[{"data":1,"prerenderedAt":10598},["ShallowReactive",2],{"learn-\u002Flearn\u002Ftools\u002Fbearing-distance":3,"learn-\u002Flearn\u002Ftools\u002Fbearing-distance-surround":10591},{"id":4,"title":5,"body":6,"category":10580,"description":10581,"extension":10582,"meta":10583,"navigation":10584,"order":10585,"path":10586,"seo":10587,"sitemap":10588,"stem":10589,"__hash__":10590},"learn\u002Flearn\u002Ftools\u002Fbearing-distance.md","Bearing & Distance",{"type":7,"value":8,"toc":10550},"minimark",[9,14,18,27,31,34,39,42,46,49,53,56,60,63,67,70,86,92,96,99,181,925,929,932,942,1342,1351,1722,1731,2056,2060,2063,2190,2200,2261,2264,2427,2431,2560,2715,2844,3138,3460,3463,3471,3546,3553,3628,3635,3796,3803,3964,3968,4306,4309,4631,4663,4872,4906,5105,5204,5242,5389,5687,6274,6351,6355,6359,6362,6510,6571,6723,6728,6999,7002,7073,7235,7238,7403,7406,7411,7537,7542,7772,7775,8020,8023,8029,8033,8036,8041,8269,8272,8505,8508,8541,8910,9129,9136,9169,9175,9334,9337,9606,9613,9618,9621,9624,9844,9847,9850,9854,9857,9861,10117,10121,10362,10366,10372,10493,10496,10500,10503,10507,10510,10524,10527,10531,10534,10537,10541,10544],[10,11,13],"h2",{"id":12},"introduction","Introduction",[15,16,17],"p",{},"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.",[15,19,20,21,26],{},"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 ",[22,23,25],"a",{"href":24},"\u002Ftools\u002Fbearing-distance","Bearing & distance calculator",". This article provides a comprehensive guide to the mathematical and practical principles that govern these coordinate transformations.",[10,28,30],{"id":29},"the-practical-role-of-coordinate-geometry-in-surveying","The Practical Role of Coordinate Geometry in Surveying",[15,32,33],{},"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:",[35,36,38],"h3",{"id":37},"polar-data-acquisition-with-modern-instruments","Polar Data Acquisition with Modern Instruments",[15,40,41],{},"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.",[35,43,45],{"id":44},"traverse-computations-and-quality-control","Traverse Computations and Quality Control",[15,47,48],{},"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.",[35,50,52],{"id":51},"construction-layout-and-stakeout-offsets","Construction Layout and Stakeout Offsets",[15,54,55],{},"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.",[10,57,59],{"id":58},"azimuth-and-bearing-conventions","Azimuth and Bearing Conventions",[15,61,62],{},"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).",[35,64,66],{"id":65},"whole-circle-bearings-azimuths","Whole Circle Bearings (Azimuths)",[15,68,69],{},"An azimuth is an angular direction measured continuously clockwise from Grid North, ranging from 0° to 360°. Under this convention:",[71,72,73,77,80,83],"ul",{},[74,75,76],"li",{},"North is represented by 0° (or 360°).",[74,78,79],{},"East is represented by 90°.",[74,81,82],{},"South is represented by 180°.",[74,84,85],{},"West is represented by 270°.",[15,87,88,89,91],{},"Azimuths are widely used in programmatic computations, digital data collectors, and the ",[22,90,25],{"href":24}," because they represent direction as a single, continuous numerical value, which simplifies algorithmic processing.",[35,93,95],{"id":94},"quadrant-bearings-reduced-bearings","Quadrant Bearings (Reduced Bearings)",[15,97,98],{},"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).",[15,100,101,102,150,151,180],{},"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 (",[103,104,107,130],"span",{"className":105},[106],"katex",[103,108,111],{"className":109},[110],"katex-mathml",[112,113,115],"math",{"xmlns":114},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[116,117,118,125],"semantics",{},[119,120,121],"mrow",{},[122,123,124],"mi",{},"θ",[126,127,129],"annotation",{"encoding":128},"application\u002Fx-tex","\\theta",[103,131,135],{"className":132,"ariaHidden":134},[133],"katex-html","true",[103,136,139,144],{"className":137},[138],"base",[103,140],{"className":141,"style":143},[142],"strut","height:0.6944em;",[103,145,124],{"className":146,"style":149},[147,148],"mord","mathnormal","margin-right:0.0278em;",") and quadrant bearings (",[103,152,154,167],{"className":153},[106],[103,155,157],{"className":156},[110],[112,158,159],{"xmlns":114},[116,160,161,165],{},[119,162,163],{},[122,164,22],{},[126,166,22],{"encoding":128},[103,168,170],{"className":169,"ariaHidden":134},[133],[103,171,173,177],{"className":172},[138],[103,174],{"className":175,"style":176},[142],"height:0.4306em;",[103,178,22],{"className":179},[147,148],") across the four geographic quadrants:",[182,183,184,206],"table",{},[185,186,187],"thead",{},[188,189,190,194,197,200,203],"tr",{},[191,192,193],"th",{},"Quadrant",[191,195,196],{},"Azimuth range",[191,198,199],{},"Quadrant notation",[191,201,202],{},"Azimuth → quadrant bearing",[191,204,205],{},"Quadrant bearing → azimuth",[207,208,209,352,551,738],"tbody",{},[188,210,211,215,218,258,294],{},[212,213,214],"td",{},"I (NE)",[212,216,217],{},"0°–90°",[212,219,220,221,257],{},"N ",[103,222,224,242],{"className":223},[106],[103,225,227],{"className":226},[110],[112,228,229],{"xmlns":114},[116,230,231,239],{},[119,232,233,235],{},[122,234,22],{},[122,236,238],{"mathvariant":237},"normal","°",[126,240,241],{"encoding":128},"a°",[103,243,245],{"className":244,"ariaHidden":134},[133],[103,246,248,251,254],{"className":247},[138],[103,249],{"className":250,"style":143},[142],[103,252,22],{"className":253},[147,148],[103,255,238],{"className":256},[147]," E",[212,259,220,260,257],{},[103,261,263,279],{"className":262},[106],[103,264,266],{"className":265},[110],[112,267,268],{"xmlns":114},[116,269,270,276],{},[119,271,272,274],{},[122,273,124],{},[122,275,238],{"mathvariant":237},[126,277,278],{"encoding":128},"\\theta°",[103,280,282],{"className":281,"ariaHidden":134},[133],[103,283,285,288,291],{"className":284},[138],[103,286],{"className":287,"style":143},[142],[103,289,124],{"className":290,"style":149},[147,148],[103,292,238],{"className":293},[147],[212,295,296],{},[103,297,299,319],{"className":298},[106],[103,300,302],{"className":301},[110],[112,303,304],{"xmlns":114},[116,305,306,316],{},[119,307,308,310,314],{},[122,309,124],{},[311,312,313],"mo",{},"=",[122,315,22],{},[126,317,318],{"encoding":128},"\\theta = a",[103,320,322,343],{"className":321,"ariaHidden":134},[133],[103,323,325,328,331,336,340],{"className":324},[138],[103,326],{"className":327,"style":143},[142],[103,329,124],{"className":330,"style":149},[147,148],[103,332],{"className":333,"style":335},[334],"mspace","margin-right:0.2778em;",[103,337,313],{"className":338},[339],"mrel",[103,341],{"className":342,"style":335},[334],[103,344,346,349],{"className":345},[138],[103,347],{"className":348,"style":176},[142],[103,350,22],{"className":351},[147,148],[188,353,354,357,360,396,473],{},[212,355,356],{},"II (SE)",[212,358,359],{},"90°–180°",[212,361,362,363,257],{},"S ",[103,364,366,381],{"className":365},[106],[103,367,369],{"className":368},[110],[112,370,371],{"xmlns":114},[116,372,373,379],{},[119,374,375,377],{},[122,376,22],{},[122,378,238],{"mathvariant":237},[126,380,241],{"encoding":128},[103,382,384],{"className":383,"ariaHidden":134},[133],[103,385,387,390,393],{"className":386},[138],[103,388],{"className":389,"style":143},[142],[103,391,22],{"className":392},[147,148],[103,394,238],{"className":395},[147],[212,397,362,398,257],{},[103,399,401,431],{"className":400},[106],[103,402,404],{"className":403},[110],[112,405,406],{"xmlns":114},[116,407,408,428],{},[119,409,410,414,418,420,423,425],{},[311,411,413],{"stretchy":412},"false","(",[415,416,417],"mn",{},"180",[122,419,238],{"mathvariant":237},[311,421,422],{},"−",[122,424,124],{},[311,426,427],{"stretchy":412},")",[126,429,430],{"encoding":128},"(180° - \\theta)",[103,432,434,460],{"className":433,"ariaHidden":134},[133],[103,435,437,441,445,449,453,457],{"className":436},[138],[103,438],{"className":439,"style":440},[142],"height:1em;vertical-align:-0.25em;",[103,442,413],{"className":443},[444],"mopen",[103,446,448],{"className":447},[147],"180°",[103,450],{"className":451,"style":452},[334],"margin-right:0.2222em;",[103,454,422],{"className":455},[456],"mbin",[103,458],{"className":459,"style":452},[334],[103,461,463,466,469],{"className":462},[138],[103,464],{"className":465,"style":440},[142],[103,467,124],{"className":468,"style":149},[147,148],[103,470,427],{"className":471},[472],"mclose",[212,474,475],{},[103,476,478,502],{"className":477},[106],[103,479,481],{"className":480},[110],[112,482,483],{"xmlns":114},[116,484,485,499],{},[119,486,487,489,491,493,495,497],{},[122,488,124],{},[311,490,313],{},[415,492,417],{},[122,494,238],{"mathvariant":237},[311,496,422],{},[122,498,22],{},[126,500,501],{"encoding":128},"\\theta = 180° - a",[103,503,505,523,542],{"className":504,"ariaHidden":134},[133],[103,506,508,511,514,517,520],{"className":507},[138],[103,509],{"className":510,"style":143},[142],[103,512,124],{"className":513,"style":149},[147,148],[103,515],{"className":516,"style":335},[334],[103,518,313],{"className":519},[339],[103,521],{"className":522,"style":335},[334],[103,524,526,530,533,536,539],{"className":525},[138],[103,527],{"className":528,"style":529},[142],"height:0.7778em;vertical-align:-0.0833em;",[103,531,448],{"className":532},[147],[103,534],{"className":535,"style":452},[334],[103,537,422],{"className":538},[456],[103,540],{"className":541,"style":452},[334],[103,543,545,548],{"className":544},[138],[103,546],{"className":547,"style":176},[142],[103,549,22],{"className":550},[147,148],[188,552,553,556,559,595,660],{},[212,554,555],{},"III (SW)",[212,557,558],{},"180°–270°",[212,560,362,561,594],{},[103,562,564,579],{"className":563},[106],[103,565,567],{"className":566},[110],[112,568,569],{"xmlns":114},[116,570,571,577],{},[119,572,573,575],{},[122,574,22],{},[122,576,238],{"mathvariant":237},[126,578,241],{"encoding":128},[103,580,582],{"className":581,"ariaHidden":134},[133],[103,583,585,588,591],{"className":584},[138],[103,586],{"className":587,"style":143},[142],[103,589,22],{"className":590},[147,148],[103,592,238],{"className":593},[147]," W",[212,596,362,597,594],{},[103,598,600,624],{"className":599},[106],[103,601,603],{"className":602},[110],[112,604,605],{"xmlns":114},[116,606,607,621],{},[119,608,609,611,613,615,617,619],{},[311,610,413],{"stretchy":412},[122,612,124],{},[311,614,422],{},[415,616,417],{},[122,618,238],{"mathvariant":237},[311,620,427],{"stretchy":412},[126,622,623],{"encoding":128},"(\\theta - 180°)",[103,625,627,648],{"className":626,"ariaHidden":134},[133],[103,628,630,633,636,639,642,645],{"className":629},[138],[103,631],{"className":632,"style":440},[142],[103,634,413],{"className":635},[444],[103,637,124],{"className":638,"style":149},[147,148],[103,640],{"className":641,"style":452},[334],[103,643,422],{"className":644},[456],[103,646],{"className":647,"style":452},[334],[103,649,651,654,657],{"className":650},[138],[103,652],{"className":653,"style":440},[142],[103,655,448],{"className":656},[147],[103,658,427],{"className":659},[472],[212,661,662],{},[103,663,665,690],{"className":664},[106],[103,666,668],{"className":667},[110],[112,669,670],{"xmlns":114},[116,671,672,687],{},[119,673,674,676,678,680,682,685],{},[122,675,124],{},[311,677,313],{},[415,679,417],{},[122,681,238],{"mathvariant":237},[311,683,684],{},"+",[122,686,22],{},[126,688,689],{"encoding":128},"\\theta = 180° + a",[103,691,693,711,729],{"className":692,"ariaHidden":134},[133],[103,694,696,699,702,705,708],{"className":695},[138],[103,697],{"className":698,"style":143},[142],[103,700,124],{"className":701,"style":149},[147,148],[103,703],{"className":704,"style":335},[334],[103,706,313],{"className":707},[339],[103,709],{"className":710,"style":335},[334],[103,712,714,717,720,723,726],{"className":713},[138],[103,715],{"className":716,"style":529},[142],[103,718,448],{"className":719},[147],[103,721],{"className":722,"style":452},[334],[103,724,684],{"className":725},[456],[103,727],{"className":728,"style":452},[334],[103,730,732,735],{"className":731},[138],[103,733],{"className":734,"style":176},[142],[103,736,22],{"className":737},[147,148],[188,739,740,743,746,781,848],{},[212,741,742],{},"IV (NW)",[212,744,745],{},"270°–360°",[212,747,220,748,594],{},[103,749,751,766],{"className":750},[106],[103,752,754],{"className":753},[110],[112,755,756],{"xmlns":114},[116,757,758,764],{},[119,759,760,762],{},[122,761,22],{},[122,763,238],{"mathvariant":237},[126,765,241],{"encoding":128},[103,767,769],{"className":768,"ariaHidden":134},[133],[103,770,772,775,778],{"className":771},[138],[103,773],{"className":774,"style":143},[142],[103,776,22],{"className":777},[147,148],[103,779,238],{"className":780},[147],[212,782,220,783,594],{},[103,784,786,811],{"className":785},[106],[103,787,789],{"className":788},[110],[112,790,791],{"xmlns":114},[116,792,793,808],{},[119,794,795,797,800,802,804,806],{},[311,796,413],{"stretchy":412},[415,798,799],{},"360",[122,801,238],{"mathvariant":237},[311,803,422],{},[122,805,124],{},[311,807,427],{"stretchy":412},[126,809,810],{"encoding":128},"(360° - \\theta)",[103,812,814,836],{"className":813,"ariaHidden":134},[133],[103,815,817,820,823,827,830,833],{"className":816},[138],[103,818],{"className":819,"style":440},[142],[103,821,413],{"className":822},[444],[103,824,826],{"className":825},[147],"360°",[103,828],{"className":829,"style":452},[334],[103,831,422],{"className":832},[456],[103,834],{"className":835,"style":452},[334],[103,837,839,842,845],{"className":838},[138],[103,840],{"className":841,"style":440},[142],[103,843,124],{"className":844,"style":149},[147,148],[103,846,427],{"className":847},[472],[212,849,850],{},[103,851,853,877],{"className":852},[106],[103,854,856],{"className":855},[110],[112,857,858],{"xmlns":114},[116,859,860,874],{},[119,861,862,864,866,868,870,872],{},[122,863,124],{},[311,865,313],{},[415,867,799],{},[122,869,238],{"mathvariant":237},[311,871,422],{},[122,873,22],{},[126,875,876],{"encoding":128},"\\theta = 360° - a",[103,878,880,898,916],{"className":879,"ariaHidden":134},[133],[103,881,883,886,889,892,895],{"className":882},[138],[103,884],{"className":885,"style":143},[142],[103,887,124],{"className":888,"style":149},[147,148],[103,890],{"className":891,"style":335},[334],[103,893,313],{"className":894},[339],[103,896],{"className":897,"style":335},[334],[103,899,901,904,907,910,913],{"className":900},[138],[103,902],{"className":903,"style":529},[142],[103,905,826],{"className":906},[147],[103,908],{"className":909,"style":452},[334],[103,911,422],{"className":912},[456],[103,914],{"className":915,"style":452},[334],[103,917,919,922],{"className":918},[138],[103,920],{"className":921,"style":176},[142],[103,923,22],{"className":924},[147,148],[35,926,928],{"id":927},"practical-conversion-examples","Practical Conversion Examples",[15,930,931],{},"To demonstrate these conversions, consider three scenarios:",[933,934,935],"ol",{},[74,936,937,941],{},[938,939,940],"strong",{},"Southeast Quadrant (II)",": A whole-circle bearing of 143°26′ lies between 90° and 180°. The quadrant bearing is computed as:",[15,943,944],{},[103,945,947,1028],{"className":946},[106],[103,948,950],{"className":949},[110],[112,951,952],{"xmlns":114},[116,953,954,1025],{},[119,955,956,960,962,965,967,975,984,986,993,1000,1002,1005,1007,1009,1016,1023],{},[957,958,959],"mtext",{},"QB",[311,961,313],{},[957,963,964],{},"S ",[311,966,413],{"stretchy":412},[968,969,970,972],"msup",{},[415,971,417],{},[311,973,974],{},"∘",[968,976,977,980],{},[415,978,979],{},"00",[311,981,983],{"mathvariant":237,"lspace":982,"rspace":982},"0em","′",[311,985,422],{},[968,987,988,991],{},[415,989,990],{},"143",[311,992,974],{},[968,994,995,998],{},[415,996,997],{},"26",[311,999,983],{"mathvariant":237,"lspace":982,"rspace":982},[311,1001,427],{"stretchy":412},[957,1003,1004],{}," E",[311,1006,313],{},[957,1008,964],{},[968,1010,1011,1014],{},[415,1012,1013],{},"36",[311,1015,974],{},[968,1017,1018,1021],{},[415,1019,1020],{},"34",[311,1022,983],{"mathvariant":237,"lspace":982,"rspace":982},[957,1024,1004],{},[126,1026,1027],{"encoding":128},"\\text{QB}=\\text{S }(180^\\circ00'-143^\\circ26')\\text{ E}=\\text{S }36^\\circ34'\\text{ E}",[103,1029,1031,1054,1161,1256],{"className":1030,"ariaHidden":134},[133],[103,1032,1034,1038,1045,1048,1051],{"className":1033},[138],[103,1035],{"className":1036,"style":1037},[142],"height:0.8778em;vertical-align:-0.1944em;",[103,1039,1042],{"className":1040},[147,1041],"text",[103,1043,959],{"className":1044},[147],[103,1046],{"className":1047,"style":335},[334],[103,1049,313],{"className":1050},[339],[103,1052],{"className":1053,"style":335},[334],[103,1055,1057,1061,1067,1070,1074,1116,1119,1152,1155,1158],{"className":1056},[138],[103,1058],{"className":1059,"style":1060},[142],"height:1.0019em;vertical-align:-0.25em;",[103,1062,1064],{"className":1063},[147,1041],[103,1065,964],{"className":1066},[147],[103,1068,413],{"className":1069},[444],[103,1071,1073],{"className":1072},[147],"18",[103,1075,1077,1081],{"className":1076},[147],[103,1078,1080],{"className":1079},[147],"0",[103,1082,1085],{"className":1083},[1084],"msupsub",[103,1086,1089],{"className":1087},[1088],"vlist-t",[103,1090,1093],{"className":1091},[1092],"vlist-r",[103,1094,1098],{"className":1095,"style":1097},[1096],"vlist","height:0.6741em;",[103,1099,1101,1106],{"style":1100},"top:-3.063em;margin-right:0.05em;",[103,1102],{"className":1103,"style":1105},[1104],"pstrut","height:2.7em;",[103,1107,1113],{"className":1108},[1109,1110,1111,1112],"sizing","reset-size6","size3","mtight",[103,1114,974],{"className":1115},[456,1112],[103,1117,1080],{"className":1118},[147],[103,1120,1122,1125],{"className":1121},[147],[103,1123,1080],{"className":1124},[147],[103,1126,1128],{"className":1127},[1084],[103,1129,1131],{"className":1130},[1088],[103,1132,1134],{"className":1133},[1092],[103,1135,1138],{"className":1136,"style":1137},[1096],"height:0.7519em;",[103,1139,1140,1143],{"style":1100},[103,1141],{"className":1142,"style":1105},[1104],[103,1144,1146],{"className":1145},[1109,1110,1111,1112],[103,1147,1149],{"className":1148},[147,1112],[103,1150,983],{"className":1151},[147,1112],[103,1153],{"className":1154,"style":452},[334],[103,1156,422],{"className":1157},[456],[103,1159],{"className":1160,"style":452},[334],[103,1162,1164,1167,1171,1201,1205,1238,1241,1247,1250,1253],{"className":1163},[138],[103,1165],{"className":1166,"style":1060},[142],[103,1168,1170],{"className":1169},[147],"14",[103,1172,1174,1178],{"className":1173},[147],[103,1175,1177],{"className":1176},[147],"3",[103,1179,1181],{"className":1180},[1084],[103,1182,1184],{"className":1183},[1088],[103,1185,1187],{"className":1186},[1092],[103,1188,1190],{"className":1189,"style":1097},[1096],[103,1191,1192,1195],{"style":1100},[103,1193],{"className":1194,"style":1105},[1104],[103,1196,1198],{"className":1197},[1109,1110,1111,1112],[103,1199,974],{"className":1200},[456,1112],[103,1202,1204],{"className":1203},[147],"2",[103,1206,1208,1212],{"className":1207},[147],[103,1209,1211],{"className":1210},[147],"6",[103,1213,1215],{"className":1214},[1084],[103,1216,1218],{"className":1217},[1088],[103,1219,1221],{"className":1220},[1092],[103,1222,1224],{"className":1223,"style":1137},[1096],[103,1225,1226,1229],{"style":1100},[103,1227],{"className":1228,"style":1105},[1104],[103,1230,1232],{"className":1231},[1109,1110,1111,1112],[103,1233,1235],{"className":1234},[147,1112],[103,1236,983],{"className":1237},[147,1112],[103,1239,427],{"className":1240},[472],[103,1242,1244],{"className":1243},[147,1041],[103,1245,1004],{"className":1246},[147],[103,1248],{"className":1249,"style":335},[334],[103,1251,313],{"className":1252},[339],[103,1254],{"className":1255,"style":335},[334],[103,1257,1259,1262,1268,1271,1300,1303,1336],{"className":1258},[138],[103,1260],{"className":1261,"style":1137},[142],[103,1263,1265],{"className":1264},[147,1041],[103,1266,964],{"className":1267},[147],[103,1269,1177],{"className":1270},[147],[103,1272,1274,1277],{"className":1273},[147],[103,1275,1211],{"className":1276},[147],[103,1278,1280],{"className":1279},[1084],[103,1281,1283],{"className":1282},[1088],[103,1284,1286],{"className":1285},[1092],[103,1287,1289],{"className":1288,"style":1097},[1096],[103,1290,1291,1294],{"style":1100},[103,1292],{"className":1293,"style":1105},[1104],[103,1295,1297],{"className":1296},[1109,1110,1111,1112],[103,1298,974],{"className":1299},[456,1112],[103,1301,1177],{"className":1302},[147],[103,1304,1306,1310],{"className":1305},[147],[103,1307,1309],{"className":1308},[147],"4",[103,1311,1313],{"className":1312},[1084],[103,1314,1316],{"className":1315},[1088],[103,1317,1319],{"className":1318},[1092],[103,1320,1322],{"className":1321,"style":1137},[1096],[103,1323,1324,1327],{"style":1100},[103,1325],{"className":1326,"style":1105},[1104],[103,1328,1330],{"className":1329},[1109,1110,1111,1112],[103,1331,1333],{"className":1332},[147,1112],[103,1334,983],{"className":1335},[147,1112],[103,1337,1339],{"className":1338},[147,1041],[103,1340,1004],{"className":1341},[147],[933,1343,1345],{"start":1344},2,[74,1346,1347,1350],{},[938,1348,1349],{},"Southwest Quadrant (III)",": A whole-circle bearing of 234°15′ lies between 180° and 270°. The quadrant bearing is computed 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}(234^\\circ15'-180^\\circ00')\\text{ W}=\\text{S }54^\\circ15'\\text{ W}",[103,1429,1431,1452,1546,1637],{"className":1430,"ariaHidden":134},[133],[103,1432,1434,1437,1443,1446,1449],{"className":1433},[138],[103,1435],{"className":1436,"style":1037},[142],[103,1438,1440],{"className":1439},[147,1041],[103,1441,959],{"className":1442},[147],[103,1444],{"className":1445,"style":335},[334],[103,1447,313],{"className":1448},[339],[103,1450],{"className":1451,"style":335},[334],[103,1453,1455,1458,1464,1467,1471,1500,1504,1537,1540,1543],{"className":1454},[138],[103,1456],{"className":1457,"style":1060},[142],[103,1459,1461],{"className":1460},[147,1041],[103,1462,964],{"className":1463},[147],[103,1465,413],{"className":1466},[444],[103,1468,1470],{"className":1469},[147],"23",[103,1472,1474,1477],{"className":1473},[147],[103,1475,1309],{"className":1476},[147],[103,1478,1480],{"className":1479},[1084],[103,1481,1483],{"className":1482},[1088],[103,1484,1486],{"className":1485},[1092],[103,1487,1489],{"className":1488,"style":1097},[1096],[103,1490,1491,1494],{"style":1100},[103,1492],{"className":1493,"style":1105},[1104],[103,1495,1497],{"className":1496},[1109,1110,1111,1112],[103,1498,974],{"className":1499},[456,1112],[103,1501,1503],{"className":1502},[147],"1",[103,1505,1507,1511],{"className":1506},[147],[103,1508,1510],{"className":1509},[147],"5",[103,1512,1514],{"className":1513},[1084],[103,1515,1517],{"className":1516},[1088],[103,1518,1520],{"className":1519},[1092],[103,1521,1523],{"className":1522,"style":1137},[1096],[103,1524,1525,1528],{"style":1100},[103,1526],{"className":1527,"style":1105},[1104],[103,1529,1531],{"className":1530},[1109,1110,1111,1112],[103,1532,1534],{"className":1533},[147,1112],[103,1535,983],{"className":1536},[147,1112],[103,1538],{"className":1539,"style":452},[334],[103,1541,422],{"className":1542},[456],[103,1544],{"className":1545,"style":452},[334],[103,1547,1549,1552,1555,1584,1587,1619,1622,1628,1631,1634],{"className":1548},[138],[103,1550],{"className":1551,"style":1060},[142],[103,1553,1073],{"className":1554},[147],[103,1556,1558,1561],{"className":1557},[147],[103,1559,1080],{"className":1560},[147],[103,1562,1564],{"className":1563},[1084],[103,1565,1567],{"className":1566},[1088],[103,1568,1570],{"className":1569},[1092],[103,1571,1573],{"className":1572,"style":1097},[1096],[103,1574,1575,1578],{"style":1100},[103,1576],{"className":1577,"style":1105},[1104],[103,1579,1581],{"className":1580},[1109,1110,1111,1112],[103,1582,974],{"className":1583},[456,1112],[103,1585,1080],{"className":1586},[147],[103,1588,1590,1593],{"className":1589},[147],[103,1591,1080],{"className":1592},[147],[103,1594,1596],{"className":1595},[1084],[103,1597,1599],{"className":1598},[1088],[103,1600,1602],{"className":1601},[1092],[103,1603,1605],{"className":1604,"style":1137},[1096],[103,1606,1607,1610],{"style":1100},[103,1608],{"className":1609,"style":1105},[1104],[103,1611,1613],{"className":1612},[1109,1110,1111,1112],[103,1614,1616],{"className":1615},[147,1112],[103,1617,983],{"className":1618},[147,1112],[103,1620,427],{"className":1621},[472],[103,1623,1625],{"className":1624},[147,1041],[103,1626,1405],{"className":1627},[147],[103,1629],{"className":1630,"style":335},[334],[103,1632,313],{"className":1633},[339],[103,1635],{"className":1636,"style":335},[334],[103,1638,1640,1643,1649,1652,1681,1684,1716],{"className":1639},[138],[103,1641],{"className":1642,"style":1137},[142],[103,1644,1646],{"className":1645},[147,1041],[103,1647,964],{"className":1648},[147],[103,1650,1510],{"className":1651},[147],[103,1653,1655,1658],{"className":1654},[147],[103,1656,1309],{"className":1657},[147],[103,1659,1661],{"className":1660},[1084],[103,1662,1664],{"className":1663},[1088],[103,1665,1667],{"className":1666},[1092],[103,1668,1670],{"className":1669,"style":1097},[1096],[103,1671,1672,1675],{"style":1100},[103,1673],{"className":1674,"style":1105},[1104],[103,1676,1678],{"className":1677},[1109,1110,1111,1112],[103,1679,974],{"className":1680},[456,1112],[103,1682,1503],{"className":1683},[147],[103,1685,1687,1690],{"className":1686},[147],[103,1688,1510],{"className":1689},[147],[103,1691,1693],{"className":1692},[1084],[103,1694,1696],{"className":1695},[1088],[103,1697,1699],{"className":1698},[1092],[103,1700,1702],{"className":1701,"style":1137},[1096],[103,1703,1704,1707],{"style":1100},[103,1705],{"className":1706,"style":1105},[1104],[103,1708,1710],{"className":1709},[1109,1110,1111,1112],[103,1711,1713],{"className":1712},[147,1112],[103,1714,983],{"className":1715},[147,1112],[103,1717,1719],{"className":1718},[147,1041],[103,1720,1405],{"className":1721},[147],[933,1723,1725],{"start":1724},3,[74,1726,1727,1730],{},[938,1728,1729],{},"Northwest Quadrant (IV)",": A quadrant bearing of N 23°28′ W lies in the fourth quadrant. The whole-circle bearing is computed 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lassName":2052},[147,1112],[103,2054,983],{"className":2055},[147,1112],[10,2057,2059],{"id":2058},"easting-and-northing","Easting and Northing",[15,2061,2062],{},"A common source of confusion for students and technicians is the difference between mathematical Cartesian coordinate systems and plane surveying grid networks.",[15,2064,2065,2066,2097,2098,2127,2128,2158,2159,2189],{},"In pure mathematics, the coordinate axes are defined as horizontal (",[103,2067,2069,2083],{"className":2068},[106],[103,2070,2072],{"className":2071},[110],[112,2073,2074],{"xmlns":114},[116,2075,2076,2081],{},[119,2077,2078],{},[122,2079,2080],{},"X",[126,2082,2080],{"encoding":128},[103,2084,2086],{"className":2085,"ariaHidden":134},[133],[103,2087,2089,2093],{"className":2088},[138],[103,2090],{"className":2091,"style":2092},[142],"height:0.6833em;",[103,2094,2080],{"className":2095,"style":2096},[147,148],"margin-right:0.0785em;",") and vertical (",[103,2099,2101,2115],{"className":2100},[106],[103,2102,2104],{"className":2103},[110],[112,2105,2106],{"xmlns":114},[116,2107,2108,2113],{},[119,2109,2110],{},[122,2111,2112],{},"Y",[126,2114,2112],{"encoding":128},[103,2116,2118],{"className":2117,"ariaHidden":134},[133],[103,2119,2121,2124],{"className":2120},[138],[103,2122],{"className":2123,"style":2092},[142],[103,2125,2112],{"className":2126,"style":452},[147,148],"), with angles measured counterclockwise starting from the positive horizontal axis. In plane surveying, the horizontal grid axis represents Easting (",[103,2129,2131,2145],{"className":2130},[106],[103,2132,2134],{"className":2133},[110],[112,2135,2136],{"xmlns":114},[116,2137,2138,2143],{},[119,2139,2140],{},[122,2141,2142],{},"E",[126,2144,2142],{"encoding":128},[103,2146,2148],{"className":2147,"ariaHidden":134},[133],[103,2149,2151,2154],{"className":2150},[138],[103,2152],{"className":2153,"style":2092},[142],[103,2155,2142],{"className":2156,"style":2157},[147,148],"margin-right:0.0576em;",") and the vertical axis represents Northing (",[103,2160,2162,2176],{"className":2161},[106],[103,2163,2165],{"className":2164},[110],[112,2166,2167],{"xmlns":114},[116,2168,2169,2174],{},[119,2170,2171],{},[122,2172,2173],{},"N",[126,2175,2173],{"encoding":128},[103,2177,2179],{"className":2178,"ariaHidden":134},[133],[103,2180,2182,2185],{"className":2181},[138],[103,2183],{"className":2184,"style":2092},[142],[103,2186,2173],{"className":2187,"style":2188},[147,148],"margin-right:0.109em;","). Crucially, the reference direction of 0° is aligned with the vertical axis (North), and angles increase in a clockwise direction.",[2191,2192,2196],"pre",{"className":2193,"code":2195,"language":1041},[2194],"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",[2197,2198,2195],"code",{"__ignoreMap":2199},"",[15,2201,2202,2203,2231,2232,2260],{},"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: (",[103,2204,2206,2219],{"className":2205},[106],[103,2207,2209],{"className":2208},[110],[112,2210,2211],{"xmlns":114},[116,2212,2213,2217],{},[119,2214,2215],{},[122,2216,2142],{},[126,2218,2142],{"encoding":128},[103,2220,2222],{"className":2221,"ariaHidden":134},[133],[103,2223,2225,2228],{"className":2224},[138],[103,2226],{"className":2227,"style":2092},[142],[103,2229,2142],{"className":2230,"style":2157},[147,148],", ",[103,2233,2235,2248],{"className":2234},[106],[103,2236,2238],{"className":2237},[110],[112,2239,2240],{"xmlns":114},[116,2241,2242,2246],{},[119,2243,2244],{},[122,2245,2173],{},[126,2247,2173],{"encoding":128},[103,2249,2251],{"className":2250,"ariaHidden":134},[133],[103,2252,2254,2257],{"className":2253},[138],[103,2255],{"className":2256,"style":2092},[142],[103,2258,2173],{"className":2259,"style":2188},[147,148],").",[15,2262,2263],{},"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:",[182,2265,2266,2355],{},[185,2267,2268],{},[188,2269,2270,2272,2275,2278,2281,2319],{},[191,2271,193],{},[191,2273,2274],{},"General direction",[191,2276,2277],{},"Change in easting",[191,2279,2280],{},"Change in northing",[191,2282,2283,2284],{},"Sign of ",[103,2285,2287,2304],{"className":2286},[106],[103,2288,2290],{"className":2289},[110],[112,2291,2292],{"xmlns":114},[116,2293,2294,2301],{},[119,2295,2296,2299],{},[122,2297,2298],{"mathvariant":237},"Δ",[122,2300,2142],{},[126,2302,2303],{"encoding":128},"\\Delta E",[103,2305,2307],{"className":2306,"ariaHidden":134},[133],[103,2308,2310,2313,2316],{"className":2309},[138],[103,2311],{"className":2312,"style":2092},[142],[103,2314,2298],{"className":2315},[147],[103,2317,2142],{"className":2318,"style":2157},[147,148],[191,2320,2283,2321],{},[103,2322,2324,2340],{"className":2323},[106],[103,2325,2327],{"className":2326},[110],[112,2328,2329],{"xmlns":114},[116,2330,2331,2337],{},[119,2332,2333,2335],{},[122,2334,2298],{"mathvariant":237},[122,2336,2173],{},[126,2338,2339],{"encoding":128},"\\Delta N",[103,2341,2343],{"className":2342,"ariaHidden":134},[133],[103,2344,2346,2349,2352],{"className":2345},[138],[103,2347],{"className":2348,"style":2092},[142],[103,2350,2298],{"className":2351},[147],[103,2353,2173],{"className":2354,"style":2188},[147,148],[207,2356,2357,2376,2394,2411],{},[188,2358,2359,2362,2365,2368,2371,2374],{},[212,2360,2361],{},"I",[212,2363,2364],{},"Northeast (NE)",[212,2366,2367],{},"Eastward (departure)",[212,2369,2370],{},"Northward (latitude)",[212,2372,2373],{},"Positive (+)",[212,2375,2373],{},[188,2377,2378,2381,2384,2386,2389,2391],{},[212,2379,2380],{},"II",[212,2382,2383],{},"Southeast (SE)",[212,2385,2367],{},[212,2387,2388],{},"Southward (latitude)",[212,2390,2373],{},[212,2392,2393],{},"Negative (−)",[188,2395,2396,2399,2402,2405,2407,2409],{},[212,2397,2398],{},"III",[212,2400,2401],{},"Southwest (SW)",[212,2403,2404],{},"Westward (departure)",[212,2406,2388],{},[212,2408,2393],{},[212,2410,2393],{},[188,2412,2413,2416,2419,2421,2423,2425],{},[212,2414,2415],{},"IV",[212,2417,2418],{},"Northwest (NW)",[212,2420,2404],{},[212,2422,2370],{},[212,2424,2393],{},[212,2426,2373],{},[10,2428,2430],{"id":2429},"polar-to-rectangular-conversion","Polar to Rectangular Conversion",[15,2432,2433,2434,2231,2467,2500,2501,2530,2531,2559],{},"The forward coordinate calculation, known as a \"polar to rectangular\" conversion, computes the coordinate differences (",[103,2435,2437,2452],{"className":2436},[106],[103,2438,2440],{"className":2439},[110],[112,2441,2442],{"xmlns":114},[116,2443,2444,2450],{},[119,2445,2446,2448],{},[122,2447,2298],{"mathvariant":237},[122,2449,2142],{},[126,2451,2303],{"encoding":128},[103,2453,2455],{"className":2454,"ariaHidden":134},[133],[103,2456,2458,2461,2464],{"className":2457},[138],[103,2459],{"className":2460,"style":2092},[142],[103,2462,2298],{"className":2463},[147],[103,2465,2142],{"className":2466,"style":2157},[147,148],[103,2468,2470,2485],{"className":2469},[106],[103,2471,2473],{"className":2472},[110],[112,2474,2475],{"xmlns":114},[116,2476,2477,2483],{},[119,2478,2479,2481],{},[122,2480,2298],{"mathvariant":237},[122,2482,2173],{},[126,2484,2339],{"encoding":128},[103,2486,2488],{"className":2487,"ariaHidden":134},[133],[103,2489,2491,2494,2497],{"className":2490},[138],[103,2492],{"className":2493,"style":2092},[142],[103,2495,2298],{"className":2496},[147],[103,2498,2173],{"className":2499,"style":2188},[147,148],") from a known horizontal distance (",[103,2502,2504,2518],{"className":2503},[106],[103,2505,2507],{"className":2506},[110],[112,2508,2509],{"xmlns":114},[116,2510,2511,2516],{},[119,2512,2513],{},[122,2514,2515],{},"D",[126,2517,2515],{"encoding":128},[103,2519,2521],{"className":2520,"ariaHidden":134},[133],[103,2522,2524,2527],{"className":2523},[138],[103,2525],{"className":2526,"style":2092},[142],[103,2528,2515],{"className":2529,"style":149},[147,148],") and a whole-circle bearing (",[103,2532,2534,2547],{"className":2533},[106],[103,2535,2537],{"className":2536},[110],[112,2538,2539],{"xmlns":114},[116,2540,2541,2545],{},[119,2542,2543],{},[122,2544,124],{},[126,2546,129],{"encoding":128},[103,2548,2550],{"className":2549,"ariaHidden":134},[133],[103,2551,2553,2556],{"className":2552},[138],[103,2554],{"className":2555,"style":143},[142],[103,2557,124],{"className":2558,"style":149},[147,148],"). These coordinate offsets are also referred to as \"departures\" (change in Easting) and \"latitudes\" (change in Northing).",[15,2561,2562,2640],{},[103,2563,2565,2593],{"className":2564},[106],[103,2566,2568],{"className":2567},[110],[112,2569,2570],{"xmlns":114},[116,2571,2572,2590],{},[119,2573,2574,2576,2578,2580,2582,2585,2588],{},[122,2575,2298],{"mathvariant":237},[122,2577,2142],{},[311,2579,313],{},[122,2581,2515],{},[122,2583,2584],{},"sin",[311,2586,2587],{},"⁡",[122,2589,124],{},[126,2591,2592],{"encoding":128},"\\Delta E=D\\sin\\theta",[103,2594,2596,2617],{"className":2595,"ariaHidden":134},[133],[103,2597,2599,2602,2605,2608,2611,2614],{"className":2598},[138],[103,2600],{"className":2601,"style":2092},[142],[103,2603,2298],{"className":2604},[147],[103,2606,2142],{"className":2607,"style":2157},[147,148],[103,2609],{"className":2610,"style":335},[334],[103,2612,313],{"className":2613},[339],[103,2615],{"className":2616,"style":335},[334],[103,2618,2620,2623,2626,2630,2634,2637],{"className":2619},[138],[103,2621],{"className":2622,"style":143},[142],[103,2624,2515],{"className":2625,"style":149},[147,148],[103,2627],{"className":2628,"style":2629},[334],"margin-right:0.1667em;",[103,2631,2584],{"className":2632},[2633],"mop",[103,2635],{"className":2636,"style":2629},[334],[103,2638,124],{"className":2639,"style":149},[147,148],[103,2641,2643,2670],{"className":2642},[106],[103,2644,2646],{"className":2645},[110],[112,2647,2648],{"xmlns":114},[116,2649,2650,2667],{},[119,2651,2652,2654,2656,2658,2660,2663,2665],{},[122,2653,2298],{"mathvariant":237},[122,2655,2173],{},[311,2657,313],{},[122,2659,2515],{},[122,2661,2662],{},"cos",[311,2664,2587],{},[122,2666,124],{},[126,2668,2669],{"encoding":128},"\\Delta N=D\\cos\\theta",[103,2671,2673,2694],{"className":2672,"ariaHidden":134},[133],[103,2674,2676,2679,2682,2685,2688,2691],{"className":2675},[138],[103,2677],{"className":2678,"style":2092},[142],[103,2680,2298],{"className":2681},[147],[103,2683,2173],{"className":2684,"style":2188},[147,148],[103,2686],{"className":2687,"style":335},[334],[103,2689,313],{"className":2690},[339],[103,2692],{"className":2693,"style":335},[334],[103,2695,2697,2700,2703,2706,2709,2712],{"className":2696},[138],[103,2698],{"className":2699,"style":143},[142],[103,2701,2515],{"className":2702,"style":149},[147,148],[103,2704],{"className":2705,"style":2629},[334],[103,2707,2662],{"className":2708},[2633],[103,2710],{"className":2711,"style":2629},[334],[103,2713,124],{"className":2714,"style":149},[147,148],[15,2716,2717,2718,2746,2747,2775,2776,2809,2810,2843],{},"These equations use standard trigonometric functions where the whole-circle bearing (",[103,2719,2721,2734],{"className":2720},[106],[103,2722,2724],{"className":2723},[110],[112,2725,2726],{"xmlns":114},[116,2727,2728,2732],{},[119,2729,2730],{},[122,2731,124],{},[126,2733,129],{"encoding":128},[103,2735,2737],{"className":2736,"ariaHidden":134},[133],[103,2738,2740,2743],{"className":2739},[138],[103,2741],{"className":2742,"style":143},[142],[103,2744,124],{"className":2745,"style":149},[147,148],") is measured clockwise from North. If ",[103,2748,2750,2763],{"className":2749},[106],[103,2751,2753],{"className":2752},[110],[112,2754,2755],{"xmlns":114},[116,2756,2757,2761],{},[119,2758,2759],{},[122,2760,124],{},[126,2762,129],{"encoding":128},[103,2764,2766],{"className":2765,"ariaHidden":134},[133],[103,2767,2769,2772],{"className":2768},[138],[103,2770],{"className":2771,"style":143},[142],[103,2773,124],{"className":2774,"style":149},[147,148]," is input directly as a whole-circle bearing, the trigonometric functions automatically output the correct signs (positive or negative) for ",[103,2777,2779,2794],{"className":2778},[106],[103,2780,2782],{"className":2781},[110],[112,2783,2784],{"xmlns":114},[116,2785,2786,2792],{},[119,2787,2788,2790],{},[122,2789,2298],{"mathvariant":237},[122,2791,2142],{},[126,2793,2303],{"encoding":128},[103,2795,2797],{"className":2796,"ariaHidden":134},[133],[103,2798,2800,2803,2806],{"className":2799},[138],[103,2801],{"className":2802,"style":2092},[142],[103,2804,2298],{"className":2805},[147],[103,2807,2142],{"className":2808,"style":2157},[147,148]," and ",[103,2811,2813,2828],{"className":2812},[106],[103,2814,2816],{"className":2815},[110],[112,2817,2818],{"xmlns":114},[116,2819,2820,2826],{},[119,2821,2822,2824],{},[122,2823,2298],{"mathvariant":237},[122,2825,2173],{},[126,2827,2339],{"encoding":128},[103,2829,2831],{"className":2830,"ariaHidden":134},[133],[103,2832,2834,2837,2840],{"className":2833},[138],[103,2835],{"className":2836,"style":2092},[142],[103,2838,2298],{"className":2839},[147],[103,2841,2173],{"className":2842,"style":2188},[147,148],", regardless of which quadrant the line falls in.",[15,2845,2846,2847,2231,2925,2996,2997,2231,3067,3137],{},"Once these coordinate differences are calculated, the coordinates of the target point (",[103,2848,2850,2869],{"className":2849},[106],[103,2851,2853],{"className":2852},[110],[112,2854,2855],{"xmlns":114},[116,2856,2857,2866],{},[119,2858,2859],{},[2860,2861,2862,2864],"msub",{},[122,2863,2142],{},[415,2865,1204],{},[126,2867,2868],{"encoding":128},"E_2",[103,2870,2872],{"className":2871,"ariaHidden":134},[133],[103,2873,2875,2879],{"className":2874},[138],[103,2876],{"className":2877,"style":2878},[142],"height:0.8333em;vertical-align:-0.15em;",[103,2880,2882,2885],{"className":2881},[147],[103,2883,2142],{"className":2884,"style":2157},[147,148],[103,2886,2888],{"className":2887},[1084],[103,2889,2892,2916],{"className":2890},[1088,2891],"vlist-t2",[103,2893,2895,2911],{"className":2894},[1092],[103,2896,2899],{"className":2897,"style":2898},[1096],"height:0.3011em;",[103,2900,2902,2905],{"style":2901},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[103,2903],{"className":2904,"style":1105},[1104],[103,2906,2908],{"className":2907},[1109,1110,1111,1112],[103,2909,1204],{"className":2910},[147,1112],[103,2912,2915],{"className":2913},[2914],"vlist-s","​",[103,2917,2919],{"className":2918},[1092],[103,2920,2923],{"className":2921,"style":2922},[1096],"height:0.15em;",[103,2924],{},[103,2926,2928,2946],{"className":2927},[106],[103,2929,2931],{"className":2930},[110],[112,2932,2933],{"xmlns":114},[116,2934,2935,2943],{},[119,2936,2937],{},[2860,2938,2939,2941],{},[122,2940,2173],{},[415,2942,1204],{},[126,2944,2945],{"encoding":128},"N_2",[103,2947,2949],{"className":2948,"ariaHidden":134},[133],[103,2950,2952,2955],{"className":2951},[138],[103,2953],{"className":2954,"style":2878},[142],[103,2956,2958,2961],{"className":2957},[147],[103,2959,2173],{"className":2960,"style":2188},[147,148],[103,2962,2964],{"className":2963},[1084],[103,2965,2967,2988],{"className":2966},[1088,2891],[103,2968,2970,2985],{"className":2969},[1092],[103,2971,2973],{"className":2972,"style":2898},[1096],[103,2974,2976,2979],{"style":2975},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[103,2977],{"className":2978,"style":1105},[1104],[103,2980,2982],{"className":2981},[1109,1110,1111,1112],[103,2983,1204],{"className":2984},[147,1112],[103,2986,2915],{"className":2987},[2914],[103,2989,2991],{"className":2990},[1092],[103,2992,2994],{"className":2993,"style":2922},[1096],[103,2995],{},") are determined by adding the offsets to the starting point's coordinates (",[103,2998,3000,3018],{"className":2999},[106],[103,3001,3003],{"className":3002},[110],[112,3004,3005],{"xmlns":114},[116,3006,3007,3015],{},[119,3008,3009],{},[2860,3010,3011,3013],{},[122,3012,2142],{},[415,3014,1503],{},[126,3016,3017],{"encoding":128},"E_1",[103,3019,3021],{"className":3020,"ariaHidden":134},[133],[103,3022,3024,3027],{"className":3023},[138],[103,3025],{"className":3026,"style":2878},[142],[103,3028,3030,3033],{"className":3029},[147],[103,3031,2142],{"className":3032,"style":2157},[147,148],[103,3034,3036],{"className":3035},[1084],[103,3037,3039,3059],{"className":3038},[1088,2891],[103,3040,3042,3056],{"className":3041},[1092],[103,3043,3045],{"className":3044,"style":2898},[1096],[103,3046,3047,3050],{"style":2901},[103,3048],{"className":3049,"style":1105},[1104],[103,3051,3053],{"className":3052},[1109,1110,1111,1112],[103,3054,1503],{"className":3055},[147,1112],[103,3057,2915],{"className":3058},[2914],[103,3060,3062],{"className":3061},[1092],[103,3063,3065],{"className":3064,"style":2922},[1096],[103,3066],{},[103,3068,3070,3088],{"className":3069},[106],[103,3071,3073],{"className":3072},[110],[112,3074,3075],{"xmlns":114},[116,3076,3077,3085],{},[119,3078,3079],{},[2860,3080,3081,3083],{},[122,3082,2173],{},[415,3084,1503],{},[126,3086,3087],{"encoding":128},"N_1",[103,3089,3091],{"className":3090,"ariaHidden":134},[133],[103,3092,3094,3097],{"className":3093},[138],[103,3095],{"className":3096,"style":2878},[142],[103,3098,3100,3103],{"className":3099},[147],[103,3101,2173],{"className":3102,"style":2188},[147,148],[103,3104,3106],{"className":3105},[1084],[103,3107,3109,3129],{"className":3108},[1088,2891],[103,3110,3112,3126],{"className":3111},[1092],[103,3113,3115],{"className":3114,"style":2898},[1096],[103,3116,3117,3120],{"style":2975},[103,3118],{"className":3119,"style":1105},[1104],[103,3121,3123],{"className":3122},[1109,1110,1111,1112],[103,3124,1503],{"className":3125},[147,1112],[103,3127,2915],{"className":3128},[2914],[103,3130,3132],{"className":3131},[1092],[103,3133,3135],{"className":3134,"style":2922},[1096],[103,3136],{},"):",[15,3139,3140,3300],{},[103,3141,3143,3175],{"className":3142},[106],[103,3144,3146],{"className":3145},[110],[112,3147,3148],{"xmlns":114},[116,3149,3150,3172],{},[119,3151,3152,3158,3160,3166,3168,3170],{},[2860,3153,3154,3156],{},[122,3155,2142],{},[415,3157,1204],{},[311,3159,313],{},[2860,3161,3162,3164],{},[122,3163,2142],{},[415,3165,1503],{},[311,3167,684],{},[122,3169,2298],{"mathvariant":237},[122,3171,2142],{},[126,3173,3174],{"encoding":128},"E_2=E_1+\\Delta E",[103,3176,3178,3233,3288],{"className":3177,"ariaHidden":134},[133],[103,3179,3181,3184,3224,3227,3230],{"className":3180},[138],[103,3182],{"className":3183,"style":2878},[142],[103,3185,3187,3190],{"className":3186},[147],[103,3188,2142],{"className":3189,"style":2157},[147,148],[103,3191,3193],{"className":3192},[1084],[103,3194,3196,3216],{"className":3195},[1088,2891],[103,3197,3199,3213],{"className":3198},[1092],[103,3200,3202],{"className":3201,"style":2898},[1096],[103,3203,3204,3207],{"style":2901},[103,3205],{"className":3206,"style":1105},[1104],[103,3208,3210],{"className":3209},[1109,1110,1111,1112],[103,3211,1204],{"className":3212},[147,1112],[103,3214,2915],{"className":3215},[2914],[103,3217,3219],{"className":3218},[1092],[103,3220,3222],{"className":3221,"style":2922},[1096],[103,3223],{},[103,3225],{"className":3226,"style":335},[334],[103,3228,313],{"className":3229},[339],[103,3231],{"className":3232,"style":335},[334],[103,3234,3236,3239,3279,3282,3285],{"className":3235},[138],[103,3237],{"className":3238,"style":2878},[142],[103,3240,3242,3245],{"className":3241},[147],[103,3243,2142],{"className":3244,"style":2157},[147,148],[103,3246,3248],{"className":3247},[1084],[103,3249,3251,3271],{"className":3250},[1088,2891],[103,3252,3254,3268],{"className":3253},[1092],[103,3255,3257],{"className":3256,"style":2898},[1096],[103,3258,3259,3262],{"style":2901},[103,3260],{"className":3261,"style":1105},[1104],[103,3263,3265],{"className":3264},[1109,1110,1111,1112],[103,3266,1503],{"className":3267},[147,1112],[103,3269,2915],{"className":3270},[2914],[103,3272,3274],{"className":3273},[1092],[103,3275,3277],{"className":3276,"style":2922},[1096],[103,3278],{},[103,3280],{"className":3281,"style":452},[334],[103,3283,684],{"className":3284},[456],[103,3286],{"className":3287,"style":452},[334],[103,3289,3291,3294,3297],{"className":3290},[138],[103,3292],{"className":3293,"style":2092},[142],[103,3295,2298],{"className":3296},[147],[103,3298,2142],{"className":3299,"style":2157},[147,148],[103,3301,3303,3335],{"className":3302},[106],[103,3304,3306],{"className":3305},[110],[112,3307,3308],{"xmlns":114},[116,3309,3310,3332],{},[119,3311,3312,3318,3320,3326,3328,3330],{},[2860,3313,3314,3316],{},[122,3315,2173],{},[415,3317,1204],{},[311,3319,313],{},[2860,3321,3322,3324],{},[122,3323,2173],{},[415,3325,1503],{},[311,3327,684],{},[122,3329,2298],{"mathvariant":237},[122,3331,2173],{},[126,3333,3334],{"encoding":128},"N_2=N_1+\\Delta N",[103,3336,3338,3393,3448],{"className":3337,"ariaHidden":134},[133],[103,3339,3341,3344,3384,3387,3390],{"className":3340},[138],[103,3342],{"className":3343,"style":2878},[142],[103,3345,3347,3350],{"className":3346},[147],[103,3348,2173],{"className":3349,"style":2188},[147,148],[103,3351,3353],{"className":3352},[1084],[103,3354,3356,3376],{"className":3355},[1088,2891],[103,3357,3359,3373],{"className":3358},[1092],[103,3360,3362],{"className":3361,"style":2898},[1096],[103,3363,3364,3367],{"style":2975},[103,3365],{"className":3366,"style":1105},[1104],[103,3368,3370],{"className":3369},[1109,1110,1111,1112],[103,3371,1204],{"className":3372},[147,1112],[103,3374,2915],{"className":3375},[2914],[103,3377,3379],{"className":3378},[1092],[103,3380,3382],{"className":3381,"style":2922},[1096],[103,3383],{},[103,3385],{"className":3386,"style":335},[334],[103,3388,313],{"className":3389},[339],[103,3391],{"className":3392,"style":335},[334],[103,3394,3396,3399,3439,3442,3445],{"className":3395},[138],[103,3397],{"className":3398,"style":2878},[142],[103,3400,3402,3405],{"className":3401},[147],[103,3403,2173],{"className":3404,"style":2188},[147,148],[103,3406,3408],{"className":3407},[1084],[103,3409,3411,3431],{"className":3410},[1088,2891],[103,3412,3414,3428],{"className":3413},[1092],[103,3415,3417],{"className":3416,"style":2898},[1096],[103,3418,3419,3422],{"style":2975},[103,3420],{"className":3421,"style":1105},[1104],[103,3423,3425],{"className":3424},[1109,1110,1111,1112],[103,3426,1503],{"className":3427},[147,1112],[103,3429,2915],{"className":3430},[2914],[103,3432,3434],{"className":3433},[1092],[103,3435,3437],{"className":3436,"style":2922},[1096],[103,3438],{},[103,3440],{"className":3441,"style":452},[334],[103,3443,684],{"className":3444},[456],[103,3446],{"className":3447,"style":452},[334],[103,3449,3451,3454,3457],{"className":3450},[138],[103,3452],{"className":3453,"style":2092},[142],[103,3455,2298],{"className":3456},[147],[103,3458,2173],{"className":3459,"style":2188},[147,148],[15,3461,3462],{},"These forward formulas can be summarized as follows:",[71,3464,3465],{},[74,3466,3467,3470],{},[938,3468,3469],{},"Easting offset (departure)",":",[15,3472,3473],{},[103,3474,3476,3501],{"className":3475},[106],[103,3477,3479],{"className":3478},[110],[112,3480,3481],{"xmlns":114},[116,3482,3483,3499],{},[119,3484,3485,3487,3489,3491,3493,3495,3497],{},[122,3486,2298],{"mathvariant":237},[122,3488,2142],{},[311,3490,313],{},[122,3492,2515],{},[122,3494,2584],{},[311,3496,2587],{},[122,3498,124],{},[126,3500,2592],{"encoding":128},[103,3502,3504,3525],{"className":3503,"ariaHidden":134},[133],[103,3505,3507,3510,3513,3516,3519,3522],{"className":3506},[138],[103,3508],{"className":3509,"style":2092},[142],[103,3511,2298],{"className":3512},[147],[103,3514,2142],{"className":3515,"style":2157},[147,148],[103,3517],{"className":3518,"style":335},[334],[103,3520,313],{"className":3521},[339],[103,3523],{"className":3524,"style":335},[334],[103,3526,3528,3531,3534,3537,3540,3543],{"className":3527},[138],[103,3529],{"className":3530,"style":143},[142],[103,3532,2515],{"className":3533,"style":149},[147,148],[103,3535],{"className":3536,"style":2629},[334],[103,3538,2584],{"className":3539},[2633],[103,3541],{"className":3542,"style":2629},[334],[103,3544,124],{"className":3545,"style":149},[147,148],[71,3547,3548],{},[74,3549,3550,3470],{},[938,3551,3552],{},"Northing offset (latitude)",[15,3554,3555],{},[103,3556,3558,3583],{"className":3557},[106],[103,3559,3561],{"className":3560},[110],[112,3562,3563],{"xmlns":114},[116,3564,3565,3581],{},[119,3566,3567,3569,3571,3573,3575,3577,3579],{},[122,3568,2298],{"mathvariant":237},[122,3570,2173],{},[311,3572,313],{},[122,3574,2515],{},[122,3576,2662],{},[311,3578,2587],{},[122,3580,124],{},[126,3582,2669],{"encoding":128},[103,3584,3586,3607],{"className":3585,"ariaHidden":134},[133],[103,3587,3589,3592,3595,3598,3601,3604],{"className":3588},[138],[103,3590],{"className":3591,"style":2092},[142],[103,3593,2298],{"className":3594},[147],[103,3596,2173],{"className":3597,"style":2188},[147,148],[103,3599],{"className":3600,"style":335},[334],[103,3602,313],{"className":3603},[339],[103,3605],{"className":3606,"style":335},[334],[103,3608,3610,3613,3616,3619,3622,3625],{"className":3609},[138],[103,3611],{"className":3612,"style":143},[142],[103,3614,2515],{"className":3615,"style":149},[147,148],[103,3617],{"className":3618,"style":2629},[334],[103,3620,2662],{"className":3621},[2633],[103,3623],{"className":3624,"style":2629},[334],[103,3626,124],{"className":3627,"style":149},[147,148],[71,3629,3630],{},[74,3631,3632,3470],{},[938,3633,3634],{},"Target easting coordinate",[15,3636,3637],{},[103,3638,3640,3671],{"className":3639},[106],[103,3641,3643],{"className":3642},[110],[112,3644,3645],{"xmlns":114},[116,3646,3647,3669],{},[119,3648,3649,3655,3657,3663,3665,3667],{},[2860,3650,3651,3653],{},[122,3652,2142],{},[415,3654,1204],{},[311,3656,313],{},[2860,3658,3659,3661],{},[122,3660,2142],{},[415,3662,1503],{},[311,3664,684],{},[122,3666,2298],{"mathvariant":237},[122,3668,2142],{},[126,3670,3174],{"encoding":128},[103,3672,3674,3729,3784],{"className":3673,"ariaHidden":134},[133],[103,3675,3677,3680,3720,3723,3726],{"className":3676},[138],[103,3678],{"className":3679,"style":2878},[142],[103,3681,3683,3686],{"className":3682},[147],[103,3684,2142],{"className":3685,"style":2157},[147,148],[103,3687,3689],{"className":3688},[1084],[103,3690,3692,3712],{"className":3691},[1088,2891],[103,3693,3695,3709],{"className":3694},[1092],[103,3696,3698],{"className":3697,"style":2898},[1096],[103,3699,3700,3703],{"style":2901},[103,3701],{"className":3702,"style":1105},[1104],[103,3704,3706],{"className":3705},[1109,1110,1111,1112],[103,3707,1204],{"className":3708},[147,1112],[103,3710,2915],{"className":3711},[2914],[103,3713,3715],{"className":3714},[1092],[103,3716,3718],{"className":3717,"style":2922},[1096],[103,3719],{},[103,3721],{"className":3722,"style":335},[334],[103,3724,313],{"className":3725},[339],[103,3727],{"className":3728,"style":335},[334],[103,3730,3732,3735,3775,3778,3781],{"className":3731},[138],[103,3733],{"className":3734,"style":2878},[142],[103,3736,3738,3741],{"className":3737},[147],[103,3739,2142],{"className":3740,"style":2157},[147,148],[103,3742,3744],{"className":3743},[1084],[103,3745,3747,3767],{"className":3746},[1088,2891],[103,3748,3750,3764],{"className":3749},[1092],[103,3751,3753],{"className":3752,"style":2898},[1096],[103,3754,3755,3758],{"style":2901},[103,3756],{"className":3757,"style":1105},[1104],[103,3759,3761],{"className":3760},[1109,1110,1111,1112],[103,3762,1503],{"className":3763},[147,1112],[103,3765,2915],{"className":3766},[2914],[103,3768,3770],{"className":3769},[1092],[103,3771,3773],{"className":3772,"style":2922},[1096],[103,3774],{},[103,3776],{"className":3777,"style":452},[334],[103,3779,684],{"className":3780},[456],[103,3782],{"className":3783,"style":452},[334],[103,3785,3787,3790,3793],{"className":3786},[138],[103,3788],{"className":3789,"style":2092},[142],[103,3791,2298],{"className":3792},[147],[103,3794,2142],{"className":3795,"style":2157},[147,148],[71,3797,3798],{},[74,3799,3800,3470],{},[938,3801,3802],{},"Target northing coordinate",[15,3804,3805],{},[103,3806,3808,3839],{"className":3807},[106],[103,3809,3811],{"className":3810},[110],[112,3812,3813],{"xmlns":114},[116,3814,3815,3837],{},[119,3816,3817,3823,3825,3831,3833,3835],{},[2860,3818,3819,3821],{},[122,3820,2173],{},[415,3822,1204],{},[311,3824,313],{},[2860,3826,3827,3829],{},[122,3828,2173],{},[415,3830,1503],{},[311,3832,684],{},[122,3834,2298],{"mathvariant":237},[122,3836,2173],{},[126,3838,3334],{"encoding":128},[103,3840,3842,3897,3952],{"className":3841,"ariaHidden":134},[133],[103,3843,3845,3848,3888,3891,3894],{"className":3844},[138],[103,3846],{"className":3847,"style":2878},[142],[103,3849,3851,3854],{"className":3850},[147],[103,3852,2173],{"className":3853,"style":2188},[147,148],[103,3855,3857],{"className":3856},[1084],[103,3858,3860,3880],{"className":3859},[1088,2891],[103,3861,3863,3877],{"className":3862},[1092],[103,3864,3866],{"className":3865,"style":2898},[1096],[103,3867,3868,3871],{"style":2975},[103,3869],{"className":3870,"style":1105},[1104],[103,3872,3874],{"className":3873},[1109,1110,1111,1112],[103,3875,1204],{"className":3876},[147,1112],[103,3878,2915],{"className":3879},[2914],[103,3881,3883],{"className":3882},[1092],[103,3884,3886],{"className":3885,"style":2922},[1096],[103,3887],{},[103,3889],{"className":3890,"style":335},[334],[103,3892,313],{"className":3893},[339],[103,3895],{"className":3896,"style":335},[334],[103,3898,3900,3903,3943,3946,3949],{"className":3899},[138],[103,3901],{"className":3902,"style":2878},[142],[103,3904,3906,3909],{"className":3905},[147],[103,3907,2173],{"className":3908,"style":2188},[147,148],[103,3910,3912],{"className":3911},[1084],[103,3913,3915,3935],{"className":3914},[1088,2891],[103,3916,3918,3932],{"className":3917},[1092],[103,3919,3921],{"className":3920,"style":2898},[1096],[103,3922,3923,3926],{"style":2975},[103,3924],{"className":3925,"style":1105},[1104],[103,3927,3929],{"className":3928},[1109,1110,1111,1112],[103,3930,1503],{"className":3931},[147,1112],[103,3933,2915],{"className":3934},[2914],[103,3936,3938],{"className":3937},[1092],[103,3939,3941],{"className":3940,"style":2922},[1096],[103,3942],{},[103,3944],{"className":3945,"style":452},[334],[103,3947,684],{"className":3948},[456],[103,3950],{"className":3951,"style":452},[334],[103,3953,3955,3958,3961],{"className":3954},[138],[103,3956],{"className":3957,"style":2092},[142],[103,3959,2298],{"className":3960},[147],[103,3962,2173],{"className":3963,"style":2188},[147,148],[10,3965,3967],{"id":3966},"rectangular-to-polar-conversion","Rectangular to Polar Conversion",[15,3969,3970,3971,3999,4000,4028,4029,2231,4098,4167,4168,2231,4237,2260],{},"The inverse coordinate calculation, or \"rectangular to polar\" conversion, determines the horizontal distance (",[103,3972,3974,3987],{"className":3973},[106],[103,3975,3977],{"className":3976},[110],[112,3978,3979],{"xmlns":114},[116,3980,3981,3985],{},[119,3982,3983],{},[122,3984,2515],{},[126,3986,2515],{"encoding":128},[103,3988,3990],{"className":3989,"ariaHidden":134},[133],[103,3991,3993,3996],{"className":3992},[138],[103,3994],{"className":3995,"style":2092},[142],[103,3997,2515],{"className":3998,"style":149},[147,148],") and whole-circle bearing (",[103,4001,4003,4016],{"className":4002},[106],[103,4004,4006],{"className":4005},[110],[112,4007,4008],{"xmlns":114},[116,4009,4010,4014],{},[119,4011,4012],{},[122,4013,124],{},[126,4015,129],{"encoding":128},[103,4017,4019],{"className":4018,"ariaHidden":134},[133],[103,4020,4022,4025],{"className":4021},[138],[103,4023],{"className":4024,"style":143},[142],[103,4026,124],{"className":4027,"style":149},[147,148],") between two points with known coordinates: Station 1 (",[103,4030,4032,4049],{"className":4031},[106],[103,4033,4035],{"className":4034},[110],[112,4036,4037],{"xmlns":114},[116,4038,4039,4047],{},[119,4040,4041],{},[2860,4042,4043,4045],{},[122,4044,2142],{},[415,4046,1503],{},[126,4048,3017],{"encoding":128},[103,4050,4052],{"className":4051,"ariaHidden":134},[133],[103,4053,4055,4058],{"className":4054},[138],[103,4056],{"className":4057,"style":2878},[142],[103,4059,4061,4064],{"className":4060},[147],[103,4062,2142],{"className":4063,"style":2157},[147,148],[103,4065,4067],{"className":4066},[1084],[103,4068,4070,4090],{"className":4069},[1088,2891],[103,4071,4073,4087],{"className":4072},[1092],[103,4074,4076],{"className":4075,"style":2898},[1096],[103,4077,4078,4081],{"style":2901},[103,4079],{"className":4080,"style":1105},[1104],[103,4082,4084],{"className":4083},[1109,1110,1111,1112],[103,4085,1503],{"className":4086},[147,1112],[103,4088,2915],{"className":4089},[2914],[103,4091,4093],{"className":4092},[1092],[103,4094,4096],{"className":4095,"style":2922},[1096],[103,4097],{},[103,4099,4101,4118],{"className":4100},[106],[103,4102,4104],{"className":4103},[110],[112,4105,4106],{"xmlns":114},[116,4107,4108,4116],{},[119,4109,4110],{},[2860,4111,4112,4114],{},[122,4113,2173],{},[415,4115,1503],{},[126,4117,3087],{"encoding":128},[103,4119,4121],{"className":4120,"ariaHidden":134},[133],[103,4122,4124,4127],{"className":4123},[138],[103,4125],{"className":4126,"style":2878},[142],[103,4128,4130,4133],{"className":4129},[147],[103,4131,2173],{"className":4132,"style":2188},[147,148],[103,4134,4136],{"className":4135},[1084],[103,4137,4139,4159],{"className":4138},[1088,2891],[103,4140,4142,4156],{"className":4141},[1092],[103,4143,4145],{"className":4144,"style":2898},[1096],[103,4146,4147,4150],{"style":2975},[103,4148],{"className":4149,"style":1105},[1104],[103,4151,4153],{"className":4152},[1109,1110,1111,1112],[103,4154,1503],{"className":4155},[147,1112],[103,4157,2915],{"className":4158},[2914],[103,4160,4162],{"className":4161},[1092],[103,4163,4165],{"className":4164,"style":2922},[1096],[103,4166],{},") and Station 2 (",[103,4169,4171,4188],{"className":4170},[106],[103,4172,4174],{"className":4173},[110],[112,4175,4176],{"xmlns":114},[116,4177,4178,4186],{},[119,4179,4180],{},[2860,4181,4182,4184],{},[122,4183,2142],{},[415,4185,1204],{},[126,4187,2868],{"encoding":128},[103,4189,4191],{"className":4190,"ariaHidden":134},[133],[103,4192,4194,4197],{"className":4193},[138],[103,4195],{"className":4196,"style":2878},[142],[103,4198,4200,4203],{"className":4199},[147],[103,4201,2142],{"className":4202,"style":2157},[147,148],[103,4204,4206],{"className":4205},[1084],[103,4207,4209,4229],{"className":4208},[1088,2891],[103,4210,4212,4226],{"className":4211},[1092],[103,4213,4215],{"className":4214,"style":2898},[1096],[103,4216,4217,4220],{"style":2901},[103,4218],{"className":4219,"style":1105},[1104],[103,4221,4223],{"className":4222},[1109,1110,1111,1112],[103,4224,1204],{"className":4225},[147,1112],[103,4227,2915],{"className":4228},[2914],[103,4230,4232],{"className":4231},[1092],[103,4233,4235],{"className":4234,"style":2922},[1096],[103,4236],{},[103,4238,4240,4257],{"className":4239},[106],[103,4241,4243],{"className":4242},[110],[112,4244,4245],{"xmlns":114},[116,4246,4247,4255],{},[119,4248,4249],{},[2860,4250,4251,4253],{},[122,4252,2173],{},[415,4254,1204],{},[126,4256,2945],{"encoding":128},[103,4258,4260],{"className":4259,"ariaHidden":134},[133],[103,4261,4263,4266],{"className":4262},[138],[103,4264],{"className":4265,"style":2878},[142],[103,4267,4269,4272],{"className":4268},[147],[103,4270,2173],{"className":4271,"style":2188},[147,148],[103,4273,4275],{"className":4274},[1084],[103,4276,4278,4298],{"className":4277},[1088,2891],[103,4279,4281,4295],{"className":4280},[1092],[103,4282,4284],{"className":4283,"style":2898},[1096],[103,4285,4286,4289],{"style":2975},[103,4287],{"className":4288,"style":1105},[1104],[103,4290,4292],{"className":4291},[1109,1110,1111,1112],[103,4293,1204],{"className":4294},[147,1112],[103,4296,2915],{"className":4297},[2914],[103,4299,4301],{"className":4300},[1092],[103,4302,4304],{"className":4303,"style":2922},[1096],[103,4305],{},[15,4307,4308],{},"First, calculate the coordinate differences:",[15,4310,4311,4471],{},[103,4312,4314,4346],{"className":4313},[106],[103,4315,4317],{"className":4316},[110],[112,4318,4319],{"xmlns":114},[116,4320,4321,4343],{},[119,4322,4323,4325,4327,4329,4335,4337],{},[122,4324,2298],{"mathvariant":237},[122,4326,2142],{},[311,4328,313],{},[2860,4330,4331,4333],{},[122,4332,2142],{},[415,4334,1204],{},[311,4336,422],{},[2860,4338,4339,4341],{},[122,4340,2142],{},[415,4342,1503],{},[126,4344,4345],{"encoding":128},"\\Delta E=E_2-E_1",[103,4347,4349,4370,4425],{"className":4348,"ariaHidden":134},[133],[103,4350,4352,4355,4358,4361,4364,4367],{"className":4351},[138],[103,4353],{"className":4354,"style":2092},[142],[103,4356,2298],{"className":4357},[147],[103,4359,2142],{"className":4360,"style":2157},[147,148],[103,4362],{"className":4363,"style":335},[334],[103,4365,313],{"className":4366},[339],[103,4368],{"className":4369,"style":335},[334],[103,4371,4373,4376,4416,4419,4422],{"className":4372},[138],[103,4374],{"className":4375,"style":2878},[142],[103,4377,4379,4382],{"className":4378},[147],[103,4380,2142],{"className":4381,"style":2157},[147,148],[103,4383,4385],{"className":4384},[1084],[103,4386,4388,4408],{"className":4387},[1088,2891],[103,4389,4391,4405],{"className":4390},[1092],[103,4392,4394],{"className":4393,"style":2898},[1096],[103,4395,4396,4399],{"style":2901},[103,4397],{"className":4398,"style":1105},[1104],[103,4400,4402],{"className":4401},[1109,1110,1111,1112],[103,4403,1204],{"className":4404},[147,1112],[103,4406,2915],{"className":4407},[2914],[103,4409,4411],{"className":4410},[1092],[103,4412,4414],{"className":4413,"style":2922},[1096],[103,4415],{},[103,4417],{"className":4418,"style":452},[334],[103,4420,422],{"className":4421},[456],[103,4423],{"className":4424,"style":452},[334],[103,4426,4428,4431],{"className":4427},[138],[103,4429],{"className":4430,"style":2878},[142],[103,4432,4434,4437],{"className":4433},[147],[103,4435,2142],{"className":4436,"style":2157},[147,148],[103,4438,4440],{"className":4439},[1084],[103,4441,4443,4463],{"className":4442},[1088,2891],[103,4444,4446,4460],{"className":4445},[1092],[103,4447,4449],{"className":4448,"style":2898},[1096],[103,4450,4451,4454],{"style":2901},[103,4452],{"className":4453,"style":1105},[1104],[103,4455,4457],{"className":4456},[1109,1110,1111,1112],[103,4458,1503],{"className":4459},[147,1112],[103,4461,2915],{"className":4462},[2914],[103,4464,4466],{"className":4465},[1092],[103,4467,4469],{"className":4468,"style":2922},[1096],[103,4470],{},[103,4472,4474,4506],{"className":4473},[106],[103,4475,4477],{"className":4476},[110],[112,4478,4479],{"xmlns":114},[116,4480,4481,4503],{},[119,4482,4483,4485,4487,4489,4495,4497],{},[122,4484,2298],{"mathvariant":237},[122,4486,2173],{},[311,4488,313],{},[2860,4490,4491,4493],{},[122,4492,2173],{},[415,4494,1204],{},[311,4496,422],{},[2860,4498,4499,4501],{},[122,4500,2173],{},[415,4502,1503],{},[126,4504,4505],{"encoding":128},"\\Delta N=N_2-N_1",[103,4507,4509,4530,4585],{"className":4508,"ariaHidden":134},[133],[103,4510,4512,4515,4518,4521,4524,4527],{"className":4511},[138],[103,4513],{"className":4514,"style":2092},[142],[103,4516,2298],{"className":4517},[147],[103,4519,2173],{"className":4520,"style":2188},[147,148],[103,4522],{"className":4523,"style":335},[334],[103,4525,313],{"className":4526},[339],[103,4528],{"className":4529,"style":335},[334],[103,4531,4533,4536,4576,4579,4582],{"className":4532},[138],[103,4534],{"className":4535,"style":2878},[142],[103,4537,4539,4542],{"className":4538},[147],[103,4540,2173],{"className":4541,"style":2188},[147,148],[103,4543,4545],{"className":4544},[1084],[103,4546,4548,4568],{"className":4547},[1088,2891],[103,4549,4551,4565],{"className":4550},[1092],[103,4552,4554],{"className":4553,"style":2898},[1096],[103,4555,4556,4559],{"style":2975},[103,4557],{"className":4558,"style":1105},[1104],[103,4560,4562],{"className":4561},[1109,1110,1111,1112],[103,4563,1204],{"className":4564},[147,1112],[103,4566,2915],{"className":4567},[2914],[103,4569,4571],{"className":4570},[1092],[103,4572,4574],{"className":4573,"style":2922},[1096],[103,4575],{},[103,4577],{"className":4578,"style":452},[334],[103,4580,422],{"className":4581},[456],[103,4583],{"className":4584,"style":452},[334],[103,4586,4588,4591],{"className":4587},[138],[103,4589],{"className":4590,"style":2878},[142],[103,4592,4594,4597],{"className":4593},[147],[103,4595,2173],{"className":4596,"style":2188},[147,148],[103,4598,4600],{"className":4599},[1084],[103,4601,4603,4623],{"className":4602},[1088,2891],[103,4604,4606,4620],{"className":4605},[1092],[103,4607,4609],{"className":4608,"style":2898},[1096],[103,4610,4611,4614],{"style":2975},[103,4612],{"className":4613,"style":1105},[1104],[103,4615,4617],{"className":4616},[1109,1110,1111,1112],[103,4618,1503],{"className":4619},[147,1112],[103,4621,2915],{"className":4622},[2914],[103,4624,4626],{"className":4625},[1092],[103,4627,4629],{"className":4628,"style":2922},[1096],[103,4630],{},[15,4632,4633,4634,4662],{},"Using 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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 (",[103,4876,4878,4893],{"className":4877},[106],[103,4879,4881],{"className":4880},[110],[112,4882,4883],{"xmlns":114},[116,4884,4885,4890],{},[119,4886,4887],{},[122,4888,4889],{},"α",[126,4891,4892],{"encoding":128},"\\alpha",[103,4894,4896],{"className":4895,"ariaHidden":134},[133],[103,4897,4899,4902],{"className":4898},[138],[103,4900],{"className":4901,"style":176},[142],[103,4903,4889],{"className":4904,"style":4905},[147,148],"margin-right:0.0037em;",[15,4907,4908],{},[103,4909,4911,4962],{"className":4910},[106],[103,4912,4914],{"className":4913},[110],[112,4915,4916],{"xmlns":114},[116,4917,4918,4959],{},[119,4919,4920,4922,4924,4927,4929],{},[122,4921,4889],{},[311,4923,313],{},[122,4925,4926],{},"arctan",[311,4928,2587],{},[119,4930,4931,4933,4957],{},[311,4932,413],{"fence":134},[4934,4935,4936,4947],"mfrac",{},[119,4937,4938,4941,4943,4945],{},[122,4939,4940],{"mathvariant":237},"∣",[122,4942,2298],{"mathvariant":237},[122,4944,2142],{},[122,4946,4940],{"mathvariant":237},[119,4948,4949,4951,4953,4955],{},[122,4950,4940],{"mathvariant":237},[122,4952,2298],{"mathvariant":237},[122,4954,2173],{},[122,4956,4940],{"mathvariant":237},[311,4958,427],{"fence":134},[126,4960,4961],{"encoding":128},"\\alpha=\\arctan\\left(\\frac{|\\Delta 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N|}\\right)",[103,4963,4965,4983],{"className":4964,"ariaHidden":134},[133],[103,4966,4968,4971,4974,4977,4980],{"className":4967},[138],[103,4969],{"className":4970,"style":176},[142],[103,4972,4889],{"className":4973,"style":4905},[147,148],[103,4975],{"className":4976,"style":335},[334],[103,4978,313],{"className":4979},[339],[103,4981],{"className":4982,"style":335},[334],[103,4984,4986,4990,4993,4996],{"className":4985},[138],[103,4987],{"className":4988,"style":4989},[142],"height:1.8em;vertical-align:-0.65em;",[103,4991,4926],{"className":4992},[2633],[103,4994],{"className":4995,"style":2629},[334],[103,4997,5000,5010,5099],{"className":4998},[4999],"minner",[103,5001,5005],{"className":5002,"style":5004},[444,5003],"delimcenter","top:0em;",[103,5006,413],{"className":5007},[5008,5009],"delimsizing","size2",[103,5011,5013,5017,5096],{"className":5012},[147],[103,5014],{"className":5015},[444,5016],"nulldelimiter",[103,5018,5020],{"className":5019},[4934],[103,5021,5023,5087],{"className":5022},[1088,2891],[103,5024,5026,5084],{"className":5025},[1092],[103,5027,5030,5052,5063],{"className":5028,"style":5029},[1096],"height:1.01em;",[103,5031,5033,5036],{"style":5032},"top:-2.655em;",[103,5034],{"className":5035,"style":4757},[1104],[103,5037,5039],{"className":5038},[1109,1110,1111,1112],[103,5040,5042,5046,5049],{"className":5041},[147,1112],[103,5043,5045],{"className":5044},[147,1112],"∣Δ",[103,5047,2173],{"className":5048,"style":2188},[147,148,1112],[103,5050,4940],{"className":5051},[147,1112],[103,5053,5055,5058],{"style":5054},"top:-3.23em;",[103,5056],{"className":5057,"style":4757},[1104],[103,5059],{"className":5060,"style":5062},[5061],"frac-line","border-bottom-width:0.04em;",[103,5064,5066,5069],{"style":5065},"top:-3.485em;",[103,5067],{"className":5068,"style":4757},[1104],[103,5070,5072],{"className":5071},[1109,1110,1111,1112],[103,5073,5075,5078,5081],{"className":5074},[147,1112],[103,5076,5045],{"className":5077},[147,1112],[103,5079,2142],{"className":5080,"style":2157},[147,148,1112],[103,5082,4940],{"className":5083},[147,1112],[103,5085,2915],{"className":5086},[2914],[103,5088,5090],{"className":5089},[1092],[103,5091,5094],{"className":5092,"style":5093},[1096],"height:0.52em;",[103,5095],{},[103,5097],{"className":5098},[472,5016],[103,5100,5102],{"className":5101,"style":5004},[472,5003],[103,5103,427],{"className":5104},[5008,5009],[15,5106,5107,5108,5136,5137,2809,5170,5203],{},"The whole-circle bearing (",[103,5109,5111,5124],{"className":5110},[106],[103,5112,5114],{"className":5113},[110],[112,5115,5116],{"xmlns":114},[116,5117,5118,5122],{},[119,5119,5120],{},[122,5121,124],{},[126,5123,129],{"encoding":128},[103,5125,5127],{"className":5126,"ariaHidden":134},[133],[103,5128,5130,5133],{"className":5129},[138],[103,5131],{"className":5132,"style":143},[142],[103,5134,124],{"className":5135,"style":149},[147,148],") is then determined by applying the appropriate quadrant rules based on the signs of ",[103,5138,5140,5155],{"className":5139},[106],[103,5141,5143],{"className":5142},[110],[112,5144,5145],{"xmlns":114},[116,5146,5147,5153],{},[119,5148,5149,5151],{},[122,5150,2298],{"mathvariant":237},[122,5152,2142],{},[126,5154,2303],{"encoding":128},[103,5156,5158],{"className":5157,"ariaHidden":134},[133],[103,5159,5161,5164,5167],{"className":5160},[138],[103,5162],{"className":5163,"style":2092},[142],[103,5165,2298],{"className":5166},[147],[103,5168,2142],{"className":5169,"style":2157},[147,148],[103,5171,5173,5188],{"className":5172},[106],[103,5174,5176],{"className":5175},[110],[112,5177,5178],{"xmlns":114},[116,5179,5180,5186],{},[119,5181,5182,5184],{},[122,5183,2298],{"mathvariant":237},[122,5185,2173],{},[126,5187,2339],{"encoding":128},[103,5189,5191],{"className":5190,"ariaHidden":134},[133],[103,5192,5194,5197,5200],{"className":5193},[138],[103,5195],{"className":5196,"style":2092},[142],[103,5198,2298],{"className":5199},[147],[103,5201,2173],{"className":5202,"style":2188},[147,148],".",[15,5205,5206,5207,5241],{},"In programmatic applications and surveying software, the dual-argument arctangent function, ",[103,5208,5210,5225],{"className":5209},[106],[103,5211,5213],{"className":5212},[110],[112,5214,5215],{"xmlns":114},[116,5216,5217,5222],{},[119,5218,5219],{},[957,5220,5221],{},"atan2",[126,5223,5224],{"encoding":128},"\\text{atan2}",[103,5226,5228],{"className":5227,"ariaHidden":134},[133],[103,5229,5231,5235],{"className":5230},[138],[103,5232],{"className":5233,"style":5234},[142],"height:0.6444em;",[103,5236,5238],{"className":5237},[147,1041],[103,5239,5221],{"className":5240},[147],", is used instead. This function automatically evaluates the signs of both arguments to place the resulting angle in the correct quadrant:",[15,5243,5244],{},[103,5245,5247,5285],{"className":5246},[106],[103,5248,5250],{"className":5249},[110],[112,5251,5252],{"xmlns":114},[116,5253,5254,5282],{},[119,5255,5256,5263,5265,5267,5269,5271,5273,5276,5278,5280],{},[2860,5257,5258,5260],{},[122,5259,124],{},[957,5261,5262],{},"rad",[311,5264,313],{},[957,5266,5221],{},[311,5268,413],{"stretchy":412},[122,5270,2298],{"mathvariant":237},[122,5272,2142],{},[311,5274,5275],{"separator":134},",",[122,5277,2298],{"mathvariant":237},[122,5279,2173],{},[311,5281,427],{"stretchy":412},[126,5283,5284],{"encoding":128},"\\theta_{\\text{rad}}=\\text{atan2}(\\Delta E,\\Delta N)",[103,5286,5288,5352],{"className":5287,"ariaHidden":134},[133],[103,5289,5291,5295,5343,5346,5349],{"className":5290},[138],[103,5292],{"className":5293,"style":5294},[142],"height:0.8444em;vertical-align:-0.15em;",[103,5296,5298,5301],{"className":5297},[147],[103,5299,124],{"className":5300,"style":149},[147,148],[103,5302,5304],{"className":5303},[1084],[103,5305,5307,5335],{"className":5306},[1088,2891],[103,5308,5310,5332],{"className":5309},[1092],[103,5311,5314],{"className":5312,"style":5313},[1096],"height:0.3361em;",[103,5315,5317,5320],{"style":5316},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[103,5318],{"className":5319,"style":1105},[1104],[103,5321,5323],{"className":5322},[1109,1110,1111,1112],[103,5324,5326],{"className":5325},[147,1112],[103,5327,5329],{"className":5328},[147,1041,1112],[103,5330,5262],{"className":5331},[147,1112],[103,5333,2915],{"className":5334},[2914],[103,5336,5338],{"className":5337},[1092],[103,5339,5341],{"className":5340,"style":2922},[1096],[103,5342],{},[103,5344],{"className":5345,"style":335},[334],[103,5347,313],{"className":5348},[339],[103,5350],{"className":5351,"style":335},[334],[103,5353,5355,5358,5364,5367,5370,5373,5377,5380,5383,5386],{"className":5354},[138],[103,5356],{"className":5357,"style":440},[142],[103,5359,5361],{"className":5360},[147,1041],[103,5362,5221],{"className":5363},[147],[103,5365,413],{"className":5366},[444],[103,5368,2298],{"className":5369},[147],[103,5371,2142],{"className":5372,"style":2157},[147,148],[103,5374,5275],{"className":5375},[5376],"mpunct",[103,5378],{"className":5379,"style":2629},[334],[103,5381,2298],{"className":5382},[147],[103,5384,2173],{"className":5385,"style":2188},[147,148],[103,5387,427],{"className":5388},[472],[15,5390,5391,5392,5423,5424,5461,5462,5496,5497,5686],{},"The output of ",[103,5393,5395,5408],{"className":5394},[106],[103,5396,5398],{"className":5397},[110],[112,5399,5400],{"xmlns":114},[116,5401,5402,5406],{},[119,5403,5404],{},[957,5405,5221],{},[126,5407,5224],{"encoding":128},[103,5409,5411],{"className":5410,"ariaHidden":134},[133],[103,5412,5414,5417],{"className":5413},[138],[103,5415],{"className":5416,"style":5234},[142],[103,5418,5420],{"className":5419},[147,1041],[103,5421,5221],{"className":5422},[147]," is typically returned in radians ranging from ",[103,5425,5427,5444],{"className":5426},[106],[103,5428,5430],{"className":5429},[110],[112,5431,5432],{"xmlns":114},[116,5433,5434,5441],{},[119,5435,5436,5438],{},[311,5437,422],{},[122,5439,5440],{},"π",[126,5442,5443],{"encoding":128},"-\\pi",[103,5445,5447],{"className":5446,"ariaHidden":134},[133],[103,5448,5450,5454,5457],{"className":5449},[138],[103,5451],{"className":5452,"style":5453},[142],"height:0.6667em;vertical-align:-0.0833em;",[103,5455,422],{"className":5456},[147],[103,5458,5440],{"className":5459,"style":5460},[147,148],"margin-right:0.0359em;"," to ",[103,5463,5465,5481],{"className":5464},[106],[103,5466,5468],{"className":5467},[110],[112,5469,5470],{"xmlns":114},[116,5471,5472,5478],{},[119,5473,5474,5476],{},[311,5475,684],{},[122,5477,5440],{},[126,5479,5480],{"encoding":128},"+\\pi",[103,5482,5484],{"className":5483,"ariaHidden":134},[133],[103,5485,5487,5490,5493],{"className":5486},[138],[103,5488],{"className":5489,"style":5453},[142],[103,5491,684],{"className":5492},[147],[103,5494,5440],{"className":5495,"style":5460},[147,148],". To convert this value to decimal degrees and normalize it to a standard surveying azimuth range (",[103,5498,5500,5537],{"className":5499},[106],[103,5501,5503],{"className":5502},[110],[112,5504,5505],{"xmlns":114},[116,5506,5507,5534],{},[119,5508,5509,5515,5518,5525,5528],{},[968,5510,5511,5513],{},[415,5512,1080],{},[311,5514,974],{},[311,5516,5517],{},"≤",[2860,5519,5520,5522],{},[122,5521,124],{},[957,5523,5524],{},"deg",[311,5526,5527],{},"\u003C",[968,5529,5530,5532],{},[415,5531,799],{},[311,5533,974],{},[126,5535,5536],{"encoding":128},"0^\\circ \\le \\theta_{\\text{deg}} \u003C 360^\\circ",[103,5538,5540,5585,5648],{"className":5539,"ariaHidden":134},[133],[103,5541,5543,5547,5576,5579,5582],{"className":5542},[138],[103,5544],{"className":5545,"style":5546},[142],"height:0.8101em;vertical-align:-0.136em;",[103,5548,5550,5553],{"className":5549},[147],[103,5551,1080],{"className":5552},[147],[103,5554,5556],{"className":5555},[1084],[103,5557,5559],{"className":5558},[1088],[103,5560,5562],{"className":5561},[1092],[103,5563,5565],{"className":5564,"style":1097},[1096],[103,5566,5567,5570],{"style":1100},[103,5568],{"className":5569,"style":1105},[1104],[103,5571,5573],{"className":5572},[1109,1110,1111,1112],[103,5574,974],{"className":5575},[456,1112],[103,5577],{"className":5578,"style":335},[334],[103,5580,5517],{"className":5581},[339],[103,5583],{"className":5584,"style":335},[334],[103,5586,5588,5592,5639,5642,5645],{"className":5587},[138],[103,5589],{"className":5590,"style":5591},[142],"height:0.9805em;vertical-align:-0.2861em;",[103,5593,5595,5598],{"className":5594},[147],[103,5596,124],{"className":5597,"style":149},[147,148],[103,5599,5601],{"className":5600},[1084],[103,5602,5604,5630],{"className":5603},[1088,2891],[103,5605,5607,5627],{"className":5606},[1092],[103,5608,5610],{"className":5609,"style":5313},[1096],[103,5611,5612,5615],{"style":5316},[103,5613],{"className":5614,"style":1105},[1104],[103,5616,5618],{"className":5617},[1109,1110,1111,1112],[103,5619,5621],{"className":5620},[147,1112],[103,5622,5624],{"className":5623},[147,1041,1112],[103,5625,5524],{"className":5626},[147,1112],[103,5628,2915],{"className":5629},[2914],[103,5631,5633],{"className":5632},[1092],[103,5634,5637],{"className":5635,"style":5636},[1096],"height:0.2861em;",[103,5638],{},[103,5640],{"className":5641,"style":335},[334],[103,5643,5527],{"className":5644},[339],[103,5646],{"className":5647,"style":335},[334],[103,5649,5651,5654,5657],{"className":5650},[138],[103,5652],{"className":5653,"style":1097},[142],[103,5655,1013],{"className":5656},[147],[103,5658,5660,5663],{"className":5659},[147],[103,5661,1080],{"className":5662},[147],[103,5664,5666],{"className":5665},[1084],[103,5667,5669],{"className":5668},[1088],[103,5670,5672],{"className":5671},[1092],[103,5673,5675],{"className":5674,"style":1097},[1096],[103,5676,5677,5680],{"style":1100},[103,5678],{"className":5679,"style":1105},[1104],[103,5681,5683],{"className":5682},[1109,1110,1111,1112],[103,5684,974],{"className":5685},[456,1112],"), apply the following steps:",[15,5688,5689,5952],{},[103,5690,5692,5733],{"className":5691},[106],[103,5693,5695],{"className":5694},[110],[112,5696,5697],{"xmlns":114},[116,5698,5699,5730],{},[119,5700,5701,5707,5709,5715,5718],{},[2860,5702,5703,5705],{},[122,5704,124],{},[957,5706,5524],{},[311,5708,313],{},[2860,5710,5711,5713],{},[122,5712,124],{},[957,5714,5262],{},[311,5716,5717],{},"×",[119,5719,5720,5722,5728],{},[311,5721,413],{"fence":134},[4934,5723,5724,5726],{},[415,5725,417],{},[122,5727,5440],{},[311,5729,427],{"fence":134},[126,5731,5732],{"encoding":128},"\\theta_{\\text{deg}}=\\theta_{\\text{rad}}\\times\\left(\\frac{180}{\\pi}\\right)",[103,5734,5736,5797,5858],{"className":5735,"ariaHidden":134},[133],[103,5737,5739,5742,5788,5791,5794],{"className":5738},[138],[103,5740],{"className":5741,"style":5591},[142],[103,5743,5745,5748],{"className":5744},[147],[103,5746,124],{"className":5747,"style":149},[147,148],[103,5749,5751],{"className":5750},[1084],[103,5752,5754,5780],{"className":5753},[1088,2891],[103,5755,5757,5777],{"className":5756},[1092],[103,5758,5760],{"className":5759,"style":5313},[1096],[103,5761,5762,5765],{"style":5316},[103,5763],{"className":5764,"style":1105},[1104],[103,5766,5768],{"className":5767},[1109,1110,1111,1112],[103,5769,5771],{"className":5770},[147,1112],[103,5772,5774],{"className":5773},[147,1041,1112],[103,5775,5524],{"className":5776},[147,1112],[103,5778,2915],{"className":5779},[2914],[103,5781,5783],{"className":5782},[1092],[103,5784,5786],{"className":5785,"style":5636},[1096],[103,5787],{},[103,5789],{"className":5790,"style":335},[334],[103,5792,313],{"className":5793},[339],[103,5795],{"className":5796,"style":335},[334],[103,5798,5800,5803,5849,5852,5855],{"className":5799},[138],[103,5801],{"className":5802,"style":5294},[142],[103,5804,5806,5809],{"className":5805},[147],[103,5807,124],{"className":5808,"style":149},[147,148],[103,5810,5812],{"className":5811},[1084],[103,5813,5815,5841],{"className":5814},[1088,2891],[103,5816,5818,5838],{"className":5817},[1092],[103,5819,5821],{"className":5820,"style":5313},[1096],[103,5822,5823,5826],{"style":5316},[103,5824],{"className":5825,"style":1105},[1104],[103,5827,5829],{"className":5828},[1109,1110,1111,1112],[103,5830,5832],{"className":5831},[147,1112],[103,5833,5835],{"className":5834},[147,1041,1112],[103,5836,5262],{"className":5837},[147,1112],[103,5839,2915],{"className":5840},[2914],[103,5842,5844],{"className":5843},[1092],[103,5845,5847],{"className":5846,"style":2922},[1096],[103,5848],{},[103,5850],{"className":5851,"style":452},[334],[103,5853,5717],{"className":5854},[456],[103,5856],{"className":5857,"style":452},[334],[103,5859,5861,5865],{"className":5860},[138],[103,5862],{"className":5863,"style":5864},[142],"height:1.2em;vertical-align:-0.35em;",[103,5866,5868,5875,5946],{"className":5867},[4999],[103,5869,5871],{"className":5870,"style":5004},[444,5003],[103,5872,413],{"className":5873},[5008,5874],"size1",[103,5876,5878,5881,5943],{"className":5877},[147],[103,5879],{"className":5880},[444,5016],[103,5882,5884],{"className":5883},[4934],[103,5885,5887,5934],{"className":5886},[1088,2891],[103,5888,5890,5931],{"className":5889},[1092],[103,5891,5894,5908,5916],{"className":5892,"style":5893},[1096],"height:0.8451em;",[103,5895,5896,5899],{"style":5032},[103,5897],{"className":5898,"style":4757},[1104],[103,5900,5902],{"className":5901},[1109,1110,1111,1112],[103,5903,5905],{"className":5904},[147,1112],[103,5906,5440],{"className":5907,"style":5460},[147,148,1112],[103,5909,5910,5913],{"style":5054},[103,5911],{"className":5912,"style":4757},[1104],[103,5914],{"className":5915,"style":5062},[5061],[103,5917,5919,5922],{"style":5918},"top:-3.394em;",[103,5920],{"className":5921,"style":4757},[1104],[103,5923,5925],{"className":5924},[1109,1110,1111,1112],[103,5926,5928],{"className":5927},[147,1112],[103,5929,417],{"className":5930},[147,1112],[103,5932,2915],{"className":5933},[2914],[103,5935,5937],{"className":5936},[1092],[103,5938,5941],{"className":5939,"style":5940},[1096],"height:0.345em;",[103,5942],{},[103,5944],{"className":5945},[472,5016],[103,5947,5949],{"className":5948,"style":5004},[472,5003],[103,5950,427],{"className":5951},[5008,5874],[103,5953,5955,6009],{"className":5954},[106],[103,5956,5958],{"className":5957},[110],[112,5959,5960],{"xmlns":114},[116,5961,5962,6006],{},[119,5963,5964,5967,5973,5975,5981,5984,5990,5992,5998,6000],{},[957,5965,5966],{},"If ",[2860,5968,5969,5971],{},[122,5970,124],{},[957,5972,5524],{},[311,5974,5527],{},[968,5976,5977,5979],{},[415,5978,1080],{},[311,5980,974],{},[957,5982,5983],{},", then ",[2860,5985,5986,5988],{},[122,5987,124],{},[957,5989,5524],{},[311,5991,313],{},[2860,5993,5994,5996],{},[122,5995,124],{},[957,5997,5524],{},[311,5999,684],{},[968,6001,6002,6004],{},[415,6003,799],{},[311,6005,974],{},[126,6007,6008],{"encoding":128},"\\text{If }\\theta_{\\text{deg}}\u003C0^\\circ\\text{, then 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surveying crew occupies a control monument (Station A) with known coordinates:",[71,6363,6364,6437],{},[74,6365,6366,6367,6436],{},"Easting of Station A (",[103,6368,6370,6387],{"className":6369},[106],[103,6371,6373],{"className":6372},[110],[112,6374,6375],{"xmlns":114},[116,6376,6377,6385],{},[119,6378,6379],{},[2860,6380,6381,6383],{},[122,6382,2142],{},[415,6384,1503],{},[126,6386,3017],{"encoding":128},[103,6388,6390],{"className":6389,"ariaHidden":134},[133],[103,6391,6393,6396],{"className":6392},[138],[103,6394],{"className":6395,"style":2878},[142],[103,6397,6399,6402],{"className":6398},[147],[103,6400,2142],{"className":6401,"style":2157},[147,148],[103,6403,6405],{"className":6404},[1084],[103,6406,6408,6428],{"className":6407},[1088,2891],[103,6409,6411,6425],{"className":6410},[1092],[103,6412,6414],{"className":6413,"style":2898},[1096],[103,6415,6416,6419],{"style":2901},[103,6417],{"className":6418,"style":1105},[1104],[103,6420,6422],{"className":6421},[1109,1110,1111,1112],[103,6423,1503],{"className":6424},[147,1112],[103,6426,2915],{"className":6427},[2914],[103,6429,6431],{"className":6430},[1092],[103,6432,6434],{"className":6433,"style":2922},[1096],[103,6435],{},"): 500000.000 m",[74,6438,6439,6440,6509],{},"Northing of Station A (",[103,6441,6443,6460],{"className":6442},[106],[103,6444,6446],{"className":6445},[110],[112,6447,6448],{"xmlns":114},[116,6449,6450,6458],{},[119,6451,6452],{},[2860,6453,6454,6456],{},[122,6455,2173],{},[415,6457,1503],{},[126,6459,3087],{"encoding":128},[103,6461,6463],{"className":6462,"ariaHidden":134},[133],[103,6464,6466,6469],{"className":6465},[138],[103,6467],{"className":6468,"style":2878},[142],[103,6470,6472,6475],{"className":6471},[147],[103,6473,2173],{"className":6474,"style":2188},[147,148],[103,6476,6478],{"className":6477},[1084],[103,6479,6481,6501],{"className":6480},[1088,2891],[103,6482,6484,6498],{"className":6483},[1092],[103,6485,6487],{"className":6486,"style":2898},[1096],[103,6488,6489,6492],{"style":2975},[103,6490],{"className":6491,"style":1105},[1104],[103,6493,6495],{"className":6494},[1109,1110,1111,1112],[103,6496,1503],{"className":6497},[147,1112],[103,6499,2915],{"className":6500},[2914],[103,6502,6504],{"className":6503},[1092],[103,6505,6507],{"className":6506,"style":2922},[1096],[103,6508],{},"): 150000.000 m",[15,6511,6512,6513,6541,6542,6570],{},"A total station measures a horizontal distance (",[103,6514,6516,6529],{"className":6515},[106],[103,6517,6519],{"className":6518},[110],[112,6520,6521],{"xmlns":114},[116,6522,6523,6527],{},[119,6524,6525],{},[122,6526,2515],{},[126,6528,2515],{"encoding":128},[103,6530,6532],{"className":6531,"ariaHidden":134},[133],[103,6533,6535,6538],{"className":6534},[138],[103,6536],{"className":6537,"style":2092},[142],[103,6539,2515],{"className":6540,"style":149},[147,148],") and grid azimuth (",[103,6543,6545,6558],{"className":6544},[106],[103,6546,6548],{"className":6547},[110],[112,6549,6550],{"xmlns":114},[116,6551,6552,6556],{},[119,6553,6554],{},[122,6555,124],{},[126,6557,129],{"encoding":128},[103,6559,6561],{"className":6560,"ariaHidden":134},[133],[103,6562,6564,6567],{"className":6563},[138],[103,6565],{"className":6566,"style":143},[142],[103,6568,124],{"className":6569,"style":149},[147,148],") to a proposed property corner (Station B):",[71,6572,6573,6605],{},[74,6574,6575,6576,6604],{},"Distance (",[103,6577,6579,6592],{"className":6578},[106],[103,6580,6582],{"className":6581},[110],[112,6583,6584],{"xmlns":114},[116,6585,6586,6590],{},[119,6587,6588],{},[122,6589,2515],{},[126,6591,2515],{"encoding":128},[103,6593,6595],{"className":6594,"ariaHidden":134},[133],[103,6596,6598,6601],{"className":6597},[138],[103,6599],{"className":6600,"style":2092},[142],[103,6602,2515],{"className":6603,"style":149},[147,148],"): 83.472 m",[74,6606,6607,6608,6636,6637],{},"Azimuth (",[103,6609,6611,6624],{"className":6610},[106],[103,6612,6614],{"className":6613},[110],[112,6615,6616],{"xmlns":114},[116,6617,6618,6622],{},[119,6619,6620],{},[122,6621,124],{},[126,6623,129],{"encoding":128},[103,6625,6627],{"className":6626,"ariaHidden":134},[133],[103,6628,6630,6633],{"className":6629},[138],[103,6631],{"className":6632,"style":143},[142],[103,6634,124],{"className":6635,"style":149},[147,148],"): ",[103,6638,6640,6663],{"className":6639},[106],[103,6641,6643],{"className":6642},[110],[112,6644,6645],{"xmlns":114},[116,6646,6647,6660],{},[119,6648,6649,6651,6653],{},[122,6650,124],{},[311,6652,313],{},[968,6654,6655,6658],{},[415,6656,6657],{},"127.5430",[311,6659,974],{},[126,6661,6662],{"encoding":128},"\\theta=127.5430^\\circ",[103,6664,6666,6684],{"className":6665,"ariaHidden":134},[133],[103,6667,6669,6672,6675,6678,6681],{"className":6668},[138],[103,6670],{"className":6671,"style":143},[142],[103,6673,124],{"className":6674,"style":149},[147,148],[103,6676],{"className":6677,"style":335},[334],[103,6679,313],{"className":6680},[339],[103,6682],{"className":6683,"style":335},[334],[103,6685,6687,6690,6694],{"className":6686},[138],[103,6688],{"className":6689,"style":1097},[142],[103,6691,6693],{"className":6692},[147],"127.543",[103,6695,6697,6700],{"className":6696},[147],[103,6698,1080],{"className":6699},[147],[103,6701,6703],{"className":6702},[1084],[103,6704,6706],{"className":6705},[1088],[103,6707,6709],{"className":6708},[1092],[103,6710,6712],{"className":6711,"style":1097},[1096],[103,6713,6714,6717],{"style":1100},[103,6715],{"className":6716,"style":1105},[1104],[103,6718,6720],{"className":6719},[1109,1110,1111,1112],[103,6721,974],{"className":6722},[456,1112],[15,6724,6725],{},[938,6726,6727],{},"Step 1: Convert the azimuth from decimal degrees to radians",[15,6729,6730],{},[103,6731,6733,6779],{"className":6732},[106],[103,6734,6736],{"className":6735},[110],[112,6737,6738],{"xmlns":114},[116,6739,6740,6776],{},[119,6741,6742,6748,6750,6756,6758,6770,6773],{},[2860,6743,6744,6746],{},[122,6745,124],{},[957,6747,5262],{},[311,6749,313],{},[968,6751,6752,6754],{},[415,6753,6657],{},[311,6755,974],{},[311,6757,5717],{},[119,6759,6760,6762,6768],{},[311,6761,413],{"fence":134},[4934,6763,6764,6766],{},[122,6765,5440],{},[415,6767,417],{},[311,6769,427],{"fence":134},[311,6771,6772],{},"≈",[415,6774,6775],{},"2.226045952",[126,6777,6778],{"encoding":128},"\\theta_{\\text{rad}}=127.5430^\\circ\\times\\left(\\frac{\\pi}{180}\\right)\\approx2.226045952",[103,6780,6782,6843,6891,6990],{"className":6781,"ariaHidden":134},[133],[103,6783,6785,6788,6834,6837,6840],{"className":6784},[138],[103,6786],{"className":6787,"style":5294},[142],[103,6789,6791,6794],{"className":6790},[147],[103,6792,124],{"className":6793,"style":149},[147,148],[103,6795,6797],{"className":6796},[1084],[103,6798,6800,6826],{"className":6799},[1088,2891],[103,6801,6803,6823],{"className":6802},[1092],[103,6804,6806],{"className":6805,"style":5313},[1096],[103,6807,6808,6811],{"style":5316},[103,6809],{"className":6810,"style":1105},[1104],[103,6812,6814],{"className":6813},[1109,1110,1111,1112],[103,6815,6817],{"className":6816},[147,1112],[103,6818,6820],{"className":6819},[147,1041,1112],[103,6821,5262],{"className":6822},[147,1112],[103,6824,2915],{"className":6825},[2914],[103,6827,6829],{"className":6828},[1092],[103,6830,6832],{"className":6831,"style":2922},[1096],[103,6833],{},[103,6835],{"className":6836,"style":335},[334],[103,6838,313],{"className":6839},[339],[103,6841],{"className":6842,"style":335},[334],[103,6844,6846,6850,6853,6882,6885,6888],{"className":6845},[138],[103,6847],{"className":6848,"style":6849},[142],"height:0.7574em;vertical-align:-0.0833em;",[103,6851,6693],{"className":6852},[147],[103,6854,6856,6859],{"className":6855},[147],[103,6857,1080],{"className":6858},[147],[103,6860,6862],{"className":6861},[1084],[103,6863,6865],{"className":6864},[1088],[103,6866,6868],{"className":6867},[1092],[103,6869,6871],{"className":6870,"style":1097},[1096],[103,6872,6873,6876],{"style":1100},[103,6874],{"className":6875,"style":1105},[1104],[103,6877,6879],{"className":6878},[1109,1110,1111,1112],[103,6880,974],{"className":6881},[456,1112],[103,6883],{"className":6884,"style":452},[334],[103,6886,5717],{"className":6887},[456],[103,6889],{"className":6890,"style":452},[334],[103,6892,6894,6897,6981,6984,6987],{"className":6893},[138],[103,6895],{"className":6896,"style":5864},[142],[103,6898,6900,6906,6975],{"className":6899},[4999],[103,6901,6903],{"className":6902,"style":5004},[444,5003],[103,6904,413],{"className":6905},[5008,5874],[103,6907,6909,6912,6972],{"className":6908},[147],[103,6910],{"className":6911},[444,5016],[103,6913,6915],{"className":6914},[4934],[103,6916,6918,6964],{"className":6917},[1088,2891],[103,6919,6921,6961],{"className":6920},[1092],[103,6922,6925,6939,6947],{"className":6923,"style":6924},[1096],"height:0.6954em;",[103,6926,6927,6930],{"style":5032},[103,6928],{"className":6929,"style":4757},[1104],[103,6931,6933],{"className":6932},[1109,1110,1111,1112],[103,6934,6936],{"className":6935},[147,1112],[103,6937,417],{"className":6938},[147,1112],[103,6940,6941,6944],{"style":5054},[103,6942],{"className":6943,"style":4757},[1104],[103,6945],{"className":6946,"style":5062},[5061],[103,6948,6949,6952],{"style":5918},[103,6950],{"className":6951,"style":4757},[1104],[103,6953,6955],{"className":6954},[1109,1110,1111,1112],[103,6956,6958],{"className":6957},[147,1112],[103,6959,5440],{"className":6960,"style":5460},[147,148,1112],[103,6962,2915],{"className":6963},[2914],[103,6965,6967],{"className":6966},[1092],[103,6968,6970],{"className":6969,"style":5940},[1096],[103,6971],{},[103,6973],{"className":6974},[472,5016],[103,6976,6978],{"className":6977,"style":5004},[472,5003],[103,6979,427],{"className":6980},[5008,5874],[103,6982],{"className":6983,"style":335},[334],[103,6985,6772],{"className":6986},[339],[103,6988],{"className":6989,"style":335},[334],[103,6991,6993,6996],{"className":6992},[138],[103,6994],{"className":6995,"style":5234},[142],[103,6997,6775],{"className":6998},[147],[15,7000,7001],{},"The converted angle is approximately 2.226045952 radians.",[15,7003,7004],{},[938,7005,7006,7007,2809,7040,427],{},"Step 2: Calculate the coordinate offsets (",[103,7008,7010,7025],{"className":7009},[106],[103,7011,7013],{"className":7012},[110],[112,7014,7015],{"xmlns":114},[116,7016,7017,7023],{},[119,7018,7019,7021],{},[122,7020,2298],{"mathvariant":237},[122,7022,2142],{},[126,7024,2303],{"encoding":128},[103,7026,7028],{"className":7027,"ariaHidden":134},[133],[103,7029,7031,7034,7037],{"className":7030},[138],[103,7032],{"className":7033,"style":2092},[142],[103,7035,2298],{"className":7036},[147],[103,7038,2142],{"className":7039,"style":2157},[147,148],[103,7041,7043,7058],{"className":7042},[106],[103,7044,7046],{"className":7045},[110],[112,7047,7048],{"xmlns":114},[116,7049,7050,7056],{},[119,7051,7052,7054],{},[122,7053,2298],{"mathvariant":237},[122,7055,2173],{},[126,7057,2339],{"encoding":128},[103,7059,7061],{"className":7060,"ariaHidden":134},[133],[103,7062,7064,7067,7070],{"className":7063},[138],[103,7065],{"className":7066,"style":2092},[142],[103,7068,2298],{"className":7069},[147],[103,7071,2173],{"className":7072,"style":2188},[147,148],[15,7074,7075],{},[103,7076,7078,7126],{"className":7077},[106],[103,7079,7081],{"className":7080},[110],[112,7082,7083],{"xmlns":114},[116,7084,7085,7123],{},[119,7086,7087,7089,7091,7093,7095,7097,7099,7101,7103,7106,7108,7110,7112,7114,7116,7118,7120],{},[122,7088,2298],{"mathvariant":237},[122,7090,2142],{},[311,7092,313],{},[122,7094,2515],{},[122,7096,2584],{},[311,7098,2587],{},[122,7100,124],{},[311,7102,313],{},[415,7104,7105],{},"83.472",[311,7107,5717],{},[122,7109,2584],{},[311,7111,2587],{},[311,7113,413],{"stretchy":412},[415,7115,6775],{},[311,7117,427],{"stretchy":412},[311,7119,6772],{},[415,7121,7122],{},"66.185987",[126,7124,7125],{"encoding":128},"\\Delta E=D\\sin\\theta=83.472\\times\\sin(2.226045952)\\approx66.185987",[103,7127,7129,7150,7180,7199,7226],{"className":7128,"ariaHidden":134},[133],[103,7130,7132,7135,7138,7141,7144,7147],{"className":7131},[138],[103,7133],{"className":7134,"style":2092},[142],[103,7136,2298],{"className":7137},[147],[103,7139,2142],{"className":7140,"style":2157},[147,148],[103,7142],{"className":7143,"style":335},[334],[103,7145,313],{"className":7146},[339],[103,7148],{"className":7149,"style":335},[334],[103,7151,7153,7156,7159,7162,7165,7168,7171,7174,7177],{"className":7152},[138],[103,7154],{"className":7155,"style":143},[142],[103,7157,2515],{"className":7158,"style":149},[147,148],[103,7160],{"className":7161,"style":2629},[334],[103,7163,2584],{"className":7164},[2633],[103,7166],{"className":7167,"style":2629},[334],[103,7169,124],{"className":7170,"style":149},[147,148],[103,7172],{"className":7173,"style":335},[334],[103,7175,313],{"className":7176},[339],[103,7178],{"className":7179,"style":335},[334],[103,7181,7183,7187,7190,7193,7196],{"className":7182},[138],[103,7184],{"className":7185,"style":7186},[142],"height:0.7278em;vertical-align:-0.0833em;",[103,7188,7105],{"className":7189},[147],[103,7191],{"className":7192,"style":452},[334],[103,7194,5717],{"className":7195},[456],[103,7197],{"className":7198,"style":452},[334],[103,7200,7202,7205,7208,7211,7214,7217,7220,7223],{"className":7201},[138],[103,7203],{"className":7204,"style":440},[142],[103,7206,2584],{"className":7207},[2633],[103,7209,413],{"className":7210},[444],[103,7212,6775],{"className":7213},[147],[103,7215,427],{"className":7216},[472],[103,7218],{"className":7219,"style":335},[334],[103,7221,6772],{"className":7222},[339],[103,7224],{"className":7225,"style":335},[334],[103,7227,7229,7232],{"className":7228},[138],[103,7230],{"className":7231,"style":5234},[142],[103,7233,7122],{"className":7234},[147],[15,7236,7237],{},"The Easting offset is approximately 66.185987 m.",[15,7239,7240],{},[103,7241,7243,7292],{"className":7242},[106],[103,7244,7246],{"className":7245},[110],[112,7247,7248],{"xmlns":114},[116,7249,7250,7289],{},[119,7251,7252,7254,7256,7258,7260,7262,7264,7266,7268,7270,7272,7274,7276,7278,7280,7282,7284,7286],{},[122,7253,2298],{"mathvariant":237},[122,7255,2173],{},[311,7257,313],{},[122,7259,2515],{},[122,7261,2662],{},[311,7263,2587],{},[122,7265,124],{},[311,7267,313],{},[415,7269,7105],{},[311,7271,5717],{},[122,7273,2662],{},[311,7275,2587],{},[311,7277,413],{"stretchy":412},[415,7279,6775],{},[311,7281,427],{"stretchy":412},[311,7283,6772],{},[311,7285,422],{},[415,7287,7288],{},"50.861997",[126,7290,7291],{"encoding":128},"\\Delta N=D\\cos\\theta=83.472\\times\\cos(2.226045952)\\approx-50.861997",[103,7293,7295,7316,7346,7364,7391],{"className":7294,"ariaHidden":134},[133],[103,7296,7298,7301,7304,7307,7310,7313],{"className":7297},[138],[103,7299],{"className":7300,"style":2092},[142],[103,7302,2298],{"className":7303},[147],[103,7305,2173],{"className":7306,"style":2188},[147,148],[103,7308],{"className":7309,"style":335},[334],[103,7311,313],{"className":7312},[339],[103,7314],{"className":7315,"style":335},[334],[103,7317,7319,7322,7325,7328,7331,7334,7337,7340,7343],{"className":7318},[138],[103,7320],{"className":7321,"style":143},[142],[103,7323,2515],{"className":7324,"style":149},[147,148],[103,7326],{"className":7327,"style":2629},[334],[103,7329,2662],{"className":7330},[2633],[103,7332],{"className":7333,"style":2629},[334],[103,7335,124],{"className":7336,"style":149},[147,148],[103,7338],{"className":7339,"style":335},[334],[103,7341,313],{"className":7342},[339],[103,7344],{"className":7345,"style":335},[334],[103,7347,7349,7352,7355,7358,7361],{"className":7348},[138],[103,7350],{"className":7351,"style":7186},[142],[103,7353,7105],{"className":7354},[147],[103,7356],{"className":7357,"style":452},[334],[103,7359,5717],{"className":7360},[456],[103,7362],{"className":7363,"style":452},[334],[103,7365,7367,7370,7373,7376,7379,7382,7385,7388],{"className":7366},[138],[103,7368],{"className":7369,"style":440},[142],[103,7371,2662],{"className":7372},[2633],[103,7374,413],{"className":7375},[444],[103,7377,6775],{"className":7378},[147],[103,7380,427],{"className":7381},[472],[103,7383],{"className":7384,"style":335},[334],[103,7386,6772],{"className":7387},[339],[103,7389],{"className":7390,"style":335},[334],[103,7392,7394,7397,7400],{"className":7393},[138],[103,7395],{"className":7396,"style":7186},[142],[103,7398,422],{"className":7399},[147],[103,7401,7288],{"className":7402},[147],[15,7404,7405],{},"The Northing offset is approximately −50.861997 m.",[15,7407,7408],{},[938,7409,7410],{},"Step 3: Round the offsets to millimeter precision (3 decimal places)",[71,7412,7413,7473],{},[74,7414,7415,7472],{},[103,7416,7418,7439],{"className":7417},[106],[103,7419,7421],{"className":7420},[110],[112,7422,7423],{"xmlns":114},[116,7424,7425,7436],{},[119,7426,7427,7429,7431,7433],{},[122,7428,2298],{"mathvariant":237},[122,7430,2142],{},[311,7432,6772],{},[415,7434,7435],{},"66.186",[126,7437,7438],{"encoding":128},"\\Delta E\\approx66.186",[103,7440,7442,7463],{"className":7441,"ariaHidden":134},[133],[103,7443,7445,7448,7451,7454,7457,7460],{"className":7444},[138],[103,7446],{"className":7447,"style":2092},[142],[103,7449,2298],{"className":7450},[147],[103,7452,2142],{"className":7453,"style":2157},[147,148],[103,7455],{"className":7456,"style":335},[334],[103,7458,6772],{"className":7459},[339],[103,7461],{"className":7462,"style":335},[334],[103,7464,7466,7469],{"className":7465},[138],[103,7467],{"className":7468,"style":5234},[142],[103,7470,7435],{"className":7471},[147]," m",[74,7474,7475,7472],{},[103,7476,7478,7501],{"className":7477},[106],[103,7479,7481],{"className":7480},[110],[112,7482,7483],{"xmlns":114},[116,7484,7485,7498],{},[119,7486,7487,7489,7491,7493,7495],{},[122,7488,2298],{"mathvariant":237},[122,7490,2173],{},[311,7492,6772],{},[311,7494,422],{},[415,7496,7497],{},"50.862",[126,7499,7500],{"encoding":128},"\\Delta N\\approx-50.862",[103,7502,7504,7525],{"className":7503,"ariaHidden":134},[133],[103,7505,7507,7510,7513,7516,7519,7522],{"className":7506},[138],[103,7508],{"className":7509,"style":2092},[142],[103,7511,2298],{"className":7512},[147],[103,7514,2173],{"className":7515,"style":2188},[147,148],[103,7517],{"className":7518,"style":335},[334],[103,7520,6772],{"className":7521},[339],[103,7523],{"className":7524,"style":335},[334],[103,7526,7528,7531,7534],{"className":7527},[138],[103,7529],{"className":7530,"style":7186},[142],[103,7532,422],{"className":7533},[147],[103,7535,7497],{"className":7536},[147],[15,7538,7539],{},[938,7540,7541],{},"Step 4: Calculate the absolute coordinates of Station B",[15,7543,7544],{},[103,7545,7547,7593],{"className":7546},[106],[103,7548,7550],{"className":7549},[110],[112,7551,7552],{"xmlns":114},[116,7553,7554,7590],{},[119,7555,7556,7562,7564,7570,7572,7574,7576,7578,7581,7583,7585,7587],{},[2860,7557,7558,7560],{},[122,7559,2142],{},[415,7561,1204],{},[311,7563,313],{},[2860,7565,7566,7568],{},[122,7567,2142],{},[415,7569,1503],{},[311,7571,684],{},[122,7573,2298],{"mathvariant":237},[122,7575,2142],{},[311,7577,313],{},[415,7579,7580],{},"500000.000",[311,7582,684],{},[415,7584,7435],{},[311,7586,313],{},[415,7588,7589],{},"500066.186",[126,7591,7592],{"encoding":128},"E_2=E_1+\\Delta E=500000.000+66.186=500066.186",[103,7594,7596,7651,7706,7727,7745,7763],{"className":7595,"ariaHidden":134},[133],[103,7597,7599,7602,7642,7645,7648],{"className":7598},[138],[103,7600],{"className":7601,"style":2878},[142],[103,7603,7605,7608],{"className":7604},[147],[103,7606,2142],{"className":7607,"style":2157},[147,148],[103,7609,7611],{"className":7610},[1084],[103,7612,7614,7634],{"className":7613},[1088,2891],[103,7615,7617,7631],{"className":7616},[1092],[103,7618,7620],{"className":7619,"style":2898},[1096],[103,7621,7622,7625],{"style":2901},[103,7623],{"className":7624,"style":1105},[1104],[103,7626,7628],{"className":7627},[1109,1110,1111,1112],[103,7629,1204],{"className":7630},[147,1112],[103,7632,2915],{"className":7633},[2914],[103,7635,7637],{"className":7636},[1092],[103,7638,7640],{"className":7639,"style":2922},[1096],[103,7641],{},[103,7643],{"className":7644,"style":335},[334],[103,7646,313],{"className":7647},[339],[103,7649],{"className":7650,"style":335},[334],[103,7652,7654,7657,7697,7700,7703],{"className":7653},[138],[103,7655],{"className":7656,"style":2878},[142],[103,7658,7660,7663],{"className":7659},[147],[103,7661,2142],{"className":7662,"style":2157},[147,148],[103,7664,7666],{"className":7665},[1084],[103,7667,7669,7689],{"className":7668},[1088,2891],[103,7670,7672,7686],{"className":7671},[1092],[103,7673,7675],{"className":7674,"style":2898},[1096],[103,7676,7677,7680],{"style":2901},[103,7678],{"className":7679,"style":1105},[1104],[103,7681,7683],{"className":7682},[1109,1110,1111,1112],[103,7684,1503],{"className":7685},[147,1112],[103,7687,2915],{"className":7688},[2914],[103,7690,7692],{"className":7691},[1092],[103,7693,7695],{"className":7694,"style":2922},[1096],[103,7696],{},[103,7698],{"className":7699,"style":452},[334],[103,7701,684],{"className":7702},[456],[103,7704],{"className":7705,"style":452},[334],[103,7707,7709,7712,7715,7718,7721,7724],{"className":7708},[138],[103,7710],{"className":7711,"style":2092},[142],[103,7713,2298],{"className":7714},[147],[103,7716,2142],{"className":7717,"style":2157},[147,148],[103,7719],{"className":7720,"style":335},[334],[103,7722,313],{"className":7723},[339],[103,7725],{"className":7726,"style":335},[334],[103,7728,7730,7733,7736,7739,7742],{"className":7729},[138],[103,7731],{"className":7732,"style":7186},[142],[103,7734,7580],{"className":7735},[147],[103,7737],{"className":7738,"style":452},[334],[103,7740,684],{"className":7741},[456],[103,7743],{"className":7744,"style":452},[334],[103,7746,7748,7751,7754,7757,7760],{"className":7747},[138],[103,7749],{"className":7750,"style":5234},[142],[103,7752,7435],{"className":7753},[147],[103,7755],{"className":7756,"style":335},[334],[103,7758,313],{"className":7759},[339],[103,7761],{"className":7762,"style":335},[334],[103,7764,7766,7769],{"className":7765},[138],[103,7767],{"className":7768,"style":5234},[142],[103,7770,7589],{"className":7771},[147],[15,7773,7774],{},"The Easting of Station B is 500066.186 m.",[15,7776,7777],{},[103,7778,7780,7832],{"className":7779},[106],[103,7781,7783],{"className":7782},[110],[112,7784,7785],{"xmlns":114},[116,7786,7787,7829],{},[119,7788,7789,7795,7797,7803,7805,7807,7809,7811,7814,7816,7818,7820,7822,7824,7826],{},[2860,7790,7791,7793],{},[122,7792,2173],{},[415,7794,1204],{},[311,7796,313],{},[2860,7798,7799,7801],{},[122,7800,2173],{},[415,7802,1503],{},[311,7804,684],{},[122,7806,2298],{"mathvariant":237},[122,7808,2173],{},[311,7810,313],{},[415,7812,7813],{},"150000.000",[311,7815,684],{},[311,7817,413],{"stretchy":412},[311,7819,422],{},[415,7821,7497],{},[311,7823,427],{"stretchy":412},[311,7825,313],{},[415,7827,7828],{},"149949.138",[126,7830,7831],{"encoding":128},"N_2=N_1+\\Delta N=150000.000+(-50.862)=149949.138",[103,7833,7835,7890,7945,7966,7984,8011],{"className":7834,"ariaHidden":134},[133],[103,7836,7838,7841,7881,7884,7887],{"className":7837},[138],[103,7839],{"className":7840,"style":2878},[142],[103,7842,7844,7847],{"className":7843},[147],[103,7845,2173],{"className":7846,"style":2188},[147,148],[103,7848,7850],{"className":7849},[1084],[103,7851,7853,7873],{"className":7852},[1088,2891],[103,7854,7856,7870],{"className":7855},[1092],[103,7857,7859],{"className":7858,"style":2898},[1096],[103,7860,7861,7864],{"style":2975},[103,7862],{"className":7863,"style":1105},[1104],[103,7865,7867],{"className":7866},[1109,1110,1111,1112],[103,7868,1204],{"className":7869},[147,1112],[103,7871,2915],{"className":7872},[2914],[103,7874,7876],{"className":7875},[1092],[103,7877,7879],{"className":7878,"style":2922},[1096],[103,7880],{},[103,7882],{"className":7883,"style":335},[334],[103,7885,313],{"className":7886},[339],[103,7888],{"className":7889,"style":335},[334],[103,7891,7893,7896,7936,7939,7942],{"className":7892},[138],[103,7894],{"className":7895,"style":2878},[142],[103,7897,7899,7902],{"className":7898},[147],[103,7900,2173],{"className":7901,"style":2188},[147,148],[103,7903,7905],{"className":7904},[1084],[103,7906,7908,7928],{"className":7907},[1088,2891],[103,7909,7911,7925],{"className":7910},[1092],[103,7912,7914],{"className":7913,"style":2898},[1096],[103,7915,7916,7919],{"style":2975},[103,7917],{"className":7918,"style":1105},[1104],[103,7920,7922],{"className":7921},[1109,1110,1111,1112],[103,7923,1503],{"className":7924},[147,1112],[103,7926,2915],{"className":7927},[2914],[103,7929,7931],{"className":7930},[1092],[103,7932,7934],{"className":7933,"style":2922},[1096],[103,7935],{},[103,7937],{"className":7938,"style":452},[334],[103,7940,684],{"className":7941},[456],[103,7943],{"className":7944,"style":452},[334],[103,7946,7948,7951,7954,7957,7960,7963],{"className":7947},[138],[103,7949],{"className":7950,"style":2092},[142],[103,7952,2298],{"className":7953},[147],[103,7955,2173],{"className":7956,"style":2188},[147,148],[103,7958],{"className":7959,"style":335},[334],[103,7961,313],{"className":7962},[339],[103,7964],{"className":7965,"style":335},[334],[103,7967,7969,7972,7975,7978,7981],{"className":7968},[138],[103,7970],{"className":7971,"style":7186},[142],[103,7973,7813],{"className":7974},[147],[103,7976],{"className":7977,"style":452},[334],[103,7979,684],{"className":7980},[456],[103,7982],{"className":7983,"style":452},[334],[103,7985,7987,7990,7993,7996,7999,8002,8005,8008],{"className":7986},[138],[103,7988],{"className":7989,"style":440},[142],[103,7991,413],{"className":7992},[444],[103,7994,422],{"className":7995},[147],[103,7997,7497],{"className":7998},[147],[103,8000,427],{"className":8001},[472],[103,8003],{"className":8004,"style":335},[334],[103,8006,313],{"className":8007},[339],[103,8009],{"className":8010,"style":335},[334],[103,8012,8014,8017],{"className":8013},[138],[103,8015],{"className":8016,"style":5234},[142],[103,8018,7828],{"className":8019},[147],[15,8021,8022],{},"The Northing of Station B is 149949.138 m.",[15,8024,8025,8026,5203],{},"The final coordinates of Station B are ",[938,8027,8028],{},"500066.186 E, 149949.138 N",[35,8030,8032],{"id":8031},"worked-example-2-rectangular-to-polar-check-back-verification","Worked Example 2: Rectangular to Polar (Check-Back Verification)",[15,8034,8035],{},"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).",[15,8037,8038],{},[938,8039,8040],{},"Step 1: Calculate the coordinate differences",[15,8042,8043],{},[103,8044,8046,8090],{"className":8045},[106],[103,8047,8049],{"className":8048},[110],[112,8050,8051],{"xmlns":114},[116,8052,8053,8087],{},[119,8054,8055,8057,8059,8061,8067,8069,8075,8077,8079,8081,8083,8085],{},[122,8056,2298],{"mathvariant":237},[122,8058,2142],{},[311,8060,313],{},[2860,8062,8063,8065],{},[122,8064,2142],{},[415,8066,1204],{},[311,8068,422],{},[2860,8070,8071,8073],{},[122,8072,2142],{},[415,8074,1503],{},[311,8076,313],{},[415,8078,7589],{},[311,8080,422],{},[415,8082,7580],{},[311,8084,313],{},[415,8086,7435],{},[126,8088,8089],{"encoding":128},"\\Delta E=E_2-E_1=500066.186-500000.000=66.186",[103,8091,8093,8114,8169,8224,8242,8260],{"className":8092,"ariaHidden":134},[133],[103,8094,8096,8099,8102,8105,8108,8111],{"className":8095},[138],[103,8097],{"className":8098,"style":2092},[142],[103,8100,2298],{"className":8101},[147],[103,8103,2142],{"className":8104,"style":2157},[147,148],[103,8106],{"className":8107,"style":335},[334],[103,8109,313],{"className":8110},[339],[103,8112],{"className":8113,"style":335},[334],[103,8115,8117,8120,8160,8163,8166],{"className":8116},[138],[103,8118],{"className":8119,"style":2878},[142],[103,8121,8123,8126],{"className":8122},[147],[103,8124,2142],{"className":8125,"style":2157},[147,148],[103,8127,8129],{"className":8128},[1084],[103,8130,8132,8152],{"className":8131},[1088,2891],[103,8133,8135,8149],{"className":8134},[1092],[103,8136,8138],{"className":8137,"style":2898},[1096],[103,8139,8140,8143],{"style":2901},[103,8141],{"className":8142,"style":1105},[1104],[103,8144,8146],{"className":8145},[1109,1110,1111,1112],[103,8147,1204],{"className":8148},[147,1112],[103,8150,2915],{"className":8151},[2914],[103,8153,8155],{"className":8154},[1092],[103,8156,8158],{"className":8157,"style":2922},[1096],[103,8159],{},[103,8161],{"className":8162,"style":452},[334],[103,8164,422],{"className":8165},[456],[103,8167],{"className":8168,"style":452},[334],[103,8170,8172,8175,8215,8218,8221],{"className":8171},[138],[103,8173],{"className":8174,"style":2878},[142],[103,8176,8178,8181],{"className":8177},[147],[103,8179,2142],{"className":8180,"style":2157},[147,148],[103,8182,8184],{"className":8183},[1084],[103,8185,8187,8207],{"className":8186},[1088,2891],[103,8188,8190,8204],{"className":8189},[1092],[103,8191,8193],{"className":8192,"style":2898},[1096],[103,8194,8195,8198],{"style":2901},[103,8196],{"className":8197,"style":1105},[1104],[103,8199,8201],{"className":8200},[1109,1110,1111,1112],[103,8202,1503],{"className":8203},[147,1112],[103,8205,2915],{"className":8206},[2914],[103,8208,8210],{"className":8209},[1092],[103,8211,8213],{"className":8212,"style":2922},[1096],[103,8214],{},[103,8216],{"className":8217,"style":335},[334],[103,8219,313],{"className":8220},[339],[103,8222],{"className":8223,"style":335},[334],[103,8225,8227,8230,8233,8236,8239],{"className":8226},[138],[103,8228],{"className":8229,"style":7186},[142],[103,8231,7589],{"className":8232},[147],[103,8234],{"className":8235,"style":452},[334],[103,8237,422],{"className":8238},[456],[103,8240],{"className":8241,"style":452},[334],[103,8243,8245,8248,8251,8254,8257],{"className":8244},[138],[103,8246],{"className":8247,"style":5234},[142],[103,8249,7580],{"className":8250},[147],[103,8252],{"className":8253,"style":335},[334],[103,8255,313],{"className":8256},[339],[103,8258],{"className":8259,"style":335},[334],[103,8261,8263,8266],{"className":8262},[138],[103,8264],{"className":8265,"style":5234},[142],[103,8267,7435],{"className":8268},[147],[15,8270,8271],{},"The difference in Easting is 66.186 m.",[15,8273,8274],{},[103,8275,8277,8323],{"className":8276},[106],[103,8278,8280],{"className":8279},[110],[112,8281,8282],{"xmlns":114},[116,8283,8284,8320],{},[119,8285,8286,8288,8290,8292,8298,8300,8306,8308,8310,8312,8314,8316,8318],{},[122,8287,2298],{"mathvariant":237},[122,8289,2173],{},[311,8291,313],{},[2860,8293,8294,8296],{},[122,8295,2173],{},[415,8297,1204],{},[311,8299,422],{},[2860,8301,8302,8304],{},[122,8303,2173],{},[415,8305,1503],{},[311,8307,313],{},[415,8309,7828],{},[311,8311,422],{},[415,8313,7813],{},[311,8315,313],{},[311,8317,422],{},[415,8319,7497],{},[126,8321,8322],{"encoding":128},"\\Delta N=N_2-N_1=149949.138-150000.000=-50.862",[103,8324,8326,8347,8402,8457,8475,8493],{"className":8325,"ariaHidden":134},[133],[103,8327,8329,8332,8335,8338,8341,8344],{"className":8328},[138],[103,8330],{"className":8331,"style":2092},[142],[103,8333,2298],{"className":8334},[147],[103,8336,2173],{"className":8337,"style":2188},[147,148],[103,8339],{"className":8340,"style":335},[334],[103,8342,313],{"className":8343},[339],[103,8345],{"className":8346,"style":335},[334],[103,8348,8350,8353,8393,8396,8399],{"className":8349},[138],[103,8351],{"className":8352,"style":2878},[142],[103,8354,8356,8359],{"className":8355},[147],[103,8357,2173],{"className":8358,"style":2188},[147,148],[103,8360,8362],{"className":8361},[1084],[103,8363,8365,8385],{"className":8364},[1088,2891],[103,8366,8368,8382],{"className":8367},[1092],[103,8369,8371],{"className":8370,"style":2898},[1096],[103,8372,8373,8376],{"style":2975},[103,8374],{"className":8375,"style":1105},[1104],[103,8377,8379],{"className":8378},[1109,1110,1111,1112],[103,8380,1204],{"className":8381},[147,1112],[103,8383,2915],{"className":8384},[2914],[103,8386,8388],{"className":8387},[1092],[103,8389,8391],{"className":8390,"style":2922},[1096],[103,8392],{},[103,8394],{"className":8395,"style":452},[334],[103,8397,422],{"className":8398},[456],[103,8400],{"className":8401,"style":452},[334],[103,8403,8405,8408,8448,8451,8454],{"className":8404},[138],[103,8406],{"className":8407,"style":2878},[142],[103,8409,8411,8414],{"className":8410},[147],[103,8412,2173],{"className":8413,"style":2188},[147,148],[103,8415,8417],{"className":8416},[1084],[103,8418,8420,8440],{"className":8419},[1088,2891],[103,8421,8423,8437],{"className":8422},[1092],[103,8424,8426],{"className":8425,"style":2898},[1096],[103,8427,8428,8431],{"style":2975},[103,8429],{"className":8430,"style":1105},[1104],[103,8432,8434],{"className":8433},[1109,1110,1111,1112],[103,8435,1503],{"className":8436},[147,1112],[103,8438,2915],{"className":8439},[2914],[103,8441,8443],{"className":8442},[1092],[103,8444,8446],{"className":8445,"style":2922},[1096],[103,8447],{},[103,8449],{"className":8450,"style":335},[334],[103,8452,313],{"className":8453},[339],[103,8455],{"className":8456,"style":335},[334],[103,8458,8460,8463,8466,8469,8472],{"className":8459},[138],[103,8461],{"className":8462,"style":7186},[142],[103,8464,7828],{"className":8465},[147],[103,8467],{"className":8468,"style":452},[334],[103,8470,422],{"className":8471},[456],[103,8473],{"className":8474,"style":452},[334],[103,8476,8478,8481,8484,8487,8490],{"className":8477},[138],[103,8479],{"className":8480,"style":5234},[142],[103,8482,7813],{"className":8483},[147],[103,8485],{"className":8486,"style":335},[334],[103,8488,313],{"className":8489},[339],[103,8491],{"className":8492,"style":335},[334],[103,8494,8496,8499,8502],{"className":8495},[138],[103,8497],{"className":8498,"style":7186},[142],[103,8500,422],{"className":8501},[147],[103,8503,7497],{"className":8504},[147],[15,8506,8507],{},"The difference in Northing is −50.862 m.",[15,8509,8510],{},[938,8511,8512,8513,427],{},"Step 2: Calculate the horizontal distance (",[103,8514,8516,8529],{"className":8515},[106],[103,8517,8519],{"className":8518},[110],[112,8520,8521],{"xmlns":114},[116,8522,8523,8527],{},[119,8524,8525],{},[122,8526,2515],{},[126,8528,2515],{"encoding":128},[103,8530,8532],{"className":8531,"ariaHidden":134},[133],[103,8533,8535,8538],{"className":8534},[138],[103,8536],{"className":8537,"style":2092},[142],[103,8539,2515],{"className":8540,"style":149},[147,148],[15,8542,8543],{},[103,8544,8546,8614],{"className":8545},[106],[103,8547,8549],{"className":8548},[110],[112,8550,8551],{"xmlns":114},[116,8552,8553,8611],{},[119,8554,8555,8557,8559,8581,8583],{},[122,8556,2515],{},[311,8558,313],{},[4682,8560,8561],{},[119,8562,8563,8565,8571,8573,8575],{},[122,8564,2298],{"mathvariant":237},[968,8566,8567,8569],{},[122,8568,2142],{},[415,8570,1204],{},[311,8572,684],{},[122,8574,2298],{"mathvariant":237},[968,8576,8577,8579],{},[122,8578,2173],{},[415,8580,1204],{},[311,8582,313],{},[4682,8584,8585],{},[119,8586,8587,8589,8591,8597,8599,8601,8603,8605],{},[311,8588,413],{"stretchy":412},[415,8590,7435],{},[968,8592,8593,8595],{},[311,8594,427],{"stretchy":412},[415,8596,1204],{},[311,8598,684],{},[311,8600,413],{"stretchy":412},[311,8602,422],{},[415,8604,7497],{},[968,8606,8607,8609],{},[311,8608,427],{"stretchy":412},[415,8610,1204],{},[126,8612,8613],{"encoding":128},"D=\\sqrt{\\Delta E^2+\\Delta N^2}=\\sqrt{(66.186)^2+(-50.862)^2}",[103,8615,8617,8635,8767],{"className":8616,"ariaHidden":134},[133],[103,8618,8620,8623,8626,8629,8632],{"className":8619},[138],[103,8621],{"className":8622,"style":2092},[142],[103,8624,2515],{"className":8625,"style":149},[147,148],[103,8627],{"className":8628,"style":335},[334],[103,8630,313],{"className":8631},[339],[103,8633],{"className":8634,"style":335},[334],[103,8636,8638,8641,8758,8761,8764],{"className":8637},[138],[103,8639],{"className":8640,"style":4734},[142],[103,8642,8644],{"className":8643},[147,4738],[103,8645,8647,8750],{"className":8646},[1088,2891],[103,8648,8650,8747],{"className":8649},[1092],[103,8651,8653,8735],{"className":8652,"style":4748},[1096],[103,8654,8656,8659],{"className":8655,"style":4753},[4752],[103,8657],{"className":8658,"style":4757},[1104],[103,8660,8662,8665,8694,8697,8700,8703,8706],{"className":8661,"style":4761},[147],[103,8663,2298],{"className":8664},[147],[103,8666,8668,8671],{"className":8667},[147],[103,8669,2142],{"className":8670,"style":2157},[147,148],[103,8672,8674],{"className":8673},[1084],[103,8675,8677],{"className":8676},[1088],[103,8678,8680],{"className":8679},[1092],[103,8681,8683],{"className":8682,"style":4783},[1096],[103,8684,8685,8688],{"style":4786},[103,8686],{"className":8687,"style":1105},[1104],[103,8689,8691],{"className":8690},[1109,1110,1111,1112],[103,8692,1204],{"className":8693},[147,1112],[103,8695],{"className":8696,"style":452},[334],[103,8698,684],{"className":8699},[456],[103,8701],{"className":8702,"style":452},[334],[103,8704,2298],{"className":8705},[147],[103,8707,8709,8712],{"className":8708},[147],[103,8710,2173],{"className":8711,"style":2188},[147,148],[103,8713,8715],{"className":8714},[1084],[103,8716,8718],{"className":8717},[1088],[103,8719,8721],{"className":8720},[1092],[103,8722,8724],{"className":8723,"style":4783},[1096],[103,8725,8726,8729],{"style":4786},[103,8727],{"className":8728,"style":1105},[1104],[103,8730,8732],{"className":8731},[1109,1110,1111,1112],[103,8733,1204],{"className":8734},[147,1112],[103,8736,8737,8740],{"style":4839},[103,8738],{"className":8739,"style":4757},[1104],[103,8741,8743],{"className":8742,"style":4847},[4846],[4849,8744,8745],{"xmlns":4851,"width":4852,"height":4853,"viewBox":4854,"preserveAspectRatio":4855},[4857,8746],{"d":4859},[103,8748,2915],{"className":8749},[2914],[103,8751,8753],{"className":8752},[1092],[103,8754,8756],{"className":8755,"style":4869},[1096],[103,8757],{},[103,8759],{"className":8760,"style":335},[334],[103,8762,313],{"className":8763},[339],[103,8765],{"className":8766,"style":335},[334],[103,8768,8770,8774],{"className":8769},[138],[103,8771],{"className":8772,"style":8773},[142],"height:1.24em;vertical-align:-0.305em;",[103,8775,8777],{"className":8776},[147,4738],[103,8778,8780,8901],{"className":8779},[1088,2891],[103,8781,8783,8898],{"className":8782},[1092],[103,8784,8787,8881],{"className":8785,"style":8786},[1096],"height:0.935em;",[103,8788,8791,8795],{"className":8789,"style":8790},[4752],"top:-3.2em;",[103,8792],{"className":8793,"style":8794},[1104],"height:3.2em;",[103,8796,8799,8802,8805,8834,8837,8840,8843,8846,8849,8852],{"className":8797,"style":8798},[147],"padding-left:1em;",[103,8800,413],{"className":8801},[444],[103,8803,7435],{"className":8804},[147],[103,8806,8808,8811],{"className":8807},[472],[103,8809,427],{"className":8810},[472],[103,8812,8814],{"className":8813},[1084],[103,8815,8817],{"className":8816},[1088],[103,8818,8820],{"className":8819},[1092],[103,8821,8823],{"className":8822,"style":4783},[1096],[103,8824,8825,8828],{"style":4786},[103,8826],{"className":8827,"style":1105},[1104],[103,8829,8831],{"className":8830},[1109,1110,1111,1112],[103,8832,1204],{"className":8833},[147,1112],[103,8835],{"className":8836,"style":452},[334],[103,8838,684],{"className":8839},[456],[103,8841],{"className":8842,"style":452},[334],[103,8844,413],{"className":8845},[444],[103,8847,422],{"className":8848},[147],[103,8850,7497],{"className":8851},[147],[103,8853,8855,8858],{"className":8854},[472],[103,8856,427],{"className":8857},[472],[103,8859,8861],{"className":8860},[1084],[103,8862,8864],{"className":8863},[1088],[103,8865,8867],{"className":8866},[1092],[103,8868,8870],{"className":8869,"style":4783},[1096],[103,8871,8872,8875],{"style":4786},[103,8873],{"className":8874,"style":1105},[1104],[103,8876,8878],{"className":8877},[1109,1110,1111,1112],[103,8879,1204],{"className":8880},[147,1112],[103,8882,8884,8887],{"style":8883},"top:-2.895em;",[103,8885],{"className":8886,"style":8794},[1104],[103,8888,8891],{"className":8889,"style":8890},[4846],"min-width:1.02em;height:1.28em;",[4849,8892,8895],{"xmlns":4851,"width":4852,"height":8893,"viewBox":8894,"preserveAspectRatio":4855},"1.28em","0 0 400000 1296",[4857,8896],{"d":8897},"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 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to three decimal places yields ",[938,9133,9134],{},"83.472 m",", which matches the original measured distance.",[15,9137,9138],{},[938,9139,9140,9141,427],{},"Step 3: Calculate the whole-circle bearing (",[103,9142,9144,9157],{"className":9143},[106],[103,9145,9147],{"className":9146},[110],[112,9148,9149],{"xmlns":114},[116,9150,9151,9155],{},[119,9152,9153],{},[122,9154,124],{},[126,9156,129],{"encoding":128},[103,9158,9160],{"className":9159,"ariaHidden":134},[133],[103,9161,9163,9166],{"className":9162},[138],[103,9164],{"className":9165,"style":143},[142],[103,9167,124],{"className":9168,"style":149},[147,148],[15,9170,9171,9172,9174],{},"Using the programmatic ",[2197,9173,5221],{}," function, substitute the coordinate differences:",[15,9176,9177],{},[103,9178,9180,9219],{"className":9179},[106],[103,9181,9183],{"className":9182},[110],[112,9184,9185],{"xmlns":114},[116,9186,9187,9216],{},[119,9188,9189,9195,9197,9199,9201,9203,9205,9207,9209,9211,9213],{},[2860,9190,9191,9193],{},[122,9192,124],{},[957,9194,5262],{},[311,9196,313],{},[957,9198,5221],{},[311,9200,413],{"stretchy":412},[415,9202,7435],{},[311,9204,5275],{"separator":134},[311,9206,422],{},[415,9208,7497],{},[311,9210,427],{"stretchy":412},[311,9212,6772],{},[415,9214,9215],{},"2.226011974",[126,9217,9218],{"encoding":128},"\\theta_{\\text{rad}}=\\text{atan2}(66.186,-50.862)\\approx2.226011974",[103,9220,9222,9283,9325],{"className":9221,"ariaHidden":134},[133],[103,9223,9225,9228,9274,9277,9280],{"className":9224},[138],[103,9226],{"className":9227,"style":5294},[142],[103,9229,9231,9234],{"className":9230},[147],[103,9232,124],{"className":9233,"style":149},[147,148],[103,9235,9237],{"className":9236},[1084],[103,9238,9240,9266],{"className":9239},[1088,2891],[103,9241,9243,9263],{"className":9242},[1092],[103,9244,9246],{"className":9245,"style":5313},[1096],[103,9247,9248,9251],{"style":5316},[103,9249],{"className":9250,"style":1105},[1104],[103,9252,9254],{"className":9253},[1109,1110,1111,1112],[103,9255,9257],{"className":9256},[147,1112],[103,9258,9260],{"className":9259},[147,1041,1112],[103,9261,5262],{"className":9262},[147,1112],[103,9264,2915],{"className":9265},[2914],[103,9267,9269],{"className":9268},[1092],[103,9270,9272],{"className":9271,"style":2922},[1096],[103,9273],{},[103,9275],{"className":9276,"style":335},[334],[103,9278,313],{"className":9279},[339],[103,9281],{"className":9282,"style":335},[334],[103,9284,9286,9289,9295,9298,9301,9304,9307,9310,9313,9316,9319,9322],{"className":9285},[138],[103,9287],{"className":9288,"style":440},[142],[103,9290,9292],{"className":9291},[147,1041],[103,9293,5221],{"className":9294},[147],[103,9296,413],{"className":9297},[444],[103,9299,7435],{"className":9300},[147],[103,9302,5275],{"className":9303},[5376],[103,9305],{"className":9306,"style":2629},[334],[103,9308,422],{"className":9309},[147],[103,9311,7497],{"className":9312},[147],[103,9314,427],{"className":9315},[472],[103,9317],{"className":9318,"style":335},[334],[103,9320,6772],{"className":9321},[339],[103,9323],{"className":9324,"style":335},[334],[103,9326,9328,9331],{"className":9327},[138],[103,9329],{"className":9330,"style":5234},[142],[103,9332,9215],{"className":9333},[147],[15,9335,9336],{},"Convert the radian value to decimal degrees:",[15,9338,9339],{},[103,9340,9342,9387],{"className":9341},[106],[103,9343,9345],{"className":9344},[110],[112,9346,9347],{"xmlns":114},[116,9348,9349,9384],{},[119,9350,9351,9357,9359,9361,9363,9375,9377],{},[2860,9352,9353,9355],{},[122,9354,124],{},[957,9356,5524],{},[311,9358,313],{},[415,9360,9215],{},[311,9362,5717],{},[119,9364,9365,9367,9373],{},[311,9366,413],{"fence":134},[4934,9368,9369,9371],{},[415,9370,417],{},[122,9372,5440],{},[311,9374,427],{"fence":134},[311,9376,6772],{},[968,9378,9379,9382],{},[415,9380,9381],{},"127.539818",[311,9383,974],{},[126,9385,9386],{"encoding":128},"\\theta_{\\text{deg}}=2.226011974\\times\\left(\\frac{180}{\\pi}\\right)\\approx127.539818^\\circ",[103,9388,9390,9451,9469,9567],{"className":9389,"ariaHidden":134},[133],[103,9391,9393,9396,9442,9445,9448],{"className":9392},[138],[103,9394],{"className":9395,"style":5591},[142],[103,9397,9399,9402],{"className":9398},[147],[103,9400,124],{"className":9401,"style":149},[147,148],[103,9403,9405],{"className":9404},[1084],[103,9406,9408,9434],{"className":9407},[1088,2891],[103,9409,9411,9431],{"className":9410},[1092],[103,9412,9414],{"className":9413,"style":5313},[1096],[103,9415,9416,9419],{"style":5316},[103,9417],{"className":9418,"style":1105},[1104],[103,9420,9422],{"className":9421},[1109,1110,1111,1112],[103,9423,9425],{"className":9424},[147,1112],[103,9426,9428],{"className":9427},[147,1041,1112],[103,9429,5524],{"className":9430},[147,1112],[103,9432,2915],{"className":9433},[2914],[103,9435,9437],{"className":9436},[1092],[103,9438,9440],{"className":9439,"style":5636},[1096],[103,9441],{},[103,9443],{"className":9444,"style":335},[334],[103,9446,313],{"className":9447},[339],[103,9449],{"className":9450,"style":335},[334],[103,9452,9454,9457,9460,9463,9466],{"className":9453},[138],[103,9455],{"className":9456,"style":7186},[142],[103,9458,9215],{"className":9459},[147],[103,9461],{"className":9462,"style":452},[334],[103,9464,5717],{"className":9465},[456],[103,9467],{"className":9468,"style":452},[334],[103,9470,9472,9475,9558,9561,9564],{"className":9471},[138],[103,9473],{"className":9474,"style":5864},[142],[103,9476,9478,9484,9552],{"className":9477},[4999],[103,9479,9481],{"className":9480,"style":5004},[444,5003],[103,9482,413],{"className":9483},[5008,5874],[103,9485,9487,9490,9549],{"className":9486},[147],[103,9488],{"className":9489},[444,5016],[103,9491,9493],{"className":9492},[4934],[103,9494,9496,9541],{"className":9495},[1088,2891],[103,9497,9499,9538],{"className":9498},[1092],[103,9500,9502,9516,9524],{"className":9501,"style":5893},[1096],[103,9503,9504,9507],{"style":5032},[103,9505],{"className":9506,"style":4757},[1104],[103,9508,9510],{"className":9509},[1109,1110,1111,1112],[103,9511,9513],{"className":9512},[147,1112],[103,9514,5440],{"className":9515,"style":5460},[147,148,1112],[103,9517,9518,9521],{"style":5054},[103,9519],{"className":9520,"style":4757},[1104],[103,9522],{"className":9523,"style":5062},[5061],[103,9525,9526,9529],{"style":5918},[103,9527],{"className":9528,"style":4757},[1104],[103,9530,9532],{"className":9531},[1109,1110,1111,1112],[103,9533,9535],{"className":9534},[147,1112],[103,9536,417],{"className":9537},[147,1112],[103,9539,2915],{"className":9540},[2914],[103,9542,9544],{"className":9543},[1092],[103,9545,9547],{"className":9546,"style":5940},[1096],[103,9548],{},[103,9550],{"className":9551},[472,5016],[103,9553,9555],{"className":9554,"style":5004},[472,5003],[103,9556,427],{"className":9557},[5008,5874],[103,9559],{"className":9560,"style":335},[334],[103,9562,6772],{"className":9563},[339],[103,9565],{"className":9566,"style":335},[334],[103,9568,9570,9573,9577],{"className":9569},[138],[103,9571],{"className":9572,"style":1097},[142],[103,9574,9576],{"className":9575},[147],"127.53981",[103,9578,9580,9583],{"className":9579},[147],[103,9581,1946],{"className":9582},[147],[103,9584,9586],{"className":9585},[1084],[103,9587,9589],{"className":9588},[1088],[103,9590,9592],{"className":9591},[1092],[103,9593,9595],{"className":9594,"style":1097},[1096],[103,9596,9597,9600],{"style":1100},[103,9598],{"className":9599,"style":1105},[1104],[103,9601,9603],{"className":9602},[1109,1110,1111,1112],[103,9604,974],{"className":9605},[456,1112],[15,9607,9608,9609,9612],{},"Rounding the azimuth to four decimal places yields ",[938,9610,9611],{},"127.5401°"," (or 127°32′24″ in DMS format).",[15,9614,9615],{},[938,9616,9617],{},"Step 4: Analyze the rounding effect",[15,9619,9620],{},"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).",[15,9622,9623],{},"A coordinate rounding shift of just 1 mm over a distance of 83.472 m introduces an angular variation of:",[15,9625,9626],{},[103,9627,9629,9675],{"className":9628},[106],[103,9630,9632],{"className":9631},[110],[112,9633,9634],{"xmlns":114},[116,9635,9636,9672],{},[119,9637,9638,9641,9643,9645,9648,9650,9663,9665],{},[122,9639,9640],{},"δ",[122,9642,124],{},[311,9644,313],{},[122,9646,9647],{},"arcsin",[311,9649,2587],{},[119,9651,9652,9654,9661],{},[311,9653,413],{"fence":134},[4934,9655,9656,9659],{},[415,9657,9658],{},"0.001",[415,9660,7105],{},[311,9662,427],{"fence":134},[311,9664,6772],{},[968,9666,9667,9670],{},[415,9668,9669],{},"0.000687",[311,9671,974],{},[126,9673,9674],{"encoding":128},"\\delta\\theta=\\arcsin\\left(\\frac{0.001}{83.472}\\right)\\approx0.000687^\\circ",[103,9676,9678,9700,9804],{"className":9677,"ariaHidden":134},[133],[103,9679,9681,9684,9688,9691,9694,9697],{"className":9680},[138],[103,9682],{"className":9683,"style":143},[142],[103,9685,9640],{"className":9686,"style":9687},[147,148],"margin-right:0.0379em;",[103,9689,124],{"className":9690,"style":149},[147,148],[103,9692],{"className":9693,"style":335},[334],[103,9695,313],{"className":9696},[339],[103,9698],{"className":9699,"style":335},[334],[103,9701,9703,9706,9709,9712,9795,9798,9801],{"className":9702},[138],[103,9704],{"className":9705,"style":5864},[142],[103,9707,9647],{"className":9708},[2633],[103,9710],{"className":9711,"style":2629},[334],[103,9713,9715,9721,9789],{"className":9714},[4999],[103,9716,9718],{"className":9717,"style":5004},[444,5003],[103,9719,413],{"className":9720},[5008,5874],[103,9722,9724,9727,9786],{"className":9723},[147],[103,9725],{"className":9726},[444,5016],[103,9728,9730],{"className":9729},[4934],[103,9731,9733,9778],{"className":9732},[1088,2891],[103,9734,9736,9775],{"className":9735},[1092],[103,9737,9739,9753,9761],{"className":9738,"style":5893},[1096],[103,9740,9741,9744],{"style":5032},[103,9742],{"className":9743,"style":4757},[1104],[103,9745,9747],{"className":9746},[1109,1110,1111,1112],[103,9748,9750],{"className":9749},[147,1112],[103,9751,7105],{"className":9752},[147,1112],[103,9754,9755,9758],{"style":5054},[103,9756],{"className":9757,"style":4757},[1104],[103,9759],{"className":9760,"style":5062},[5061],[103,9762,9763,9766],{"style":5918},[103,9764],{"className":9765,"style":4757},[1104],[103,9767,9769],{"className":9768},[1109,1110,1111,1112],[103,9770,9772],{"className":9771},[147,1112],[103,9773,9658],{"className":9774},[147,1112],[103,9776,2915],{"className":9777},[2914],[103,9779,9781],{"className":9780},[1092],[103,9782,9784],{"className":9783,"style":5940},[1096],[103,9785],{},[103,9787],{"className":9788},[472,5016],[103,9790,9792],{"className":9791,"style":5004},[472,5003],[103,9793,427],{"className":9794},[5008,5874],[103,9796],{"className":9797,"style":335},[334],[103,9799,6772],{"className":9800},[339],[103,9802],{"className":9803,"style":335},[334],[103,9805,9807,9810,9814],{"className":9806},[138],[103,9808],{"className":9809,"style":1097},[142],[103,9811,9813],{"className":9812},[147],"0.00068",[103,9815,9817,9821],{"className":9816},[147],[103,9818,9820],{"className":9819},[147],"7",[103,9822,9824],{"className":9823},[1084],[103,9825,9827],{"className":9826},[1088],[103,9828,9830],{"className":9829},[1092],[103,9831,9833],{"className":9832,"style":1097},[1096],[103,9834,9835,9838],{"style":1100},[103,9836],{"className":9837,"style":1105},[1104],[103,9839,9841],{"className":9840},[1109,1110,1111,1112],[103,9842,974],{"className":9843},[456,1112],[15,9845,9846],{},"This variation is approximately equal to 0.000687 degrees (or 2.5 seconds).",[15,9848,9849],{},"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.",[10,9851,9853],{"id":9852},"common-computational-errors-in-coordinate-geometry","Common Computational Errors in Coordinate Geometry",[15,9855,9856],{},"When setting up custom spreadsheets, writing automation scripts, or performing manual calculations, civil engineering technicians and surveyors frequently encounter standard computational errors.",[35,9858,9860],{"id":9859},"swapping-trigonometric-functions-eastingnorthing-vs-xy","Swapping trigonometric functions (Easting\u002FNorthing vs. X\u002FY)",[15,9862,9863,9864,9892,9893,9967,9968,2809,10042,10116],{},"Because standard mathematics defines the horizontal axis as ",[103,9865,9867,9880],{"className":9866},[106],[103,9868,9870],{"className":9869},[110],[112,9871,9872],{"xmlns":114},[116,9873,9874,9878],{},[119,9875,9876],{},[122,9877,2080],{},[126,9879,2080],{"encoding":128},[103,9881,9883],{"className":9882,"ariaHidden":134},[133],[103,9884,9886,9889],{"className":9885},[138],[103,9887],{"className":9888,"style":2092},[142],[103,9890,2080],{"className":9891,"style":2096},[147,148]," and uses the cosine function for horizontal projections, beginners often compute ",[103,9894,9896,9922],{"className":9895},[106],[103,9897,9899],{"className":9898},[110],[112,9900,9901],{"xmlns":114},[116,9902,9903,9919],{},[119,9904,9905,9907,9909,9911,9913,9915,9917],{},[122,9906,2298],{"mathvariant":237},[122,9908,2142],{},[311,9910,313],{},[122,9912,2515],{},[122,9914,2662],{},[311,9916,2587],{},[122,9918,124],{},[126,9920,9921],{"encoding":128},"\\Delta E = D \\cos\\theta",[103,9923,9925,9946],{"className":9924,"ariaHidden":134},[133],[103,9926,9928,9931,9934,9937,9940,9943],{"className":9927},[138],[103,9929],{"className":9930,"style":2092},[142],[103,9932,2298],{"className":9933},[147],[103,9935,2142],{"className":9936,"style":2157},[147,148],[103,9938],{"className":9939,"style":335},[334],[103,9941,313],{"className":9942},[339],[103,9944],{"className":9945,"style":335},[334],[103,9947,9949,9952,9955,9958,9961,9964],{"className":9948},[138],[103,9950],{"className":9951,"style":143},[142],[103,9953,2515],{"className":9954,"style":149},[147,148],[103,9956],{"className":9957,"style":2629},[334],[103,9959,2662],{"className":9960},[2633],[103,9962],{"className":9963,"style":2629},[334],[103,9965,124],{"className":9966,"style":149},[147,148],". In plane surveying, because the angle is measured from the vertical axis (North), the trigonometric functions are reversed: ",[103,9969,9971,9997],{"className":9970},[106],[103,9972,9974],{"className":9973},[110],[112,9975,9976],{"xmlns":114},[116,9977,9978,9994],{},[119,9979,9980,9982,9984,9986,9988,9990,9992],{},[122,9981,2298],{"mathvariant":237},[122,9983,2142],{},[311,9985,313],{},[122,9987,2515],{},[122,9989,2584],{},[311,9991,2587],{},[122,9993,124],{},[126,9995,9996],{"encoding":128},"\\Delta E = D \\sin\\theta",[103,9998,10000,10021],{"className":9999,"ariaHidden":134},[133],[103,10001,10003,10006,10009,10012,10015,10018],{"className":10002},[138],[103,10004],{"className":10005,"style":2092},[142],[103,10007,2298],{"className":10008},[147],[103,10010,2142],{"className":10011,"style":2157},[147,148],[103,10013],{"className":10014,"style":335},[334],[103,10016,313],{"className":10017},[339],[103,10019],{"className":10020,"style":335},[334],[103,10022,10024,10027,10030,10033,10036,10039],{"className":10023},[138],[103,10025],{"className":10026,"style":143},[142],[103,10028,2515],{"className":10029,"style":149},[147,148],[103,10031],{"className":10032,"style":2629},[334],[103,10034,2584],{"className":10035},[2633],[103,10037],{"className":10038,"style":2629},[334],[103,10040,124],{"className":10041,"style":149},[147,148],[103,10043,10045,10071],{"className":10044},[106],[103,10046,10048],{"className":10047},[110],[112,10049,10050],{"xmlns":114},[116,10051,10052,10068],{},[119,10053,10054,10056,10058,10060,10062,10064,10066],{},[122,10055,2298],{"mathvariant":237},[122,10057,2173],{},[311,10059,313],{},[122,10061,2515],{},[122,10063,2662],{},[311,10065,2587],{},[122,10067,124],{},[126,10069,10070],{"encoding":128},"\\Delta N = D \\cos\\theta",[103,10072,10074,10095],{"className":10073,"ariaHidden":134},[133],[103,10075,10077,10080,10083,10086,10089,10092],{"className":10076},[138],[103,10078],{"className":10079,"style":2092},[142],[103,10081,2298],{"className":10082},[147],[103,10084,2173],{"className":10085,"style":2188},[147,148],[103,10087],{"className":10088,"style":335},[334],[103,10090,313],{"className":10091},[339],[103,10093],{"className":10094,"style":335},[334],[103,10096,10098,10101,10104,10107,10110,10113],{"className":10097},[138],[103,10099],{"className":10100,"style":143},[142],[103,10102,2515],{"className":10103,"style":149},[147,148],[103,10105],{"className":10106,"style":2629},[334],[103,10108,2662],{"className":10109},[2633],[103,10111],{"className":10112,"style":2629},[334],[103,10114,124],{"className":10115,"style":149},[147,148],". Swapping these functions mirrors the calculated coordinates across the 45° diagonal of the quadrant, resulting in incorrect point locations.",[35,10118,10120],{"id":10119},"radian-vs-degree-discrepancies","Radian vs. degree discrepancies",[15,10122,10123,10124,10361],{},"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 (",[103,10125,10127,10161],{"className":10126},[106],[103,10128,10130],{"className":10129},[110],[112,10131,10132],{"xmlns":114},[116,10133,10134,10158],{},[119,10135,10136,10142,10144,10150,10152],{},[2860,10137,10138,10140],{},[122,10139,124],{},[957,10141,5262],{},[311,10143,313],{},[2860,10145,10146,10148],{},[122,10147,124],{},[957,10149,5524],{},[311,10151,5717],{},[4934,10153,10154,10156],{},[122,10155,5440],{},[415,10157,417],{},[126,10159,10160],{"encoding":128},"\\theta_{\\text{rad}} = \\theta_{\\text{deg}} \\times \\frac{\\pi}{180}",[103,10162,10164,10225,10286],{"className":10163,"ariaHidden":134},[133],[103,10165,10167,10170,10216,10219,10222],{"className":10166},[138],[103,10168],{"className":10169,"style":5294},[142],[103,10171,10173,10176],{"className":10172},[147],[103,10174,124],{"className":10175,"style":149},[147,148],[103,10177,10179],{"className":10178},[1084],[103,10180,10182,10208],{"className":10181},[1088,2891],[103,10183,10185,10205],{"className":10184},[1092],[103,10186,10188],{"className":10187,"style":5313},[1096],[103,10189,10190,10193],{"style":5316},[103,10191],{"className":10192,"style":1105},[1104],[103,10194,10196],{"className":10195},[1109,1110,1111,1112],[103,10197,10199],{"className":10198},[147,1112],[103,10200,10202],{"className":10201},[147,1041,1112],[103,10203,5262],{"className":10204},[147,1112],[103,10206,2915],{"className":10207},[2914],[103,10209,10211],{"className":10210},[1092],[103,10212,10214],{"className":10213,"style":2922},[1096],[103,10215],{},[103,10217],{"className":10218,"style":335},[334],[103,10220,313],{"className":10221},[339],[103,10223],{"className":10224,"style":335},[334],[103,10226,10228,10231,10277,10280,10283],{"className":10227},[138],[103,10229],{"className":10230,"style":5591},[142],[103,10232,10234,10237],{"className":10233},[147],[103,10235,124],{"className":10236,"style":149},[147,148],[103,10238,10240],{"className":10239},[1084],[103,10241,10243,10269],{"className":10242},[1088,2891],[103,10244,10246,10266],{"className":10245},[1092],[103,10247,10249],{"className":10248,"style":5313},[1096],[103,10250,10251,10254],{"style":5316},[103,10252],{"className":10253,"style":1105},[1104],[103,10255,10257],{"className":10256},[1109,1110,1111,1112],[103,10258,10260],{"className":10259},[147,1112],[103,10261,10263],{"className":10262},[147,1041,1112],[103,10264,5524],{"className":10265},[147,1112],[103,10267,2915],{"className":10268},[2914],[103,10270,10272],{"className":10271},[1092],[103,10273,10275],{"className":10274,"style":5636},[1096],[103,10276],{},[103,10278],{"className":10279,"style":452},[334],[103,10281,5717],{"className":10282},[456],[103,10284],{"className":10285,"style":452},[334],[103,10287,10289,10293],{"className":10288},[138],[103,10290],{"className":10291,"style":10292},[142],"height:1.0404em;vertical-align:-0.345em;",[103,10294,10296,10299,10358],{"className":10295},[147],[103,10297],{"className":10298},[444,5016],[103,10300,10302],{"className":10301},[4934],[103,10303,10305,10350],{"className":10304},[1088,2891],[103,10306,10308,10347],{"className":10307},[1092],[103,10309,10311,10325,10333],{"className":10310,"style":6924},[1096],[103,10312,10313,10316],{"style":5032},[103,10314],{"className":10315,"style":4757},[1104],[103,10317,10319],{"className":10318},[1109,1110,1111,1112],[103,10320,10322],{"className":10321},[147,1112],[103,10323,417],{"className":10324},[147,1112],[103,10326,10327,10330],{"style":5054},[103,10328],{"className":10329,"style":4757},[1104],[103,10331],{"className":10332,"style":5062},[5061],[103,10334,10335,10338],{"style":5918},[103,10336],{"className":10337,"style":4757},[1104],[103,10339,10341],{"className":10340},[1109,1110,1111,1112],[103,10342,10344],{"className":10343},[147,1112],[103,10345,5440],{"className":10346,"style":5460},[147,148,1112],[103,10348,2915],{"className":10349},[2914],[103,10351,10353],{"className":10352},[1092],[103,10354,10356],{"className":10355,"style":5940},[1096],[103,10357],{},[103,10359],{"className":10360},[472,5016],") yields incorrect coordinate deltas.",[35,10363,10365],{"id":10364},"parameter-order-in-dual-argument-arctangent-functions","Parameter order in dual-argument arctangent functions",[15,10367,10368,10369,10371],{},"The ",[2197,10370,5221],{}," function argument order varies across environments. For grid azimuth from North (clockwise), Surveying Core uses the same convention as JavaScript and Python:",[182,10373,10374,10387],{},[185,10375,10376],{},[188,10377,10378,10381,10384],{},[191,10379,10380],{},"Environment",[191,10382,10383],{},"Function syntax",[191,10385,10386],{},"Surveying mapping",[207,10388,10389,10404,10480],{},[188,10390,10391,10394,10399],{},[212,10392,10393],{},"JavaScript, Python, Surveying Core",[212,10395,10396],{},[2197,10397,10398],{},"atan2(y, x)",[212,10400,10401],{},[2197,10402,10403],{},"atan2(ΔE, ΔN)",[188,10405,10406,10409,10414],{},[212,10407,10408],{},"Spreadsheets (Excel, Google Sheets)",[212,10410,10411],{},[2197,10412,10413],{},"ATAN2(x, y)",[212,10415,10416,10419,10420,10449,10450],{},[2197,10417,10418],{},"ATAN2(ΔE, ΔN)"," when Easting = ",[103,10421,10423,10437],{"className":10422},[106],[103,10424,10426],{"className":10425},[110],[112,10427,10428],{"xmlns":114},[116,10429,10430,10435],{},[119,10431,10432],{},[122,10433,10434],{},"x",[126,10436,10434],{"encoding":128},[103,10438,10440],{"className":10439,"ariaHidden":134},[133],[103,10441,10443,10446],{"className":10442},[138],[103,10444],{"className":10445,"style":176},[142],[103,10447,10434],{"className":10448},[147,148]," and Northing = ",[103,10451,10453,10467],{"className":10452},[106],[103,10454,10456],{"className":10455},[110],[112,10457,10458],{"xmlns":114},[116,10459,10460,10465],{},[119,10461,10462],{},[122,10463,10464],{},"y",[126,10466,10464],{"encoding":128},[103,10468,10470],{"className":10469,"ariaHidden":134},[133],[103,10471,10473,10477],{"className":10472},[138],[103,10474],{"className":10475,"style":10476},[142],"height:0.625em;vertical-align:-0.1944em;",[103,10478,10464],{"className":10479,"style":5460},[147,148],[188,10481,10482,10485,10490],{},[212,10483,10484],{},"Standard calculators",[212,10486,10487],{},[2197,10488,10489],{},"atan(y\u002Fx)",[212,10491,10492],{},"`atan(",[15,10494,10495],{},"Always confirm argument order in your spreadsheet or language documentation before inverting coordinates.",[10,10497,10499],{"id":10498},"field-practices-datums-and-coordinate-reference-systems","Field Practices, Datums, and Coordinate Reference Systems",[15,10501,10502],{},"Applying coordinate geometry in the field requires translating theoretical formulas onto the physical, curved surface of the Earth.",[35,10504,10506],{"id":10505},"grid-azimuth-vs-magnetic-azimuth","Grid Azimuth vs. Magnetic Azimuth",[15,10508,10509],{},"A critical field distinction is the difference between grid azimuths and magnetic azimuths:",[71,10511,10512,10518],{},[74,10513,10514,10517],{},[938,10515,10516],{},"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.",[74,10519,10520,10523],{},[938,10521,10522],{},"Grid azimuth",": Tied to a flat map projection (such as State Plane or UTM) where Grid North is parallel to the central meridian.",[15,10525,10526],{},"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.",[35,10528,10530],{"id":10529},"coordinate-reference-systems-crs-and-scale-factors","Coordinate Reference Systems (CRS) and Scale Factors",[15,10532,10533],{},"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.",[15,10535,10536],{},"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.",[10,10538,10540],{"id":10539},"next-steps","Next Steps",[15,10542,10543],{},"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.",[15,10545,10546,10547,10549],{},"To quickly verify field measurements, compute coordinates, or check manual traverse calculations, use the ",[22,10548,25],{"href":24},". This dedicated tool automates forward and inverse coordinate calculations, handles degree-to-radian conversions, and ensures millimeter-level accuracy for project quality control.",{"title":2199,"searchDepth":1344,"depth":1344,"links":10551},[10552,10553,10558,10563,10564,10565,10566,10570,10575,10579],{"id":12,"depth":1344,"text":13},{"id":29,"depth":1344,"text":30,"children":10554},[10555,10556,10557],{"id":37,"depth":1724,"text":38},{"id":44,"depth":1724,"text":45},{"id":51,"depth":1724,"text":52},{"id":58,"depth":1344,"text":59,"children":10559},[10560,10561,10562],{"id":65,"depth":1724,"text":66},{"id":94,"depth":1724,"text":95},{"id":927,"depth":1724,"text":928},{"id":2058,"depth":1344,"text":2059},{"id":2429,"depth":1344,"text":2430},{"id":3966,"depth":1344,"text":3967},{"id":6353,"depth":1344,"text":6354,"children":10567},[10568,10569],{"id":6357,"depth":1724,"text":6358},{"id":8031,"depth":1724,"text":8032},{"id":9852,"depth":1344,"text":9853,"children":10571},[10572,10573,10574],{"id":9859,"depth":1724,"text":9860},{"id":10119,"depth":1724,"text":10120},{"id":10364,"depth":1724,"text":10365},{"id":10498,"depth":1344,"text":10499,"children":10576},[10577,10578],{"id":10505,"depth":1724,"text":10506},{"id":10529,"depth":1724,"text":10530},{"id":10539,"depth":1344,"text":10540},"tools","Coordinate geometry for plane surveying — azimuth and quadrant bearings, polar ↔ rectangular conversion, worked examples, and common errors.","md",{},true,1,"\u002Flearn\u002Ftools\u002Fbearing-distance",{"title":5,"description":10581},{"loc":10586},"learn\u002Ftools\u002Fbearing-distance","ox0hasMcQmIJtaGlJFXk361ZthJlirgnSUGw1AAdmBE",[10592,10593],null,{"title":10594,"path":10595,"stem":10596,"description":10597,"category":10580,"children":-1},"Traverse Closure & Bowditch Adjustment","\u002Flearn\u002Ftools\u002Ftraverse-and-bowditch","learn\u002Ftools\u002Ftraverse-and-bowditch","Closed traverses — departures and latitudes, linear misclosure, relative precision, Bowditch compass-rule adjustment, and a full worked example.",1783851946336]