693,36745],{},[788,36694,36695,36707,36709,36711,36713,36715,36727,36729,36731,36743],{},[3525,36696,36697,36699],{},[791,36698,793],{},[788,36700,36701,36703,36705],{},[791,36702,13887],{},[980,36704,5940],{"separator":803},[1625,36706,35206],{},[980,36708,982],{},[1625,36710,5886],{},[980,36712,1082],{"stretchy":1081},[791,36714,2963],{"mathvariant":906},[3525,36716,36717,36719],{},[791,36718,2808],{},[788,36720,36721,36723,36725],{},[791,36722,13887],{},[980,36724,5940],{"separator":803},[1625,36726,35206],{},[980,36728,5940],{"separator":803},[791,36730,2963],{"mathvariant":906},[3525,36732,36733,36735],{},[791,36734,2839],{},[788,36736,36737,36739,36741],{},[791,36738,13887],{},[980,36740,5940],{"separator":803},[1625,36742,35206],{},[980,36744,1096],{"stretchy":1081},[795,36746,36747],{"encoding":797},"\\theta_{i,\\text{adj}} = \\text{atan2}(\\Delta E_{i,\\text{adj}}, \\Delta N_{i,\\text{adj}})",[772,36749,36751,36818],{"className":36750,"ariaHidden":803},[802],[772,36752,36754,36757,36809,36812,36815],{"className":36753},[807],[772,36755],{"className":36756,"style":6256},[811],[772,36758,36760,36763],{"className":36759},[816],[772,36761,793],{"className":36762,"style":818},[816,817],[772,36764,36766],{"className":36765},[1752],[772,36767,36769,36801],{"className":36768},[1756,3556],[772,36770,36772,36798],{"className":36771},[1760],[772,36773,36775],{"className":36774,"style":5978},[1764],[772,36776,36777,36780],{"style":5981},[772,36778],{"className":36779,"style":1773},[1772],[772,36781,36783],{"className":36782},[1777,1778,1779,1780],[772,36784,36786,36789,36792],{"className":36785},[816,1780],[772,36787,13887],{"className":36788},[816,817,1780],[772,36790,5940],{"className":36791},[6041,1780],[772,36793,36795],{"className":36794},[816,1709,1780],[772,36796,35206],{"className":36797},[816,1780],[772,36799,3580],{"className":36800},[3579],[772,36802,36804],{"className":36803},[1760],[772,36805,36807],{"className":36806,"style":6301},[1764],[772,36808],{},[772,36810],{"className":36811,"style":1004},[1003],[772,36813,982],{"className":36814},[1008],[772,36816],{"className":36817,"style":1004},[1003],[772,36819,36821,36824,36830,36833,36836,36888,36891,36894,36897,36949],{"className":36820},[807],[772,36822],{"className":36823,"style":35720},[811],[772,36825,36827],{"className":36826},[816,1709],[772,36828,5886],{"className":36829},[816],[772,36831,1082],{"className":36832},[1113],[772,36834,2963],{"className":36835},[816],[772,36837,36839,36842],{"className":36838},[816],[772,36840,2808],{"className":36841,"style":2823},[816,817],[772,36843,36845],{"className":36844},[1752],[772,36846,36848,36880],{"className":36847},[1756,3556],[772,36849,36851,36877],{"className":36850},[1760],[772,36852,36854],{"className":36853,"style":5978},[1764],[772,36855,36856,36859],{"style":3566},[772,36857],{"className":36858,"style":1773},[1772],[772,36860,36862],{"className":36861},[1777,1778,1779,1780],[772,36863,36865,36868,36871],{"className":36864},[816,1780],[772,36866,13887],{"className":36867},[816,817,1780],[772,36869,5940],{"className":36870},[6041,1780],[772,36872,36874],{"className":36873},[816,1709,1780],[772,36875,35206],{"className":36876},[816,1780],[772,36878,3580],{"className":36879},[3579],[772,36881,36883],{"className":36882},[1760],[772,36884,36886],{"className":36885,"style":6301},[1764],[772,36887],{},[772,36889,5940],{"className":36890},[6041],[772,36892],{"className":36893,"style":3294},[1003],[772,36895,2963],{"className":36896},[816],[772,36898,36900,36903],{"className":36899},[816],[772,36901,2839],{"className":36902,"style":2854},[816,817],[772,36904,36906],{"className":36905},[1752],[772,36907,36909,36941],{"className":36908},[1756,3556],[772,36910,36912,36938],{"className":36911},[1760],[772,36913,36915],{"className":36914,"style":5978},[1764],[772,36916,36917,36920],{"style":3640},[772,36918],{"className":36919,"style":1773},[1772],[772,36921,36923],{"className":36922},[1777,1778,1779,1780],[772,36924,36926,36929,36932],{"className":36925},[816,1780],[772,36927,13887],{"className":36928},[816,817,1780],[772,36930,5940],{"className":36931},[6041,1780],[772,36933,36935],{"className":36934},[816,1709,1780],[772,36936,35206],{"className":36937},[816,1780],[772,36939,3580],{"className":36940},[3579],[772,36942,36944],{"className":36943},[1760],[772,36945,36947],{"className":36946,"style":6301},[1764],[772,36948],{},[772,36950,1096],{"className":36951},[1141],[11,36953,36954,36955,5868],{},"Run the full workflow in the ",[693,36956,32100],{"href":32099},[323,36958,36960],{"id":36959},"worked-example-4-leg-closed-loop","Worked Example: 4-Leg Closed Loop",[11,36962,36963,36964,36966,36967,17739,37060,37153],{},"Control station ",[318,36965,13735],{},": ",[772,36968,36970,36993],{"className":36969},[775],[772,36971,36973],{"className":36972},[779],[781,36974,36975],{"xmlns":783},[785,36976,36977,36990],{},[788,36978,36979,36985,36987],{},[3525,36980,36981,36983],{},[791,36982,2808],{},[791,36984,13735],{},[980,36986,982],{},[1084,36988,36989],{},"1000.0000",[795,36991,36992],{"encoding":797},"E_A = 1000.0000",[772,36994,36996,37051],{"className":36995,"ariaHidden":803},[802],[772,36997,36999,37002,37042,37045,37048],{"className":36998},[807],[772,37000],{"className":37001,"style":3543},[811],[772,37003,37005,37008],{"className":37004},[816],[772,37006,2808],{"className":37007,"style":2823},[816,817],[772,37009,37011],{"className":37010},[1752],[772,37012,37014,37034],{"className":37013},[1756,3556],[772,37015,37017,37031],{"className":37016},[1760],[772,37018,37020],{"className":37019,"style":23077},[1764],[772,37021,37022,37025],{"style":3566},[772,37023],{"className":37024,"style":1773},[1772],[772,37026,37028],{"className":37027},[1777,1778,1779,1780],[772,37029,13735],{"className":37030},[816,817,1780],[772,37032,3580],{"className":37033},[3579],[772,37035,37037],{"className":37036},[1760],[772,37038,37040],{"className":37039,"style":3587},[1764],[772,37041],{},[772,37043],{"className":37044,"style":1004},[1003],[772,37046,982],{"className":37047},[1008],[772,37049],{"className":37050,"style":1004},[1003],[772,37052,37054,37057],{"className":37053},[807],[772,37055],{"className":37056,"style":5899},[811],[772,37058,36989],{"className":37059},[816],[772,37061,37063,37086],{"className":37062},[775],[772,37064,37066],{"className":37065},[779],[781,37067,37068],{"xmlns":783},[785,37069,37070,37083],{},[788,37071,37072,37078,37080],{},[3525,37073,37074,37076],{},[791,37075,2839],{},[791,37077,13735],{},[980,37079,982],{},[1084,37081,37082],{},"5000.0000",[795,37084,37085],{"encoding":797},"N_A = 5000.0000",[772,37087,37089,37144],{"className":37088,"ariaHidden":803},[802],[772,37090,37092,37095,37135,37138,37141],{"className":37091},[807],[772,37093],{"className":37094,"style":3543},[811],[772,37096,37098,37101],{"className":37097},[816],[772,37099,2839],{"className":37100,"style":2854},[816,817],[772,37102,37104],{"className":37103},[1752],[772,37105,37107,37127],{"className":37106},[1756,3556],[772,37108,37110,37124],{"className":37109},[1760],[772,37111,37113],{"className":37112,"style":23077},[1764],[772,37114,37115,37118],{"style":3640},[772,37116],{"className":37117,"style":1773},[1772],[772,37119,37121],{"className":37120},[1777,1778,1779,1780],[772,37122,13735],{"className":37123},[816,817,1780],[772,37125,3580],{"className":37126},[3579],[772,37128,37130],{"className":37129},[1760],[772,37131,37133],{"className":37132,"style":3587},[1764],[772,37134],{},[772,37136],{"className":37137,"style":1004},[1003],[772,37139,982],{"className":37140},[1008],[772,37142],{"className":37143,"style":1004},[1003],[772,37145,37147,37150],{"className":37146},[807],[772,37148],{"className":37149,"style":5899},[811],[772,37151,37082],{"className":37152},[816]," m.",[706,37155,37157],{"id":37156},"raw-field-observations","Raw field observations",[851,37159,37160,37173],{},[854,37161,37162],{},[857,37163,37164,37167,37170],{},[860,37165,37166],{},"Leg",[860,37168,37169],{},"Azimuth (°)",[860,37171,37172],{},"Distance (m)",[876,37174,37175,37186,37197,37208],{},[857,37176,37177,37180,37183],{},[881,37178,37179],{},"AB",[881,37181,37182],{},"45.000",[881,37184,37185],{},"120.00",[857,37187,37188,37191,37194],{},[881,37189,37190],{},"BC",[881,37192,37193],{},"135.000",[881,37195,37196],{},"150.00",[857,37198,37199,37202,37205],{},[881,37200,37201],{},"CD",[881,37203,37204],{},"225.000",[881,37206,37207],{},"110.00",[857,37209,37210,37213,37216],{},[881,37211,37212],{},"DA",[881,37214,37215],{},"311.170",[881,37217,37218],{},"150.30",[706,37220,37222],{"id":37221},"step-1-raw-components","Step 1: Raw components",[851,37224,37225,37301],{},[854,37226,37227],{},[857,37228,37229,37231,37266],{},[860,37230,37166],{},[860,37232,37233,19532],{},[772,37234,37236,37251],{"className":37235},[775],[772,37237,37239],{"className":37238},[779],[781,37240,37241],{"xmlns":783},[785,37242,37243,37249],{},[788,37244,37245,37247],{},[791,37246,2963],{"mathvariant":906},[791,37248,2808],{},[795,37250,2968],{"encoding":797},[772,37252,37254],{"className":37253,"ariaHidden":803},[802],[772,37255,37257,37260,37263],{"className":37256},[807],[772,37258],{"className":37259,"style":2758},[811],[772,37261,2963],{"className":37262},[816],[772,37264,2808],{"className":37265,"style":2823},[816,817],[860,37267,37268,19532],{},[772,37269,37271,37286],{"className":37270},[775],[772,37272,37274],{"className":37273},[779],[781,37275,37276],{"xmlns":783},[785,37277,37278,37284],{},[788,37279,37280,37282],{},[791,37281,2963],{"mathvariant":906},[791,37283,2839],{},[795,37285,3004],{"encoding":797},[772,37287,37289],{"className":37288,"ariaHidden":803},[802],[772,37290,37292,37295,37298],{"className":37291},[807],[772,37293],{"className":37294,"style":2758},[811],[772,37296,2963],{"className":37297},[816],[772,37299,2839],{"className":37300,"style":2854},[816,817],[876,37302,37303,37312,37322,37331],{},[857,37304,37305,37307,37310],{},[881,37306,37179],{},[881,37308,37309],{},"84.8528",[881,37311,37309],{},[857,37313,37314,37316,37319],{},[881,37315,37190],{},[881,37317,37318],{},"106.0660",[881,37320,37321],{},"−106.0660",[857,37323,37324,37326,37329],{},[881,37325,37201],{},[881,37327,37328],{},"−77.7817",[881,37330,37328],{},[857,37332,37333,37335,37338],{},[881,37334,37212],{},[881,37336,37337],{},"−113.1398",[881,37339,37340],{},"98.9418",[706,37342,37344],{"id":37343},"step-2-misclosure-and-precision","Step 2: Misclosure and precision",[335,37346,37347,37401,37599,37653],{},[338,37348,37349,8137],{},[772,37350,37352,37371],{"className":37351},[775],[772,37353,37355],{"className":37354},[779],[781,37356,37357],{"xmlns":783},[785,37358,37359,37368],{},[788,37360,37361,37363,37365],{},[791,37362,14943],{},[980,37364,982],{},[1084,37366,37367],{},"530.30",[795,37369,37370],{"encoding":797},"P = 530.30",[772,37372,37374,37392],{"className":37373,"ariaHidden":803},[802],[772,37375,37377,37380,37383,37386,37389],{"className":37376},[807],[772,37378],{"className":37379,"style":2758},[811],[772,37381,14943],{"className":37382,"style":14964},[816,817],[772,37384],{"className":37385,"style":1004},[1003],[772,37387,982],{"className":37388},[1008],[772,37390],{"className":37391,"style":1004},[1003],[772,37393,37395,37398],{"className":37394},[807],[772,37396],{"className":37397,"style":5899},[811],[772,37399,37367],{"className":37400},[816],[338,37402,37403,17739,37501,8137],{},[772,37404,37406,37431],{"className":37405},[775],[772,37407,37409],{"className":37408},[779],[781,37410,37411],{"xmlns":783},[785,37412,37413,37428],{},[788,37414,37415,37421,37423,37425],{},[3525,37416,37417,37419],{},[791,37418,32893],{},[791,37420,2808],{},[980,37422,982],{},[980,37424,1091],{},[1084,37426,37427],{},"0.0027",[795,37429,37430],{"encoding":797},"e_E = -0.0027",[772,37432,37434,37489],{"className":37433,"ariaHidden":803},[802],[772,37435,37437,37440,37480,37483,37486],{"className":37436},[807],[772,37438],{"className":37439,"style":26076},[811],[772,37441,37443,37446],{"className":37442},[816],[772,37444,32893],{"className":37445},[816,817],[772,37447,37449],{"className":37448},[1752],[772,37450,37452,37472],{"className":37451},[1756,3556],[772,37453,37455,37469],{"className":37454},[1760],[772,37456,37458],{"className":37457,"style":23077},[1764],[772,37459,37460,37463],{"style":16341},[772,37461],{"className":37462,"style":1773},[1772],[772,37464,37466],{"className":37465},[1777,1778,1779,1780],[772,37467,2808],{"className":37468,"style":2823},[816,817,1780],[772,37470,3580],{"className":37471},[3579],[772,37473,37475],{"className":37474},[1760],[772,37476,37478],{"className":37477,"style":3587},[1764],[772,37479],{},[772,37481],{"className":37482,"style":1004},[1003],[772,37484,982],{"className":37485},[1008],[772,37487],{"className":37488,"style":1004},[1003],[772,37490,37492,37495,37498],{"className":37491},[807],[772,37493],{"className":37494,"style":7851},[811],[772,37496,1091],{"className":37497},[816],[772,37499,37427],{"className":37500},[816],[772,37502,37504,37529],{"className":37503},[775],[772,37505,37507],{"className":37506},[779],[781,37508,37509],{"xmlns":783},[785,37510,37511,37526],{},[788,37512,37513,37519,37521,37523],{},[3525,37514,37515,37517],{},[791,37516,32893],{},[791,37518,2839],{},[980,37520,982],{},[980,37522,1091],{},[1084,37524,37525],{},"0.0531",[795,37527,37528],{"encoding":797},"e_N = -0.0531",[772,37530,37532,37587],{"className":37531,"ariaHidden":803},[802],[772,37533,37535,37538,37578,37581,37584],{"className":37534},[807],[772,37536],{"className":37537,"style":26076},[811],[772,37539,37541,37544],{"className":37540},[816],[772,37542,32893],{"className":37543},[816,817],[772,37545,37547],{"className":37546},[1752],[772,37548,37550,37570],{"className":37549},[1756,3556],[772,37551,37553,37567],{"className":37552},[1760],[772,37554,37556],{"className":37555,"style":23077},[1764],[772,37557,37558,37561],{"style":16341},[772,37559],{"className":37560,"style":1773},[1772],[772,37562,37564],{"className":37563},[1777,1778,1779,1780],[772,37565,2839],{"className":37566,"style":2854},[816,817,1780],[772,37568,3580],{"className":37569},[3579],[772,37571,37573],{"className":37572},[1760],[772,37574,37576],{"className":37575,"style":3587},[1764],[772,37577],{},[772,37579],{"className":37580,"style":1004},[1003],[772,37582,982],{"className":37583},[1008],[772,37585],{"className":37586,"style":1004},[1003],[772,37588,37590,37593,37596],{"className":37589},[807],[772,37591],{"className":37592,"style":7851},[811],[772,37594,1091],{"className":37595},[816],[772,37597,37525],{"className":37598},[816],[338,37600,37601,8137],{},[772,37602,37604,37623],{"className":37603},[775],[772,37605,37607],{"className":37606},[779],[781,37608,37609],{"xmlns":783},[785,37610,37611,37620],{},[788,37612,37613,37615,37617],{},[791,37614,32893],{},[980,37616,982],{},[1084,37618,37619],{},"0.0532",[795,37621,37622],{"encoding":797},"e = 0.0532",[772,37624,37626,37644],{"className":37625,"ariaHidden":803},[802],[772,37627,37629,37632,37635,37638,37641],{"className":37628},[807],[772,37630],{"className":37631,"style":845},[811],[772,37633,32893],{"className":37634},[816,817],[772,37636],{"className":37637,"style":1004},[1003],[772,37639,982],{"className":37640},[1008],[772,37642],{"className":37643,"style":1004},[1003],[772,37645,37647,37650],{"className":37646},[807],[772,37648],{"className":37649,"style":5899},[811],[772,37651,37619],{"className":37652},[816],[338,37654,37655,37656,12019,37685,37688],{},"Relative precision ",[772,37657,37659,37673],{"className":37658},[775],[772,37660,37662],{"className":37661},[779],[781,37663,37664],{"xmlns":783},[785,37665,37666,37670],{},[788,37667,37668],{},[980,37669,7437],{},[795,37671,37672],{"encoding":797},"\\approx",[772,37674,37676],{"className":37675,"ariaHidden":803},[802],[772,37677,37679,37682],{"className":37678},[807],[772,37680],{"className":37681,"style":25626},[811],[772,37683,7437],{"className":37684},[1008],[318,37686,37687],{},"1 : 9,965"," (acceptable for urban design control)",[706,37690,37692],{"id":37691},"step-3-bowditch-corrections","Step 3: Bowditch corrections",[11,37694,37695,37696,37730,37731,37765],{},"Corrections oppose misclosure (",[772,37697,37699,37715],{"className":37698},[775],[772,37700,37702],{"className":37701},[779],[781,37703,37704],{"xmlns":783},[785,37705,37706,37712],{},[788,37707,37708,37710],{},[980,37709,1353],{},[1084,37711,37427],{},[795,37713,37714],{"encoding":797},"+0.0027",[772,37716,37718],{"className":37717,"ariaHidden":803},[802],[772,37719,37721,37724,37727],{"className":37720},[807],[772,37722],{"className":37723,"style":7851},[811],[772,37725,1353],{"className":37726},[816],[772,37728,37427],{"className":37729},[816]," m easting, ",[772,37732,37734,37750],{"className":37733},[775],[772,37735,37737],{"className":37736},[779],[781,37738,37739],{"xmlns":783},[785,37740,37741,37747],{},[788,37742,37743,37745],{},[980,37744,1353],{},[1084,37746,37525],{},[795,37748,37749],{"encoding":797},"+0.0531",[772,37751,37753],{"className":37752,"ariaHidden":803},[802],[772,37754,37756,37759,37762],{"className":37755},[807],[772,37757],{"className":37758,"style":7851},[811],[772,37760,1353],{"className":37761},[816],[772,37763,37525],{"className":37764},[816]," m northing total):",[851,37767,37768,37918],{},[854,37769,37770],{},[857,37771,37772,37774,37846],{},[860,37773,37166],{},[860,37775,37776,19532],{},[772,37777,37779,37797],{"className":37778},[775],[772,37780,37782],{"className":37781},[779],[781,37783,37784],{"xmlns":783},[785,37785,37786,37794],{},[788,37787,37788],{},[3525,37789,37790,37792],{},[791,37791,22471],{},[791,37793,2808],{},[795,37795,37796],{"encoding":797},"C_E",[772,37798,37800],{"className":37799,"ariaHidden":803},[802],[772,37801,37803,37806],{"className":37802},[807],[772,37804],{"className":37805,"style":3543},[811],[772,37807,37809,37812],{"className":37808},[816],[772,37810,22471],{"className":37811,"style":22486},[816,817],[772,37813,37815],{"className":37814},[1752],[772,37816,37818,37838],{"className":37817},[1756,3556],[772,37819,37821,37835],{"className":37820},[1760],[772,37822,37824],{"className":37823,"style":23077},[1764],[772,37825,37826,37829],{"style":34731},[772,37827],{"className":37828,"style":1773},[1772],[772,37830,37832],{"className":37831},[1777,1778,1779,1780],[772,37833,2808],{"className":37834,"style":2823},[816,817,1780],[772,37836,3580],{"className":37837},[3579],[772,37839,37841],{"className":37840},[1760],[772,37842,37844],{"className":37843,"style":3587},[1764],[772,37845],{},[860,37847,37848,19532],{},[772,37849,37851,37869],{"className":37850},[775],[772,37852,37854],{"className":37853},[779],[781,37855,37856],{"xmlns":783},[785,37857,37858,37866],{},[788,37859,37860],{},[3525,37861,37862,37864],{},[791,37863,22471],{},[791,37865,2839],{},[795,37867,37868],{"encoding":797},"C_N",[772,37870,37872],{"className":37871,"ariaHidden":803},[802],[772,37873,37875,37878],{"className":37874},[807],[772,37876],{"className":37877,"style":3543},[811],[772,37879,37881,37884],{"className":37880},[816],[772,37882,22471],{"className":37883,"style":22486},[816,817],[772,37885,37887],{"className":37886},[1752],[772,37888,37890,37910],{"className":37889},[1756,3556],[772,37891,37893,37907],{"className":37892},[1760],[772,37894,37896],{"className":37895,"style":23077},[1764],[772,37897,37898,37901],{"style":34731},[772,37899],{"className":37900,"style":1773},[1772],[772,37902,37904],{"className":37903},[1777,1778,1779,1780],[772,37905,2839],{"className":37906,"style":2854},[816,817,1780],[772,37908,3580],{"className":37909},[3579],[772,37911,37913],{"className":37912},[1760],[772,37914,37916],{"className":37915,"style":3587},[1764],[772,37917],{},[876,37919,37920,37930,37940,37949],{},[857,37921,37922,37924,37927],{},[881,37923,37179],{},[881,37925,37926],{},"+0.0006",[881,37928,37929],{},"+0.0120",[857,37931,37932,37934,37937],{},[881,37933,37190],{},[881,37935,37936],{},"+0.0008",[881,37938,37939],{},"+0.0150",[857,37941,37942,37944,37946],{},[881,37943,37201],{},[881,37945,37926],{},[881,37947,37948],{},"+0.0110",[857,37950,37951,37953,37956],{},[881,37952,37212],{},[881,37954,37955],{},"+0.0007",[881,37957,37958],{},"+0.0151",[706,37960,37962],{"id":37961},"step-4-adjusted-coordinates","Step 4: Adjusted coordinates",[851,37964,37965,37976],{},[854,37966,37967],{},[857,37968,37969,37972,37974],{},[860,37970,37971],{},"Station",[860,37973,29061],{},[860,37975,29064],{},[876,37977,37978,37986,37996,38006,38016],{},[857,37979,37980,37982,37984],{},[881,37981,13735],{},[881,37983,36989],{},[881,37985,37082],{},[857,37987,37988,37990,37993],{},[881,37989,22441],{},[881,37991,37992],{},"1084.8534",[881,37994,37995],{},"5084.8648",[857,37997,37998,38000,38003],{},[881,37999,22471],{},[881,38001,38002],{},"1190.9202",[881,38004,38005],{},"4978.8138",[857,38007,38008,38010,38013],{},[881,38009,3180],{},[881,38011,38012],{},"1113.1391",[881,38014,38015],{},"4901.0431",[857,38017,38018,38021,38023],{},[881,38019,38020],{},"A (closed)",[881,38022,36989],{},[881,38024,37082],{},[706,38026,38028],{"id":38027},"step-5-adjusted-azimuths-and-distances","Step 5: Adjusted azimuths and distances",[851,38030,38031,38043],{},[854,38032,38033],{},[857,38034,38035,38037,38040],{},[860,38036,37166],{},[860,38038,38039],{},"Adj. azimuth (°)",[860,38041,38042],{},"Adj. distance (m)",[876,38044,38045,38055,38065,38075],{},[857,38046,38047,38049,38052],{},[881,38048,37179],{},[881,38050,38051],{},"44.996",[881,38053,38054],{},"120.009",[857,38056,38057,38059,38062],{},[881,38058,37190],{},[881,38060,38061],{},"134.996",[881,38063,38064],{},"149.990",[857,38066,38067,38069,38072],{},[881,38068,37201],{},[881,38070,38071],{},"225.004",[881,38073,38074],{},"109.992",[857,38076,38077,38079,38082],{},[881,38078,37212],{},[881,38080,38081],{},"311.175",[881,38083,38084],{},"150.309",[706,38086,38088],{"id":38087},"field-book-summary","Field book summary",[851,38090,38091,38437],{},[854,38092,38093],{},[857,38094,38095,38097,38099,38102,38133,38168,38203,38239,38274,38310,38345,38348,38378],{},[860,38096,37971],{},[860,38098,37166],{},[860,38100,38101],{},"Raw θ (°)",[860,38103,38104,38105,19532],{},"Raw ",[772,38106,38108,38121],{"className":38107},[775],[772,38109,38111],{"className":38110},[779],[781,38112,38113],{"xmlns":783},[785,38114,38115,38119],{},[788,38116,38117],{},[791,38118,3180],{},[795,38120,3180],{"encoding":797},[772,38122,38124],{"className":38123,"ariaHidden":803},[802],[772,38125,38127,38130],{"className":38126},[807],[772,38128],{"className":38129,"style":2758},[811],[772,38131,3180],{"className":38132,"style":818},[816,817],[860,38134,38104,38135],{},[772,38136,38138,38153],{"className":38137},[775],[772,38139,38141],{"className":38140},[779],[781,38142,38143],{"xmlns":783},[785,38144,38145,38151],{},[788,38146,38147,38149],{},[791,38148,2963],{"mathvariant":906},[791,38150,2808],{},[795,38152,2968],{"encoding":797},[772,38154,38156],{"className":38155,"ariaHidden":803},[802],[772,38157,38159,38162,38165],{"className":38158},[807],[772,38160],{"className":38161,"style":2758},[811],[772,38163,2963],{"className":38164},[816],[772,38166,2808],{"className":38167,"style":2823},[816,817],[860,38169,38104,38170],{},[772,38171,38173,38188],{"className":38172},[775],[772,38174,38176],{"className":38175},[779],[781,38177,38178],{"xmlns":783},[785,38179,38180,38186],{},[788,38181,38182,38184],{},[791,38183,2963],{"mathvariant":906},[791,38185,2839],{},[795,38187,3004],{"encoding":797},[772,38189,38191],{"className":38190,"ariaHidden":803},[802],[772,38192,38194,38197,38200],{"className":38193},[807],[772,38195],{"className":38196,"style":2758},[811],[772,38198,2963],{"className":38199},[816],[772,38201,2839],{"className":38202,"style":2854},[816,817],[860,38204,38205,38206],{},"Corr ",[772,38207,38209,38224],{"className":38208},[775],[772,38210,38212],{"className":38211},[779],[781,38213,38214],{"xmlns":783},[785,38215,38216,38222],{},[788,38217,38218,38220],{},[791,38219,2963],{"mathvariant":906},[791,38221,2808],{},[795,38223,2968],{"encoding":797},[772,38225,38227],{"className":38226,"ariaHidden":803},[802],[772,38228,38230,38233,38236],{"className":38229},[807],[772,38231],{"className":38232,"style":2758},[811],[772,38234,2963],{"className":38235},[816],[772,38237,2808],{"className":38238,"style":2823},[816,817],[860,38240,38205,38241],{},[772,38242,38244,38259],{"className":38243},[775],[772,38245,38247],{"className":38246},[779],[781,38248,38249],{"xmlns":783},[785,38250,38251,38257],{},[788,38252,38253,38255],{},[791,38254,2963],{"mathvariant":906},[791,38256,2839],{},[795,38258,3004],{"encoding":797},[772,38260,38262],{"className":38261,"ariaHidden":803},[802],[772,38263,38265,38268,38271],{"className":38264},[807],[772,38266],{"className":38267,"style":2758},[811],[772,38269,2963],{"className":38270},[816],[772,38272,2839],{"className":38273,"style":2854},[816,817],[860,38275,38276,38277],{},"Adj ",[772,38278,38280,38295],{"className":38279},[775],[772,38281,38283],{"className":38282},[779],[781,38284,38285],{"xmlns":783},[785,38286,38287,38293],{},[788,38288,38289,38291],{},[791,38290,2963],{"mathvariant":906},[791,38292,2808],{},[795,38294,2968],{"encoding":797},[772,38296,38298],{"className":38297,"ariaHidden":803},[802],[772,38299,38301,38304,38307],{"className":38300},[807],[772,38302],{"className":38303,"style":2758},[811],[772,38305,2963],{"className":38306},[816],[772,38308,2808],{"className":38309,"style":2823},[816,817],[860,38311,38276,38312],{},[772,38313,38315,38330],{"className":38314},[775],[772,38316,38318],{"className":38317},[779],[781,38319,38320],{"xmlns":783},[785,38321,38322,38328],{},[788,38323,38324,38326],{},[791,38325,2963],{"mathvariant":906},[791,38327,2839],{},[795,38329,3004],{"encoding":797},[772,38331,38333],{"className":38332,"ariaHidden":803},[802],[772,38334,38336,38339,38342],{"className":38335},[807],[772,38337],{"className":38338,"style":2758},[811],[772,38340,2963],{"className":38341},[816],[772,38343,2839],{"className":38344,"style":2854},[816,817],[860,38346,38347],{},"Adj θ (°)",[860,38349,38276,38350,19532],{},[772,38351,38353,38366],{"className":38352},[775],[772,38354,38356],{"className":38355},[779],[781,38357,38358],{"xmlns":783},[785,38359,38360,38364],{},[788,38361,38362],{},[791,38363,3180],{},[795,38365,3180],{"encoding":797},[772,38367,38369],{"className":38368,"ariaHidden":803},[802],[772,38370,38372,38375],{"className":38371},[807],[772,38373],{"className":38374,"style":2758},[811],[772,38376,3180],{"className":38377,"style":818},[816,817],[860,38379,38380,38381,2896,38409,1096],{},"Coordinates (",[772,38382,38384,38397],{"className":38383},[775],[772,38385,38387],{"className":38386},[779],[781,38388,38389],{"xmlns":783},[785,38390,38391,38395],{},[788,38392,38393],{},[791,38394,2808],{},[795,38396,2808],{"encoding":797},[772,38398,38400],{"className":38399,"ariaHidden":803},[802],[772,38401,38403,38406],{"className":38402},[807],[772,38404],{"className":38405,"style":2758},[811],[772,38407,2808],{"className":38408,"style":2823},[816,817],[772,38410,38412,38425],{"className":38411},[775],[772,38413,38415],{"className":38414},[779],[781,38416,38417],{"xmlns":783},[785,38418,38419,38423],{},[788,38420,38421],{},[791,38422,2839],{},[795,38424,2839],{"encoding":797},[772,38426,38428],{"className":38427,"ariaHidden":803},[802],[772,38429,38431,38434],{"className":38430},[807],[772,38432],{"className":38433,"style":2758},[811],[772,38435,2839],{"className":38436,"style":2854},[816,817],[876,38438,38439,38468,38500,38529,38561,38590,38621,38650,38682,38710],{},[857,38440,38441,38443,38445,38447,38449,38451,38453,38455,38457,38459,38461,38463,38465],{},[881,38442,13735],{},[881,38444],{},[881,38446],{},[881,38448],{},[881,38450],{},[881,38452],{},[881,38454],{},[881,38456],{},[881,38458],{},[881,38460],{},[881,38462],{},[881,38464],{},[881,38466,38467],{},"1000.0000, 5000.0000",[857,38469,38470,38472,38474,38476,38478,38480,38482,38485,38488,38491,38494,38496,38498],{},[881,38471],{},[881,38473,37179],{},[881,38475,37182],{},[881,38477,37185],{},[881,38479,37309],{},[881,38481,37309],{},[881,38483,38484],{},"0.0006",[881,38486,38487],{},"0.0120",[881,38489,38490],{},"84.8534",[881,38492,38493],{},"84.8648",[881,38495,38051],{},[881,38497,38054],{},[881,38499],{},[857,38501,38502,38504,38506,38508,38510,38512,38514,38516,38518,38520,38522,38524,38526],{},[881,38503,22441],{},[881,38505],{},[881,38507],{},[881,38509],{},[881,38511],{},[881,38513],{},[881,38515],{},[881,38517],{},[881,38519],{},[881,38521],{},[881,38523],{},[881,38525],{},[881,38527,38528],{},"1084.8534, 5084.8648",[857,38530,38531,38533,38535,38537,38539,38541,38543,38546,38549,38552,38555,38557,38559],{},[881,38532],{},[881,38534,37190],{},[881,38536,37193],{},[881,38538,37196],{},[881,38540,37318],{},[881,38542,37321],{},[881,38544,38545],{},"0.0008",[881,38547,38548],{},"0.0150",[881,38550,38551],{},"106.0668",[881,38553,38554],{},"−106.0510",[881,38556,38061],{},[881,38558,38064],{},[881,38560],{},[857,38562,38563,38565,38567,38569,38571,38573,38575,38577,38579,38581,38583,38585,38587],{},[881,38564,22471],{},[881,38566],{},[881,38568],{},[881,38570],{},[881,38572],{},[881,38574],{},[881,38576],{},[881,38578],{},[881,38580],{},[881,38582],{},[881,38584],{},[881,38586],{},[881,38588,38589],{},"1190.9202, 4978.8138",[857,38591,38592,38594,38596,38598,38600,38602,38604,38606,38609,38612,38615,38617,38619],{},[881,38593],{},[881,38595,37201],{},[881,38597,37204],{},[881,38599,37207],{},[881,38601,37328],{},[881,38603,37328],{},[881,38605,38484],{},[881,38607,38608],{},"0.0110",[881,38610,38611],{},"−77.7811",[881,38613,38614],{},"−77.7707",[881,38616,38071],{},[881,38618,38074],{},[881,38620],{},[857,38622,38623,38625,38627,38629,38631,38633,38635,38637,38639,38641,38643,38645,38647],{},[881,38624,3180],{},[881,38626],{},[881,38628],{},[881,38630],{},[881,38632],{},[881,38634],{},[881,38636],{},[881,38638],{},[881,38640],{},[881,38642],{},[881,38644],{},[881,38646],{},[881,38648,38649],{},"1113.1391, 4901.0431",[857,38651,38652,38654,38656,38658,38660,38662,38664,38667,38670,38673,38676,38678,38680],{},[881,38653],{},[881,38655,37212],{},[881,38657,37215],{},[881,38659,37218],{},[881,38661,37337],{},[881,38663,37340],{},[881,38665,38666],{},"0.0007",[881,38668,38669],{},"0.0151",[881,38671,38672],{},"−113.1391",[881,38674,38675],{},"98.9569",[881,38677,38081],{},[881,38679,38084],{},[881,38681],{},[857,38683,38684,38686,38688,38690,38692,38694,38696,38698,38700,38702,38704,38706,38708],{},[881,38685,13735],{},[881,38687],{},[881,38689],{},[881,38691],{},[881,38693],{},[881,38695],{},[881,38697],{},[881,38699],{},[881,38701],{},[881,38703],{},[881,38705],{},[881,38707],{},[881,38709,38467],{},[857,38711,38712,38716,38718,38720,38722,38725,38728,38730,38732,38735,38737,38739,38741],{},[881,38713,38714],{},[318,38715,20345],{},[881,38717],{},[881,38719],{},[881,38721,37367],{},[881,38723,38724],{},"−0.0027",[881,38726,38727],{},"−0.0531",[881,38729,37427],{},[881,38731,37525],{},[881,38733,38734],{},"0.0000",[881,38736,38734],{},[881,38738],{},[881,38740],{},[881,38742],{},[323,38744,38746],{"id":38745},"bowditch-vs-least-squares","Bowditch vs. Least Squares",[11,38748,38749,38750,38753],{},"Bowditch is empirical and suited to a single loop or link. ",[318,38751,38752],{},"Least squares"," minimizes weighted observation residuals, handles redundant interconnected networks, and yields error ellipses — the geodetic standard for complex control.",[851,38755,38756,38771],{},[854,38757,38758],{},[857,38759,38760,38763,38766,38769],{},[860,38761,38762],{},"Feature",[860,38764,38765],{},"Bowditch (compass)",[860,38767,38768],{},"Transit rule",[860,38770,38752],{},[876,38772,38773,38870,38884,38897,38910],{},[857,38774,38775,38778,38781,38867],{},[881,38776,38777],{},"Basis",[881,38779,38780],{},"Empirical; ∝ leg length",[881,38782,38783,38784,2896,38826],{},"Empirical; ∝ ",[772,38785,38787,38807],{"className":38786},[775],[772,38788,38790],{"className":38789},[779],[781,38791,38792],{"xmlns":783},[785,38793,38794,38804],{},[788,38795,38796,38798,38800,38802],{},[791,38797,13732],{"mathvariant":906},[791,38799,2963],{"mathvariant":906},[791,38801,2808],{},[791,38803,13732],{"mathvariant":906},[795,38805,38806],{"encoding":797},"\\|\\Delta E\\|",[772,38808,38810],{"className":38809,"ariaHidden":803},[802],[772,38811,38813,38816,38820,38823],{"className":38812},[807],[772,38814],{"className":38815,"style":1109},[811],[772,38817,38819],{"className":38818},[816],"∥Δ",[772,38821,2808],{"className":38822,"style":2823},[816,817],[772,38824,13732],{"className":38825},[816],[772,38827,38829,38849],{"className":38828},[775],[772,38830,38832],{"className":38831},[779],[781,38833,38834],{"xmlns":783},[785,38835,38836,38846],{},[788,38837,38838,38840,38842,38844],{},[791,38839,13732],{"mathvariant":906},[791,38841,2963],{"mathvariant":906},[791,38843,2839],{},[791,38845,13732],{"mathvariant":906},[795,38847,38848],{"encoding":797},"\\|\\Delta N\\|",[772,38850,38852],{"className":38851,"ariaHidden":803},[802],[772,38853,38855,38858,38861,38864],{"className":38854},[807],[772,38856],{"className":38857,"style":1109},[811],[772,38859,38819],{"className":38860},[816],[772,38862,2839],{"className":38863,"style":2854},[816,817],[772,38865,13732],{"className":38866},[816],[881,38868,38869],{},"Statistical optimization",[857,38871,38872,38875,38878,38881],{},[881,38873,38874],{},"Angle vs. distance",[881,38876,38877],{},"Assumes comparable precision",[881,38879,38880],{},"Assumes angles more precise",[881,38882,38883],{},"Custom weights",[857,38885,38886,38889,38892,38894],{},[881,38887,38888],{},"Topology",[881,38890,38891],{},"Single loop\u002Flink",[881,38893,38891],{},[881,38895,38896],{},"Arbitrary networks",[857,38898,38899,38902,38905,38907],{},[881,38900,38901],{},"Uncertainty",[881,38903,38904],{},"Relative precision only",[881,38906,38904],{},[881,38908,38909],{},"Error ellipses",[857,38911,38912,38915,38918,38921],{},[881,38913,38914],{},"Limitations",[881,38916,38917],{},"No redundancy metrics",[881,38919,38920],{},"Weak theoretical basis",[881,38922,38923],{},"Requires software",[11,38925,38926],{},"Bowditch remains appropriate for most local engineering, topo, and construction traverses.",[323,38928,38930],{"id":38929},"common-errors","Common Errors",[706,38932,38934],{"id":38933},"wrong-sign-on-corrections","Wrong sign on corrections",[11,38936,38937,39157,39158,39228,39229,39232],{},[772,38938,38940,38989],{"className":38939},[775],[772,38941,38943],{"className":38942},[779],[781,38944,38945],{"xmlns":783},[785,38946,38947,38986],{},[788,38948,38949,38961,38963,38965,38971,38973,38979,38982,38984],{},[3525,38950,38951,38953],{},[791,38952,22471],{},[788,38954,38955,38957,38959],{},[791,38956,2808],{},[980,38958,5940],{"separator":803},[791,38960,13887],{},[980,38962,982],{},[980,38964,1091],{},[3525,38966,38967,38969],{},[791,38968,32893],{},[791,38970,2808],{},[980,38972,1082],{"stretchy":1081},[3525,38974,38975,38977],{},[791,38976,3180],{},[791,38978,13887],{},[791,38980,38981],{"mathvariant":906},"\u002F",[791,38983,14943],{},[980,38985,1096],{"stretchy":1081},[795,38987,38988],{"encoding":797},"C_{E,i} = -e_E (D_i\u002FP)",[772,38990,38992,39056],{"className":38991,"ariaHidden":803},[802],[772,38993,38995,38998,39047,39050,39053],{"className":38994},[807],[772,38996],{"className":38997,"style":34710},[811],[772,38999,39001,39004],{"className":39000},[816],[772,39002,22471],{"className":39003,"style":22486},[816,817],[772,39005,39007],{"className":39006},[1752],[772,39008,39010,39039],{"className":39009},[1756,3556],[772,39011,39013,39036],{"className":39012},[1760],[772,39014,39016],{"className":39015,"style":23077},[1764],[772,39017,39018,39021],{"style":34731},[772,39019],{"className":39020,"style":1773},[1772],[772,39022,39024],{"className":39023},[1777,1778,1779,1780],[772,39025,39027,39030,39033],{"className":39026},[816,1780],[772,39028,2808],{"className":39029,"style":2823},[816,817,1780],[772,39031,5940],{"className":39032},[6041,1780],[772,39034,13887],{"className":39035},[816,817,1780],[772,39037,3580],{"className":39038},[3579],[772,39040,39042],{"className":39041},[1760],[772,39043,39045],{"className":39044,"style":6301},[1764],[772,39046],{},[772,39048],{"className":39049,"style":1004},[1003],[772,39051,982],{"className":39052},[1008],[772,39054],{"className":39055,"style":1004},[1003],[772,39057,39059,39062,39065,39105,39108,39148,39151,39154],{"className":39058},[807],[772,39060],{"className":39061,"style":1109},[811],[772,39063,1091],{"className":39064},[816],[772,39066,39068,39071],{"className":39067},[816],[772,39069,32893],{"className":39070},[816,817],[772,39072,39074],{"className":39073},[1752],[772,39075,39077,39097],{"className":39076},[1756,3556],[772,39078,39080,39094],{"className":39079},[1760],[772,39081,39083],{"className":39082,"style":23077},[1764],[772,39084,39085,39088],{"style":16341},[772,39086],{"className":39087,"style":1773},[1772],[772,39089,39091],{"className":39090},[1777,1778,1779,1780],[772,39092,2808],{"className":39093,"style":2823},[816,817,1780],[772,39095,3580],{"className":39096},[3579],[772,39098,39100],{"className":39099},[1760],[772,39101,39103],{"className":39102,"style":3587},[1764],[772,39104],{},[772,39106,1082],{"className":39107},[1113],[772,39109,39111,39114],{"className":39110},[816],[772,39112,3180],{"className":39113,"style":818},[816,817],[772,39115,39117],{"className":39116},[1752],[772,39118,39120,39140],{"className":39119},[1756,3556],[772,39121,39123,39137],{"className":39122},[1760],[772,39124,39126],{"className":39125,"style":14137},[1764],[772,39127,39128,39131],{"style":5981},[772,39129],{"className":39130,"style":1773},[1772],[772,39132,39134],{"className":39133},[1777,1778,1779,1780],[772,39135,13887],{"className":39136},[816,817,1780],[772,39138,3580],{"className":39139},[3579],[772,39141,39143],{"className":39142},[1760],[772,39144,39146],{"className":39145,"style":3587},[1764],[772,39147],{},[772,39149,38981],{"className":39150},[816],[772,39152,14943],{"className":39153,"style":14964},[816,817],[772,39155,1096],{"className":39156},[1141],". Negative ",[772,39159,39161,39179],{"className":39160},[775],[772,39162,39164],{"className":39163},[779],[781,39165,39166],{"xmlns":783},[785,39167,39168,39176],{},[788,39169,39170],{},[3525,39171,39172,39174],{},[791,39173,32893],{},[791,39175,2808],{},[795,39177,39178],{"encoding":797},"e_E",[772,39180,39182],{"className":39181,"ariaHidden":803},[802],[772,39183,39185,39188],{"className":39184},[807],[772,39186],{"className":39187,"style":26076},[811],[772,39189,39191,39194],{"className":39190},[816],[772,39192,32893],{"className":39193},[816,817],[772,39195,39197],{"className":39196},[1752],[772,39198,39200,39220],{"className":39199},[1756,3556],[772,39201,39203,39217],{"className":39202},[1760],[772,39204,39206],{"className":39205,"style":23077},[1764],[772,39207,39208,39211],{"style":16341},[772,39209],{"className":39210,"style":1773},[1772],[772,39212,39214],{"className":39213},[1777,1778,1779,1780],[772,39215,2808],{"className":39216,"style":2823},[816,817,1780],[772,39218,3580],{"className":39219},[3579],[772,39221,39223],{"className":39222},[1760],[772,39224,39226],{"className":39225,"style":3587},[1764],[772,39227],{}," requires ",[318,39230,39231],{},"positive"," departure corrections. Same-sign corrections double the error.",[706,39234,39236],{"id":39235},"balancing-angles-before-checking-linear-misclosure","Balancing angles before checking linear misclosure",[11,39238,39239,39240,2896,39273,39306],{},"Angular balancing can mask distance blunders. Compute raw ",[772,39241,39243,39258],{"className":39242},[775],[772,39244,39246],{"className":39245},[779],[781,39247,39248],{"xmlns":783},[785,39249,39250,39256],{},[788,39251,39252,39254],{},[791,39253,2963],{"mathvariant":906},[791,39255,2808],{},[795,39257,2968],{"encoding":797},[772,39259,39261],{"className":39260,"ariaHidden":803},[802],[772,39262,39264,39267,39270],{"className":39263},[807],[772,39265],{"className":39266,"style":2758},[811],[772,39268,2963],{"className":39269},[816],[772,39271,2808],{"className":39272,"style":2823},[816,817],[772,39274,39276,39291],{"className":39275},[775],[772,39277,39279],{"className":39278},[779],[781,39280,39281],{"xmlns":783},[785,39282,39283,39289],{},[788,39284,39285,39287],{},[791,39286,2963],{"mathvariant":906},[791,39288,2839],{},[795,39290,3004],{"encoding":797},[772,39292,39294],{"className":39293,"ariaHidden":803},[802],[772,39295,39297,39300,39303],{"className":39296},[807],[772,39298],{"className":39299,"style":2758},[811],[772,39301,2963],{"className":39302},[816],[772,39304,2839],{"className":39305,"style":2854},[816,817]," first to detect field book errors.",[706,39308,39310],{"id":39309},"using-slope-distances","Using slope distances",[11,39312,39313],{},"Apply horizontal reduction before traverse math; raw slope distances distort perimeter and misclosure.",[706,39315,39317],{"id":39316},"uncorrected-field-errors","Uncorrected field errors",[11,39319,39320],{},"Temperature, tension, sag, meteorological EDM corrections, and similar systematic effects propagate through every leg.",[323,39322,11205],{"id":11204},[1602,39324,39325,39359,39365],{},[338,39326,39327,39328,39330,39331,5868],{},"Enter raw azimuths and horizontal distances in the ",[693,39329,32065],{"href":32064}," — check misclosure and 1 : ",[772,39332,39334,39347],{"className":39333},[775],[772,39335,39337],{"className":39336},[779],[781,39338,39339],{"xmlns":783},[785,39340,39341,39345],{},[788,39342,39343],{},[791,39344,2839],{},[795,39346,2839],{"encoding":797},[772,39348,39350],{"className":39349,"ariaHidden":803},[802],[772,39351,39353,39356],{"className":39352},[807],[772,39354],{"className":39355,"style":2758},[811],[772,39357,2839],{"className":39358,"style":2854},[816,817],[338,39360,39361,39362,39364],{},"If within tolerance, run the ",[693,39363,32100],{"href":32099}," for balanced coordinates and adjusted legs.",[338,39366,39367,39368,39370],{},"Use the ",[693,39369,696],{"href":695}," for auxiliary grid conversions.",{"title":15,"searchDepth":16,"depth":16,"links":39372},[39373,39374,39375,39376,39377,39378,39379,39380,39389,39390,39396],{"id":684,"depth":16,"text":685},{"id":32109,"depth":16,"text":32110},{"id":32143,"depth":16,"text":32144},{"id":33649,"depth":16,"text":33650},{"id":34091,"depth":16,"text":34092},{"id":34457,"depth":16,"text":34458},{"id":34467,"depth":16,"text":34468},{"id":36959,"depth":16,"text":36960,"children":39381},[39382,39383,39384,39385,39386,39387,39388],{"id":37156,"depth":73,"text":37157},{"id":37221,"depth":73,"text":37222},{"id":37343,"depth":73,"text":37344},{"id":37691,"depth":73,"text":37692},{"id":37961,"depth":73,"text":37962},{"id":38027,"depth":73,"text":38028},{"id":38087,"depth":73,"text":38088},{"id":38745,"depth":16,"text":38746},{"id":38929,"depth":16,"text":38930,"children":39391},[39392,39393,39394,39395],{"id":38933,"depth":73,"text":38934},{"id":39235,"depth":73,"text":39236},{"id":39309,"depth":73,"text":39310},{"id":39316,"depth":73,"text":39317},{"id":11204,"depth":16,"text":11205},"Closed traverses — departures and latitudes, linear misclosure, relative precision, Bowditch compass-rule adjustment, and a full worked example.",{},"\u002Flearn\u002Ftools\u002Ftraverse-and-bowditch",{"title":32047,"description":39397},{"loc":39399},"learn\u002Ftools\u002Ftraverse-and-bowditch","xdCsOQhvo7gOGwN8WTT6GA6Qxn2zrIQt-e5wqZOWMa4",1783851942723]