[{"data":1,"prerenderedAt":23626},["ShallowReactive",2],{"blog-posts":3},{"posts":4,"total":23625},[5,5847,6423,7129,9847,10450,15746,18768,20313],{"id":6,"title":7,"author":8,"body":9,"category":5807,"description":5808,"extension":5809,"image":5810,"meta":5811,"navigation":5839,"path":5840,"publishedAt":5841,"readTime":5842,"seo":5843,"sitemap":5844,"stem":5845,"__hash__":5846},"blog\u002Fblog\u002F23.tienstra-resection-danger-circle.md","Tienstra Resection: Understanding the Danger Circle and When to Reposition","Gordon McInnes",{"type":10,"value":11,"toc":5780},"minimark",[12,17,152,156,170,174,192,196,342,346,521,1737,1770,1925,2585,2822,2826,3280,3284,3762,4911,4915,5027,5298,5302,5566,5570,5650,5654,5691,5695,5754,5758,5761,5765],[13,14,16],"h2",{"id":15},"overview","Overview",[18,19,20,24,25,72,73,139,140,143,144,147,148,151],"p",{},[21,22,23],"strong",{},"Resection"," is the process of determining the coordinates of an unknown observer station (",[26,27,30,52],"span",{"className":28},[29],"katex",[26,31,34],{"className":32},[33],"katex-mathml",[35,36,38],"math",{"xmlns":37},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[39,40,41,48],"semantics",{},[42,43,44],"mrow",{},[45,46,47],"mi",{},"P",[49,50,47],"annotation",{"encoding":51},"application\u002Fx-tex",[26,53,57],{"className":54,"ariaHidden":56},[55],"katex-html","true",[26,58,61,66],{"className":59},[60],"base",[26,62],{"className":63,"style":65},[64],"strut","height:0.6833em;",[26,67,47],{"className":68,"style":71},[69,70],"mord","mathnormal","margin-right:0.1389em;",") by measuring the horizontal angles subtended by lines of sight to three known control points (",[26,74,76,103],{"className":75},[29],[26,77,79],{"className":78},[33],[35,80,81],{"xmlns":37},[39,82,83,100],{},[42,84,85,88,92,95,97],{},[45,86,87],{},"A",[89,90,91],"mo",{"separator":56},",",[45,93,94],{},"B",[89,96,91],{"separator":56},[45,98,99],{},"C",[49,101,102],{"encoding":51},"A, B, C",[26,104,106],{"className":105,"ariaHidden":56},[55],[26,107,109,113,116,120,125,129,132,135],{"className":108},[60],[26,110],{"className":111,"style":112},[64],"height:0.8778em;vertical-align:-0.1944em;",[26,114,87],{"className":115},[69,70],[26,117,91],{"className":118},[119],"mpunct",[26,121],{"className":122,"style":124},[123],"mspace","margin-right:0.1667em;",[26,126,94],{"className":127,"style":128},[69,70],"margin-right:0.0502em;",[26,130,91],{"className":131},[119],[26,133],{"className":134,"style":124},[123],[26,136,99],{"className":137,"style":138},[69,70],"margin-right:0.0715em;",") ",[26,141,142],{},"1, 2",". Among the hundreds of historical procedures to solve this \"three-point problem,\" the ",[21,145,146],{},"Tienstra method"," is widely recognized by practitioners as the most compact and elegant analytical solution ",[26,149,150],{},"1, 3",".",[13,153,155],{"id":154},"why-this-matters","Why This Matters",[18,157,158,159,162,163,166,167,151],{},"Unlike traditional resection methods that require solving intermediate bearings, distances, or triangle lengths, the Tienstra formula provides coordinates directly using a center-of-gravity type calculation ",[26,160,161],{},"1, 4",". However, it is governed by a critical geometric constraint: the ",[21,164,165],{},"Danger Circle",". If your station falls on the circumcircle of the three control points, the math fails, and your position becomes indeterminate ",[26,168,169],{},"2, 5",[13,171,173],{"id":172},"background","Background",[18,175,176,177,180,181,184,185,188,189,151],{},"The formula is named after ",[21,178,179],{},"J.M. Tienstra"," (1895–1951), a professor at the Delft University of Technology who popularized the use of ",[21,182,183],{},"barycentric coordinates"," for resection ",[26,186,187],{},"3, 6",". While the mathematical roots of the problem date back to Willebrord Snellius in 1615 and later Laurent Pothenot, Tienstra's contribution was a simplified, direct weighting system that is ideally suited for modern field computation ",[26,190,191],{},"1, 6",[13,193,195],{"id":194},"theory","Theory",[18,197,198,199,227,228,275,276,304,305,308,309,338,339,151],{},"The Tienstra method treats the unknown point ",[26,200,202,215],{"className":201},[29],[26,203,205],{"className":204},[33],[35,206,207],{"xmlns":37},[39,208,209,213],{},[42,210,211],{},[45,212,47],{},[49,214,47],{"encoding":51},[26,216,218],{"className":217,"ariaHidden":56},[55],[26,219,221,224],{"className":220},[60],[26,222],{"className":223,"style":65},[64],[26,225,47],{"className":226,"style":71},[69,70]," as the \"weighted mean\" of the three control points ",[26,229,231,251],{"className":230},[29],[26,232,234],{"className":233},[33],[35,235,236],{"xmlns":37},[39,237,238,248],{},[42,239,240,242,244,246],{},[45,241,87],{},[89,243,91],{"separator":56},[45,245,94],{},[89,247,91],{"separator":56},[49,249,250],{"encoding":51},"A, B,",[26,252,254],{"className":253,"ariaHidden":56},[55],[26,255,257,260,263,266,269,272],{"className":256},[60],[26,258],{"className":259,"style":112},[64],[26,261,87],{"className":262},[69,70],[26,264,91],{"className":265},[119],[26,267],{"className":268,"style":124},[123],[26,270,94],{"className":271,"style":128},[69,70],[26,273,91],{"className":274},[119]," and ",[26,277,279,292],{"className":278},[29],[26,280,282],{"className":281},[33],[35,283,284],{"xmlns":37},[39,285,286,290],{},[42,287,288],{},[45,289,99],{},[49,291,99],{"encoding":51},[26,293,295],{"className":294,"ariaHidden":56},[55],[26,296,298,301],{"className":297},[60],[26,299],{"className":300,"style":65},[64],[26,302,99],{"className":303,"style":138},[69,70]," ",[26,306,307],{},"6",". By assigning a specific mathematical weight (",[26,310,312,326],{"className":311},[29],[26,313,315],{"className":314},[33],[35,316,317],{"xmlns":37},[39,318,319,324],{},[42,320,321],{},[45,322,323],{},"K",[49,325,323],{"encoding":51},[26,327,329],{"className":328,"ariaHidden":56},[55],[26,330,332,335],{"className":331},[60],[26,333],{"className":334,"style":65},[64],[26,336,323],{"className":337,"style":138},[69,70],") to each control point based on the geometry of the setup, the final coordinates are derived without the need for iterative trials or complex trig sequences ",[26,340,341],{},"1",[13,343,345],{"id":344},"mathematical-principles","Mathematical Principles",[18,347,348,349,516,517,520],{},"The coordinates of the unknown point ",[26,350,352,390],{"className":351},[29],[26,353,355],{"className":354},[33],[35,356,357],{"xmlns":37},[39,358,359,387],{},[42,360,361,363,367,375,377,384],{},[45,362,47],{},[89,364,366],{"stretchy":365},"false","(",[368,369,370,373],"msub",{},[45,371,372],{},"E",[45,374,18],{},[89,376,91],{"separator":56},[368,378,379,382],{},[45,380,381],{},"N",[45,383,18],{},[89,385,386],{"stretchy":365},")",[49,388,389],{"encoding":51},"P(E_p, N_p)",[26,391,393],{"className":392,"ariaHidden":56},[55],[26,394,396,400,403,407,464,467,470,512],{"className":395},[60],[26,397],{"className":398,"style":399},[64],"height:1.0361em;vertical-align:-0.2861em;",[26,401,47],{"className":402,"style":71},[69,70],[26,404,366],{"className":405},[406],"mopen",[26,408,410,414],{"className":409},[69],[26,411,372],{"className":412,"style":413},[69,70],"margin-right:0.0576em;",[26,415,418],{"className":416},[417],"msupsub",[26,419,423,455],{"className":420},[421,422],"vlist-t","vlist-t2",[26,424,427,450],{"className":425},[426],"vlist-r",[26,428,432],{"className":429,"style":431},[430],"vlist","height:0.1514em;",[26,433,435,440],{"style":434},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[26,436],{"className":437,"style":439},[438],"pstrut","height:2.7em;",[26,441,447],{"className":442},[443,444,445,446],"sizing","reset-size6","size3","mtight",[26,448,18],{"className":449},[69,70,446],[26,451,454],{"className":452},[453],"vlist-s","​",[26,456,458],{"className":457},[426],[26,459,462],{"className":460,"style":461},[430],"height:0.2861em;",[26,463],{},[26,465,91],{"className":466},[119],[26,468],{"className":469,"style":124},[123],[26,471,473,477],{"className":472},[69],[26,474,381],{"className":475,"style":476},[69,70],"margin-right:0.109em;",[26,478,480],{"className":479},[417],[26,481,483,504],{"className":482},[421,422],[26,484,486,501],{"className":485},[426],[26,487,489],{"className":488,"style":431},[430],[26,490,492,495],{"style":491},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[26,493],{"className":494,"style":439},[438],[26,496,498],{"className":497},[443,444,445,446],[26,499,18],{"className":500},[69,70,446],[26,502,454],{"className":503},[453],[26,505,507],{"className":506},[426],[26,508,510],{"className":509,"style":461},[430],[26,511],{},[26,513,386],{"className":514},[515],"mclose"," are calculated as follows ",[26,518,519],{},"1, 7",":",[18,522,523,1146],{},[26,524,526,623],{"className":525},[29],[26,527,529],{"className":528},[33],[35,530,531],{"xmlns":37},[39,532,533,620],{},[42,534,535,541,544],{},[368,536,537,539],{},[45,538,372],{},[45,540,18],{},[89,542,543],{},"=",[545,546,547,596],"mfrac",{},[42,548,549,556,563,566,573,580,582,589],{},[368,550,551,553],{},[45,552,323],{},[554,555,341],"mn",{},[368,557,558,560],{},[45,559,372],{},[45,561,562],{},"a",[89,564,565],{},"+",[368,567,568,570],{},[45,569,323],{},[554,571,572],{},"2",[368,574,575,577],{},[45,576,372],{},[45,578,579],{},"b",[89,581,565],{},[368,583,584,586],{},[45,585,323],{},[554,587,588],{},"3",[368,590,591,593],{},[45,592,372],{},[45,594,595],{},"c",[42,597,598,604,606,612,614],{},[368,599,600,602],{},[45,601,323],{},[554,603,341],{},[89,605,565],{},[368,607,608,610],{},[45,609,323],{},[554,611,572],{},[89,613,565],{},[368,615,616,618],{},[45,617,323],{},[554,619,588],{},[49,621,622],{"encoding":51},"E_p = \\frac{K_1 E_a + K_2 E_b + K_3 E_c}{K_1 + K_2 + K_3}",[26,624,626,684],{"className":625,"ariaHidden":56},[55],[26,627,629,633,673,677,681],{"className":628},[60],[26,630],{"className":631,"style":632},[64],"height:0.9694em;vertical-align:-0.2861em;",[26,634,636,639],{"className":635},[69],[26,637,372],{"className":638,"style":413},[69,70],[26,640,642],{"className":641},[417],[26,643,645,665],{"className":644},[421,422],[26,646,648,662],{"className":647},[426],[26,649,651],{"className":650,"style":431},[430],[26,652,653,656],{"style":434},[26,654],{"className":655,"style":439},[438],[26,657,659],{"className":658},[443,444,445,446],[26,660,18],{"className":661},[69,70,446],[26,663,454],{"className":664},[453],[26,666,668],{"className":667},[426],[26,669,671],{"className":670,"style":461},[430],[26,672],{},[26,674],{"className":675,"style":676},[123],"margin-right:0.2778em;",[26,678,543],{"className":679},[680],"mrel",[26,682],{"className":683,"style":676},[123],[26,685,687,691],{"className":686},[60],[26,688],{"className":689,"style":690},[64],"height:1.3393em;vertical-align:-0.4451em;",[26,692,694,698,1143],{"className":693},[69],[26,695],{"className":696},[406,697],"nulldelimiter",[26,699,701],{"className":700},[545],[26,702,704,1134],{"className":703},[421,422],[26,705,707,1131],{"className":706},[426],[26,708,711,857,868],{"className":709,"style":710},[430],"height:0.8942em;",[26,712,714,718],{"style":713},"top:-2.655em;",[26,715],{"className":716,"style":717},[438],"height:3em;",[26,719,721],{"className":720},[443,444,445,446],[26,722,724,770,774,814,817],{"className":723},[69,446],[26,725,727,730],{"className":726},[69,446],[26,728,323],{"className":729,"style":138},[69,70,446],[26,731,733],{"className":732},[417],[26,734,736,761],{"className":735},[421,422],[26,737,739,758],{"className":738},[426],[26,740,743],{"className":741,"style":742},[430],"height:0.3173em;",[26,744,746,750],{"style":745},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[26,747],{"className":748,"style":749},[438],"height:2.5em;",[26,751,755],{"className":752},[443,753,754,446],"reset-size3","size1",[26,756,341],{"className":757},[69,446],[26,759,454],{"className":760},[453],[26,762,764],{"className":763},[426],[26,765,768],{"className":766,"style":767},[430],"height:0.143em;",[26,769],{},[26,771,565],{"className":772},[773,446],"mbin",[26,775,777,780],{"className":776},[69,446],[26,778,323],{"className":779,"style":138},[69,70,446],[26,781,783],{"className":782},[417],[26,784,786,806],{"className":785},[421,422],[26,787,789,803],{"className":788},[426],[26,790,792],{"className":791,"style":742},[430],[26,793,794,797],{"style":745},[26,795],{"className":796,"style":749},[438],[26,798,800],{"className":799},[443,753,754,446],[26,801,572],{"className":802},[69,446],[26,804,454],{"className":805},[453],[26,807,809],{"className":808},[426],[26,810,812],{"className":811,"style":767},[430],[26,813],{},[26,815,565],{"className":816},[773,446],[26,818,820,823],{"className":819},[69,446],[26,821,323],{"className":822,"style":138},[69,70,446],[26,824,826],{"className":825},[417],[26,827,829,849],{"className":828},[421,422],[26,830,832,846],{"className":831},[426],[26,833,835],{"className":834,"style":742},[430],[26,836,837,840],{"style":745},[26,838],{"className":839,"style":749},[438],[26,841,843],{"className":842},[443,753,754,446],[26,844,588],{"className":845},[69,446],[26,847,454],{"className":848},[453],[26,850,852],{"className":851},[426],[26,853,855],{"className":854,"style":767},[430],[26,856],{},[26,858,860,863],{"style":859},"top:-3.23em;",[26,861],{"className":862,"style":717},[438],[26,864],{"className":865,"style":867},[866],"frac-line","border-bottom-width:0.04em;",[26,869,871,874],{"style":870},"top:-3.4159em;",[26,872],{"className":873,"style":717},[438],[26,875,877],{"className":876},[443,444,445,446],[26,878,880,920,962,965,1005,1048,1051,1091],{"className":879},[69,446],[26,881,883,886],{"className":882},[69,446],[26,884,323],{"className":885,"style":138},[69,70,446],[26,887,889],{"className":888},[417],[26,890,892,912],{"className":891},[421,422],[26,893,895,909],{"className":894},[426],[26,896,898],{"className":897,"style":742},[430],[26,899,900,903],{"style":745},[26,901],{"className":902,"style":749},[438],[26,904,906],{"className":905},[443,753,754,446],[26,907,341],{"className":908},[69,446],[26,910,454],{"className":911},[453],[26,913,915],{"className":914},[426],[26,916,918],{"className":917,"style":767},[430],[26,919],{},[26,921,923,926],{"className":922},[69,446],[26,924,372],{"className":925,"style":413},[69,70,446],[26,927,929],{"className":928},[417],[26,930,932,954],{"className":931},[421,422],[26,933,935,951],{"className":934},[426],[26,936,939],{"className":937,"style":938},[430],"height:0.1645em;",[26,940,942,945],{"style":941},"top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;",[26,943],{"className":944,"style":749},[438],[26,946,948],{"className":947},[443,753,754,446],[26,949,562],{"className":950},[69,70,446],[26,952,454],{"className":953},[453],[26,955,957],{"className":956},[426],[26,958,960],{"className":959,"style":767},[430],[26,961],{},[26,963,565],{"className":964},[773,446],[26,966,968,971],{"className":967},[69,446],[26,969,323],{"className":970,"style":138},[69,70,446],[26,972,974],{"className":973},[417],[26,975,977,997],{"className":976},[421,422],[26,978,980,994],{"className":979},[426],[26,981,983],{"className":982,"style":742},[430],[26,984,985,988],{"style":745},[26,986],{"className":987,"style":749},[438],[26,989,991],{"className":990},[443,753,754,446],[26,992,572],{"className":993},[69,446],[26,995,454],{"className":996},[453],[26,998,1000],{"className":999},[426],[26,1001,1003],{"className":1002,"style":767},[430],[26,1004],{},[26,1006,1008,1011],{"className":1007},[69,446],[26,1009,372],{"className":1010,"style":413},[69,70,446],[26,1012,1014],{"className":1013},[417],[26,1015,1017,1039],{"className":1016},[421,422],[26,1018,1020,1036],{"className":1019},[426],[26,1021,1024],{"className":1022,"style":1023},[430],"height:0.3448em;",[26,1025,1027,1030],{"style":1026},"top:-2.3488em;margin-left:-0.0576em;margin-right:0.0714em;",[26,1028],{"className":1029,"style":749},[438],[26,1031,1033],{"className":1032},[443,753,754,446],[26,1034,579],{"className":1035},[69,70,446],[26,1037,454],{"className":1038},[453],[26,1040,1042],{"className":1041},[426],[26,1043,1046],{"className":1044,"style":1045},[430],"height:0.1512em;",[26,1047],{},[26,1049,565],{"className":1050},[773,446],[26,1052,1054,1057],{"className":1053},[69,446],[26,1055,323],{"className":1056,"style":138},[69,70,446],[26,1058,1060],{"className":1059},[417],[26,1061,1063,1083],{"className":1062},[421,422],[26,1064,1066,1080],{"className":1065},[426],[26,1067,1069],{"className":1068,"style":742},[430],[26,1070,1071,1074],{"style":745},[26,1072],{"className":1073,"style":749},[438],[26,1075,1077],{"className":1076},[443,753,754,446],[26,1078,588],{"className":1079},[69,446],[26,1081,454],{"className":1082},[453],[26,1084,1086],{"className":1085},[426],[26,1087,1089],{"className":1088,"style":767},[430],[26,1090],{},[26,1092,1094,1097],{"className":1093},[69,446],[26,1095,372],{"className":1096,"style":413},[69,70,446],[26,1098,1100],{"className":1099},[417],[26,1101,1103,1123],{"className":1102},[421,422],[26,1104,1106,1120],{"className":1105},[426],[26,1107,1109],{"className":1108,"style":938},[430],[26,1110,1111,1114],{"style":941},[26,1112],{"className":1113,"style":749},[438],[26,1115,1117],{"className":1116},[443,753,754,446],[26,1118,595],{"className":1119},[69,70,446],[26,1121,454],{"className":1122},[453],[26,1124,1126],{"className":1125},[426],[26,1127,1129],{"className":1128,"style":767},[430],[26,1130],{},[26,1132,454],{"className":1133},[453],[26,1135,1137],{"className":1136},[426],[26,1138,1141],{"className":1139,"style":1140},[430],"height:0.4451em;",[26,1142],{},[26,1144],{"className":1145},[515,697],[26,1147,1149,1237],{"className":1148},[29],[26,1150,1152],{"className":1151},[33],[35,1153,1154],{"xmlns":37},[39,1155,1156,1234],{},[42,1157,1158,1164,1166],{},[368,1159,1160,1162],{},[45,1161,381],{},[45,1163,18],{},[89,1165,543],{},[545,1167,1168,1210],{},[42,1169,1170,1176,1182,1184,1190,1196,1198,1204],{},[368,1171,1172,1174],{},[45,1173,323],{},[554,1175,341],{},[368,1177,1178,1180],{},[45,1179,381],{},[45,1181,562],{},[89,1183,565],{},[368,1185,1186,1188],{},[45,1187,323],{},[554,1189,572],{},[368,1191,1192,1194],{},[45,1193,381],{},[45,1195,579],{},[89,1197,565],{},[368,1199,1200,1202],{},[45,1201,323],{},[554,1203,588],{},[368,1205,1206,1208],{},[45,1207,381],{},[45,1209,595],{},[42,1211,1212,1218,1220,1226,1228],{},[368,1213,1214,1216],{},[45,1215,323],{},[554,1217,341],{},[89,1219,565],{},[368,1221,1222,1224],{},[45,1223,323],{},[554,1225,572],{},[89,1227,565],{},[368,1229,1230,1232],{},[45,1231,323],{},[554,1233,588],{},[49,1235,1236],{"encoding":51},"N_p = \\frac{K_1 N_a + K_2 N_b + K_3 N_c}{K_1 + K_2 + K_3}",[26,1238,1240,1295],{"className":1239,"ariaHidden":56},[55],[26,1241,1243,1246,1286,1289,1292],{"className":1242},[60],[26,1244],{"className":1245,"style":632},[64],[26,1247,1249,1252],{"className":1248},[69],[26,1250,381],{"className":1251,"style":476},[69,70],[26,1253,1255],{"className":1254},[417],[26,1256,1258,1278],{"className":1257},[421,422],[26,1259,1261,1275],{"className":1260},[426],[26,1262,1264],{"className":1263,"style":431},[430],[26,1265,1266,1269],{"style":491},[26,1267],{"className":1268,"style":439},[438],[26,1270,1272],{"className":1271},[443,444,445,446],[26,1273,18],{"className":1274},[69,70,446],[26,1276,454],{"className":1277},[453],[26,1279,1281],{"className":1280},[426],[26,1282,1284],{"className":1283,"style":461},[430],[26,1285],{},[26,1287],{"className":1288,"style":676},[123],[26,1290,543],{"className":1291},[680],[26,1293],{"className":1294,"style":676},[123],[26,1296,1298,1301],{"className":1297},[60],[26,1299],{"className":1300,"style":690},[64],[26,1302,1304,1307,1734],{"className":1303},[69],[26,1305],{"className":1306},[406,697],[26,1308,1310],{"className":1309},[545],[26,1311,1313,1726],{"className":1312},[421,422],[26,1314,1316,1723],{"className":1315},[426],[26,1317,1319,1456,1464],{"className":1318,"style":710},[430],[26,1320,1321,1324],{"style":713},[26,1322],{"className":1323,"style":717},[438],[26,1325,1327],{"className":1326},[443,444,445,446],[26,1328,1330,1370,1373,1413,1416],{"className":1329},[69,446],[26,1331,1333,1336],{"className":1332},[69,446],[26,1334,323],{"className":1335,"style":138},[69,70,446],[26,1337,1339],{"className":1338},[417],[26,1340,1342,1362],{"className":1341},[421,422],[26,1343,1345,1359],{"className":1344},[426],[26,1346,1348],{"className":1347,"style":742},[430],[26,1349,1350,1353],{"style":745},[26,1351],{"className":1352,"style":749},[438],[26,1354,1356],{"className":1355},[443,753,754,446],[26,1357,341],{"className":1358},[69,446],[26,1360,454],{"className":1361},[453],[26,1363,1365],{"className":1364},[426],[26,1366,1368],{"className":1367,"style":767},[430],[26,1369],{},[26,1371,565],{"className":1372},[773,446],[26,1374,1376,1379],{"className":1375},[69,446],[26,1377,323],{"className":1378,"style":138},[69,70,446],[26,1380,1382],{"className":1381},[417],[26,1383,1385,1405],{"className":1384},[421,422],[26,1386,1388,1402],{"className":1387},[426],[26,1389,1391],{"className":1390,"style":742},[430],[26,1392,1393,1396],{"style":745},[26,1394],{"className":1395,"style":749},[438],[26,1397,1399],{"className":1398},[443,753,754,446],[26,1400,572],{"className":1401},[69,446],[26,1403,454],{"className":1404},[453],[26,1406,1408],{"className":1407},[426],[26,1409,1411],{"className":1410,"style":767},[430],[26,1412],{},[26,1414,565],{"className":1415},[773,446],[26,1417,1419,1422],{"className":1418},[69,446],[26,1420,323],{"className":1421,"style":138},[69,70,446],[26,1423,1425],{"className":1424},[417],[26,1426,1428,1448],{"className":1427},[421,422],[26,1429,1431,1445],{"className":1430},[426],[26,1432,1434],{"className":1433,"style":742},[430],[26,1435,1436,1439],{"style":745},[26,1437],{"className":1438,"style":749},[438],[26,1440,1442],{"className":1441},[443,753,754,446],[26,1443,588],{"className":1444},[69,446],[26,1446,454],{"className":1447},[453],[26,1449,1451],{"className":1450},[426],[26,1452,1454],{"className":1453,"style":767},[430],[26,1455],{},[26,1457,1458,1461],{"style":859},[26,1459],{"className":1460,"style":717},[438],[26,1462],{"className":1463,"style":867},[866],[26,1465,1466,1469],{"style":870},[26,1467],{"className":1468,"style":717},[438],[26,1470,1472],{"className":1471},[443,444,445,446],[26,1473,1475,1515,1556,1559,1599,1640,1643,1683],{"className":1474},[69,446],[26,1476,1478,1481],{"className":1477},[69,446],[26,1479,323],{"className":1480,"style":138},[69,70,446],[26,1482,1484],{"className":1483},[417],[26,1485,1487,1507],{"className":1486},[421,422],[26,1488,1490,1504],{"className":1489},[426],[26,1491,1493],{"className":1492,"style":742},[430],[26,1494,1495,1498],{"style":745},[26,1496],{"className":1497,"style":749},[438],[26,1499,1501],{"className":1500},[443,753,754,446],[26,1502,341],{"className":1503},[69,446],[26,1505,454],{"className":1506},[453],[26,1508,1510],{"className":1509},[426],[26,1511,1513],{"className":1512,"style":767},[430],[26,1514],{},[26,1516,1518,1521],{"className":1517},[69,446],[26,1519,381],{"className":1520,"style":476},[69,70,446],[26,1522,1524],{"className":1523},[417],[26,1525,1527,1548],{"className":1526},[421,422],[26,1528,1530,1545],{"className":1529},[426],[26,1531,1533],{"className":1532,"style":938},[430],[26,1534,1536,1539],{"style":1535},"top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;",[26,1537],{"className":1538,"style":749},[438],[26,1540,1542],{"className":1541},[443,753,754,446],[26,1543,562],{"className":1544},[69,70,446],[26,1546,454],{"className":1547},[453],[26,1549,1551],{"className":1550},[426],[26,1552,1554],{"className":1553,"style":767},[430],[26,1555],{},[26,1557,565],{"className":1558},[773,446],[26,1560,1562,1565],{"className":1561},[69,446],[26,1563,323],{"className":1564,"style":138},[69,70,446],[26,1566,1568],{"className":1567},[417],[26,1569,1571,1591],{"className":1570},[421,422],[26,1572,1574,1588],{"className":1573},[426],[26,1575,1577],{"className":1576,"style":742},[430],[26,1578,1579,1582],{"style":745},[26,1580],{"className":1581,"style":749},[438],[26,1583,1585],{"className":1584},[443,753,754,446],[26,1586,572],{"className":1587},[69,446],[26,1589,454],{"className":1590},[453],[26,1592,1594],{"className":1593},[426],[26,1595,1597],{"className":1596,"style":767},[430],[26,1598],{},[26,1600,1602,1605],{"className":1601},[69,446],[26,1603,381],{"className":1604,"style":476},[69,70,446],[26,1606,1608],{"className":1607},[417],[26,1609,1611,1632],{"className":1610},[421,422],[26,1612,1614,1629],{"className":1613},[426],[26,1615,1617],{"className":1616,"style":1023},[430],[26,1618,1620,1623],{"style":1619},"top:-2.3488em;margin-left:-0.109em;margin-right:0.0714em;",[26,1621],{"className":1622,"style":749},[438],[26,1624,1626],{"className":1625},[443,753,754,446],[26,1627,579],{"className":1628},[69,70,446],[26,1630,454],{"className":1631},[453],[26,1633,1635],{"className":1634},[426],[26,1636,1638],{"className":1637,"style":1045},[430],[26,1639],{},[26,1641,565],{"className":1642},[773,446],[26,1644,1646,1649],{"className":1645},[69,446],[26,1647,323],{"className":1648,"style":138},[69,70,446],[26,1650,1652],{"className":1651},[417],[26,1653,1655,1675],{"className":1654},[421,422],[26,1656,1658,1672],{"className":1657},[426],[26,1659,1661],{"className":1660,"style":742},[430],[26,1662,1663,1666],{"style":745},[26,1664],{"className":1665,"style":749},[438],[26,1667,1669],{"className":1668},[443,753,754,446],[26,1670,588],{"className":1671},[69,446],[26,1673,454],{"className":1674},[453],[26,1676,1678],{"className":1677},[426],[26,1679,1681],{"className":1680,"style":767},[430],[26,1682],{},[26,1684,1686,1689],{"className":1685},[69,446],[26,1687,381],{"className":1688,"style":476},[69,70,446],[26,1690,1692],{"className":1691},[417],[26,1693,1695,1715],{"className":1694},[421,422],[26,1696,1698,1712],{"className":1697},[426],[26,1699,1701],{"className":1700,"style":938},[430],[26,1702,1703,1706],{"style":1535},[26,1704],{"className":1705,"style":749},[438],[26,1707,1709],{"className":1708},[443,753,754,446],[26,1710,595],{"className":1711},[69,70,446],[26,1713,454],{"className":1714},[453],[26,1716,1718],{"className":1717},[426],[26,1719,1721],{"className":1720,"style":767},[430],[26,1722],{},[26,1724,454],{"className":1725},[453],[26,1727,1729],{"className":1728},[426],[26,1730,1732],{"className":1731,"style":1140},[430],[26,1733],{},[26,1735],{"className":1736},[515,697],[1738,1739,1741,1742,386],"h3",{"id":1740},"defining-the-weights-kkk","Defining the Weights (",[26,1743,1745,1758],{"className":1744},[29],[26,1746,1748],{"className":1747},[33],[35,1749,1750],{"xmlns":37},[39,1751,1752,1756],{},[42,1753,1754],{},[45,1755,323],{},[49,1757,323],{"encoding":51},[26,1759,1761],{"className":1760,"ariaHidden":56},[55],[26,1762,1764,1767],{"className":1763},[60],[26,1765],{"className":1766,"style":65},[64],[26,1768,323],{"className":1769,"style":138},[69,70],[18,1771,1772,1773,1776,1777,1831,1832,1860,1861,139,1923,520],{},"The weights are derived from the ",[21,1774,1775],{},"cotangents"," of the interior angles of the control triangle (",[26,1778,1780,1801],{"className":1779},[29],[26,1781,1783],{"className":1782},[33],[35,1784,1785],{"xmlns":37},[39,1786,1787,1799],{},[42,1788,1789,1791,1793,1795,1797],{},[45,1790,87],{},[89,1792,91],{"separator":56},[45,1794,94],{},[89,1796,91],{"separator":56},[45,1798,99],{},[49,1800,102],{"encoding":51},[26,1802,1804],{"className":1803,"ariaHidden":56},[55],[26,1805,1807,1810,1813,1816,1819,1822,1825,1828],{"className":1806},[60],[26,1808],{"className":1809,"style":112},[64],[26,1811,87],{"className":1812},[69,70],[26,1814,91],{"className":1815},[119],[26,1817],{"className":1818,"style":124},[123],[26,1820,94],{"className":1821,"style":128},[69,70],[26,1823,91],{"className":1824},[119],[26,1826],{"className":1827,"style":124},[123],[26,1829,99],{"className":1830,"style":138},[69,70],") and the observed angles measured at station ",[26,1833,1835,1848],{"className":1834},[29],[26,1836,1838],{"className":1837},[33],[35,1839,1840],{"xmlns":37},[39,1841,1842,1846],{},[42,1843,1844],{},[45,1845,47],{},[49,1847,47],{"encoding":51},[26,1849,1851],{"className":1850,"ariaHidden":56},[55],[26,1852,1854,1857],{"className":1853},[60],[26,1855],{"className":1856,"style":65},[64],[26,1858,47],{"className":1859,"style":71},[69,70]," (",[26,1862,1864,1889],{"className":1863},[29],[26,1865,1867],{"className":1866},[33],[35,1868,1869],{"xmlns":37},[39,1870,1871,1886],{},[42,1872,1873,1876,1878,1881,1883],{},[45,1874,1875],{},"α",[89,1877,91],{"separator":56},[45,1879,1880],{},"β",[89,1882,91],{"separator":56},[45,1884,1885],{},"γ",[49,1887,1888],{"encoding":51},"\\alpha, \\beta, \\gamma",[26,1890,1892],{"className":1891,"ariaHidden":56},[55],[26,1893,1895,1899,1903,1906,1909,1913,1916,1919],{"className":1894},[60],[26,1896],{"className":1897,"style":1898},[64],"height:0.8889em;vertical-align:-0.1944em;",[26,1900,1875],{"className":1901,"style":1902},[69,70],"margin-right:0.0037em;",[26,1904,91],{"className":1905},[119],[26,1907],{"className":1908,"style":124},[123],[26,1910,1880],{"className":1911,"style":1912},[69,70],"margin-right:0.0528em;",[26,1914,91],{"className":1915},[119],[26,1917],{"className":1918,"style":124},[123],[26,1920,1885],{"className":1921,"style":1922},[69,70],"margin-right:0.0556em;",[26,1924,519],{},[1926,1927,1928,2157,2372],"ul",{},[1929,1930,1931],"li",{},[26,1932,1934,1977],{"className":1933},[29],[26,1935,1937],{"className":1936},[33],[35,1938,1939],{"xmlns":37},[39,1940,1941,1974],{},[42,1942,1943,1949,1951],{},[368,1944,1945,1947],{},[45,1946,323],{},[554,1948,341],{},[89,1950,543],{},[545,1952,1953,1955],{},[554,1954,341],{},[42,1956,1957,1960,1963,1965,1968,1970,1972],{},[45,1958,1959],{},"cot",[89,1961,1962],{},"⁡",[45,1964,87],{},[89,1966,1967],{},"−",[45,1969,1959],{},[89,1971,1962],{},[45,1973,1875],{},[49,1975,1976],{"encoding":51},"K_1 = \\frac{1}{\\cot A - \\cot \\alpha}",[26,1978,1980,2039],{"className":1979,"ariaHidden":56},[55],[26,1981,1983,1987,2030,2033,2036],{"className":1982},[60],[26,1984],{"className":1985,"style":1986},[64],"height:0.8333em;vertical-align:-0.15em;",[26,1988,1990,1993],{"className":1989},[69],[26,1991,323],{"className":1992,"style":138},[69,70],[26,1994,1996],{"className":1995},[417],[26,1997,1999,2021],{"className":1998},[421,422],[26,2000,2002,2018],{"className":2001},[426],[26,2003,2006],{"className":2004,"style":2005},[430],"height:0.3011em;",[26,2007,2009,2012],{"style":2008},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[26,2010],{"className":2011,"style":439},[438],[26,2013,2015],{"className":2014},[443,444,445,446],[26,2016,341],{"className":2017},[69,446],[26,2019,454],{"className":2020},[453],[26,2022,2024],{"className":2023},[426],[26,2025,2028],{"className":2026,"style":2027},[430],"height:0.15em;",[26,2029],{},[26,2031],{"className":2032,"style":676},[123],[26,2034,543],{"className":2035},[680],[26,2037],{"className":2038,"style":676},[123],[26,2040,2042,2046],{"className":2041},[60],[26,2043],{"className":2044,"style":2045},[64],"height:1.2484em;vertical-align:-0.4033em;",[26,2047,2049,2052,2154],{"className":2048},[69],[26,2050],{"className":2051},[406,697],[26,2053,2055],{"className":2054},[545],[26,2056,2058,2145],{"className":2057},[421,422],[26,2059,2061,2142],{"className":2060},[426],[26,2062,2065,2119,2127],{"className":2063,"style":2064},[430],"height:0.8451em;",[26,2066,2067,2070],{"style":713},[26,2068],{"className":2069,"style":717},[438],[26,2071,2073],{"className":2072},[443,444,445,446],[26,2074,2076,2091,2095,2098,2101,2113,2116],{"className":2075},[69,446],[26,2077,2080,2083,2087],{"className":2078},[2079,446],"mop",[26,2081,595],{"className":2082},[446],[26,2084,2086],{"className":2085},[446],"o",[26,2088,2090],{"className":2089},[446],"t",[26,2092],{"className":2093,"style":2094},[123,446],"margin-right:0.1952em;",[26,2096,87],{"className":2097},[69,70,446],[26,2099,1967],{"className":2100},[773,446],[26,2102,2104,2107,2110],{"className":2103},[2079,446],[26,2105,595],{"className":2106},[446],[26,2108,2086],{"className":2109},[446],[26,2111,2090],{"className":2112},[446],[26,2114],{"className":2115,"style":2094},[123,446],[26,2117,1875],{"className":2118,"style":1902},[69,70,446],[26,2120,2121,2124],{"style":859},[26,2122],{"className":2123,"style":717},[438],[26,2125],{"className":2126,"style":867},[866],[26,2128,2130,2133],{"style":2129},"top:-3.394em;",[26,2131],{"className":2132,"style":717},[438],[26,2134,2136],{"className":2135},[443,444,445,446],[26,2137,2139],{"className":2138},[69,446],[26,2140,341],{"className":2141},[69,446],[26,2143,454],{"className":2144},[453],[26,2146,2148],{"className":2147},[426],[26,2149,2152],{"className":2150,"style":2151},[430],"height:0.4033em;",[26,2153],{},[26,2155],{"className":2156},[515,697],[1929,2158,2159],{},[26,2160,2162,2202],{"className":2161},[29],[26,2163,2165],{"className":2164},[33],[35,2166,2167],{"xmlns":37},[39,2168,2169,2199],{},[42,2170,2171,2177,2179],{},[368,2172,2173,2175],{},[45,2174,323],{},[554,2176,572],{},[89,2178,543],{},[545,2180,2181,2183],{},[554,2182,341],{},[42,2184,2185,2187,2189,2191,2193,2195,2197],{},[45,2186,1959],{},[89,2188,1962],{},[45,2190,94],{},[89,2192,1967],{},[45,2194,1959],{},[89,2196,1962],{},[45,2198,1880],{},[49,2200,2201],{"encoding":51},"K_2 = \\frac{1}{\\cot B - \\cot \\beta}",[26,2203,2205,2260],{"className":2204,"ariaHidden":56},[55],[26,2206,2208,2211,2251,2254,2257],{"className":2207},[60],[26,2209],{"className":2210,"style":1986},[64],[26,2212,2214,2217],{"className":2213},[69],[26,2215,323],{"className":2216,"style":138},[69,70],[26,2218,2220],{"className":2219},[417],[26,2221,2223,2243],{"className":2222},[421,422],[26,2224,2226,2240],{"className":2225},[426],[26,2227,2229],{"className":2228,"style":2005},[430],[26,2230,2231,2234],{"style":2008},[26,2232],{"className":2233,"style":439},[438],[26,2235,2237],{"className":2236},[443,444,445,446],[26,2238,572],{"className":2239},[69,446],[26,2241,454],{"className":2242},[453],[26,2244,2246],{"className":2245},[426],[26,2247,2249],{"className":2248,"style":2027},[430],[26,2250],{},[26,2252],{"className":2253,"style":676},[123],[26,2255,543],{"className":2256},[680],[26,2258],{"className":2259,"style":676},[123],[26,2261,2263,2267],{"className":2262},[60],[26,2264],{"className":2265,"style":2266},[64],"height:1.3262em;vertical-align:-0.4811em;",[26,2268,2270,2273,2369],{"className":2269},[69],[26,2271],{"className":2272},[406,697],[26,2274,2276],{"className":2275},[545],[26,2277,2279,2360],{"className":2278},[421,422],[26,2280,2282,2357],{"className":2281},[426],[26,2283,2285,2335,2343],{"className":2284,"style":2064},[430],[26,2286,2287,2290],{"style":713},[26,2288],{"className":2289,"style":717},[438],[26,2291,2293],{"className":2292},[443,444,445,446],[26,2294,2296,2308,2311,2314,2317,2329,2332],{"className":2295},[69,446],[26,2297,2299,2302,2305],{"className":2298},[2079,446],[26,2300,595],{"className":2301},[446],[26,2303,2086],{"className":2304},[446],[26,2306,2090],{"className":2307},[446],[26,2309],{"className":2310,"style":2094},[123,446],[26,2312,94],{"className":2313,"style":128},[69,70,446],[26,2315,1967],{"className":2316},[773,446],[26,2318,2320,2323,2326],{"className":2319},[2079,446],[26,2321,595],{"className":2322},[446],[26,2324,2086],{"className":2325},[446],[26,2327,2090],{"className":2328},[446],[26,2330],{"className":2331,"style":2094},[123,446],[26,2333,1880],{"className":2334,"style":1912},[69,70,446],[26,2336,2337,2340],{"style":859},[26,2338],{"className":2339,"style":717},[438],[26,2341],{"className":2342,"style":867},[866],[26,2344,2345,2348],{"style":2129},[26,2346],{"className":2347,"style":717},[438],[26,2349,2351],{"className":2350},[443,444,445,446],[26,2352,2354],{"className":2353},[69,446],[26,2355,341],{"className":2356},[69,446],[26,2358,454],{"className":2359},[453],[26,2361,2363],{"className":2362},[426],[26,2364,2367],{"className":2365,"style":2366},[430],"height:0.4811em;",[26,2368],{},[26,2370],{"className":2371},[515,697],[1929,2373,2374],{},[26,2375,2377,2417],{"className":2376},[29],[26,2378,2380],{"className":2379},[33],[35,2381,2382],{"xmlns":37},[39,2383,2384,2414],{},[42,2385,2386,2392,2394],{},[368,2387,2388,2390],{},[45,2389,323],{},[554,2391,588],{},[89,2393,543],{},[545,2395,2396,2398],{},[554,2397,341],{},[42,2399,2400,2402,2404,2406,2408,2410,2412],{},[45,2401,1959],{},[89,2403,1962],{},[45,2405,99],{},[89,2407,1967],{},[45,2409,1959],{},[89,2411,1962],{},[45,2413,1885],{},[49,2415,2416],{"encoding":51},"K_3 = \\frac{1}{\\cot C - \\cot \\gamma}",[26,2418,2420,2475],{"className":2419,"ariaHidden":56},[55],[26,2421,2423,2426,2466,2469,2472],{"className":2422},[60],[26,2424],{"className":2425,"style":1986},[64],[26,2427,2429,2432],{"className":2428},[69],[26,2430,323],{"className":2431,"style":138},[69,70],[26,2433,2435],{"className":2434},[417],[26,2436,2438,2458],{"className":2437},[421,422],[26,2439,2441,2455],{"className":2440},[426],[26,2442,2444],{"className":2443,"style":2005},[430],[26,2445,2446,2449],{"style":2008},[26,2447],{"className":2448,"style":439},[438],[26,2450,2452],{"className":2451},[443,444,445,446],[26,2453,588],{"className":2454},[69,446],[26,2456,454],{"className":2457},[453],[26,2459,2461],{"className":2460},[426],[26,2462,2464],{"className":2463,"style":2027},[430],[26,2465],{},[26,2467],{"className":2468,"style":676},[123],[26,2470,543],{"className":2471},[680],[26,2473],{"className":2474,"style":676},[123],[26,2476,2478,2481],{"className":2477},[60],[26,2479],{"className":2480,"style":2266},[64],[26,2482,2484,2487,2582],{"className":2483},[69],[26,2485],{"className":2486},[406,697],[26,2488,2490],{"className":2489},[545],[26,2491,2493,2574],{"className":2492},[421,422],[26,2494,2496,2571],{"className":2495},[426],[26,2497,2499,2549,2557],{"className":2498,"style":2064},[430],[26,2500,2501,2504],{"style":713},[26,2502],{"className":2503,"style":717},[438],[26,2505,2507],{"className":2506},[443,444,445,446],[26,2508,2510,2522,2525,2528,2531,2543,2546],{"className":2509},[69,446],[26,2511,2513,2516,2519],{"className":2512},[2079,446],[26,2514,595],{"className":2515},[446],[26,2517,2086],{"className":2518},[446],[26,2520,2090],{"className":2521},[446],[26,2523],{"className":2524,"style":2094},[123,446],[26,2526,99],{"className":2527,"style":138},[69,70,446],[26,2529,1967],{"className":2530},[773,446],[26,2532,2534,2537,2540],{"className":2533},[2079,446],[26,2535,595],{"className":2536},[446],[26,2538,2086],{"className":2539},[446],[26,2541,2090],{"className":2542},[446],[26,2544],{"className":2545,"style":2094},[123,446],[26,2547,1885],{"className":2548,"style":1922},[69,70,446],[26,2550,2551,2554],{"style":859},[26,2552],{"className":2553,"style":717},[438],[26,2555],{"className":2556,"style":867},[866],[26,2558,2559,2562],{"style":2129},[26,2560],{"className":2561,"style":717},[438],[26,2563,2565],{"className":2564},[443,444,445,446],[26,2566,2568],{"className":2567},[69,446],[26,2569,341],{"className":2570},[69,446],[26,2572,454],{"className":2573},[453],[26,2575,2577],{"className":2576},[426],[26,2578,2580],{"className":2579,"style":2366},[430],[26,2581],{},[26,2583],{"className":2584},[515,697],[18,2586,2587],{},[2588,2589,2590,2591,2621,2622,2669,2670,2699,2700,2744,2745,2699,2775,139,2819,151],"em",{},"(Note: ",[26,2592,2594,2608],{"className":2593},[29],[26,2595,2597],{"className":2596},[33],[35,2598,2599],{"xmlns":37},[39,2600,2601,2605],{},[42,2602,2603],{},[45,2604,1875],{},[49,2606,2607],{"encoding":51},"\\alpha",[26,2609,2611],{"className":2610,"ariaHidden":56},[55],[26,2612,2614,2618],{"className":2613},[60],[26,2615],{"className":2616,"style":2617},[64],"height:0.4306em;",[26,2619,1875],{"className":2620,"style":1902},[69,70]," is the angle ",[26,2623,2625,2647],{"className":2624},[29],[26,2626,2628],{"className":2627},[33],[35,2629,2630],{"xmlns":37},[39,2631,2632,2644],{},[42,2633,2634,2638,2640,2642],{},[45,2635,2637],{"mathvariant":2636},"normal","∠",[45,2639,94],{},[45,2641,47],{},[45,2643,99],{},[49,2645,2646],{"encoding":51},"\\angle BPC",[26,2648,2650],{"className":2649,"ariaHidden":56},[55],[26,2651,2653,2657,2660,2663,2666],{"className":2652},[60],[26,2654],{"className":2655,"style":2656},[64],"height:0.6922em;",[26,2658,2637],{"className":2659},[69],[26,2661,94],{"className":2662,"style":128},[69,70],[26,2664,47],{"className":2665,"style":71},[69,70],[26,2667,99],{"className":2668,"style":138},[69,70],", ",[26,2671,2673,2687],{"className":2672},[29],[26,2674,2676],{"className":2675},[33],[35,2677,2678],{"xmlns":37},[39,2679,2680,2684],{},[42,2681,2682],{},[45,2683,1880],{},[49,2685,2686],{"encoding":51},"\\beta",[26,2688,2690],{"className":2689,"ariaHidden":56},[55],[26,2691,2693,2696],{"className":2692},[60],[26,2694],{"className":2695,"style":1898},[64],[26,2697,1880],{"className":2698,"style":1912},[69,70]," is ",[26,2701,2703,2723],{"className":2702},[29],[26,2704,2706],{"className":2705},[33],[35,2707,2708],{"xmlns":37},[39,2709,2710,2720],{},[42,2711,2712,2714,2716,2718],{},[45,2713,2637],{"mathvariant":2636},[45,2715,99],{},[45,2717,47],{},[45,2719,87],{},[49,2721,2722],{"encoding":51},"\\angle CPA",[26,2724,2726],{"className":2725,"ariaHidden":56},[55],[26,2727,2729,2732,2735,2738,2741],{"className":2728},[60],[26,2730],{"className":2731,"style":2656},[64],[26,2733,2637],{"className":2734},[69],[26,2736,99],{"className":2737,"style":138},[69,70],[26,2739,47],{"className":2740,"style":71},[69,70],[26,2742,87],{"className":2743},[69,70],", and ",[26,2746,2748,2762],{"className":2747},[29],[26,2749,2751],{"className":2750},[33],[35,2752,2753],{"xmlns":37},[39,2754,2755,2759],{},[42,2756,2757],{},[45,2758,1885],{},[49,2760,2761],{"encoding":51},"\\gamma",[26,2763,2765],{"className":2764,"ariaHidden":56},[55],[26,2766,2768,2772],{"className":2767},[60],[26,2769],{"className":2770,"style":2771},[64],"height:0.625em;vertical-align:-0.1944em;",[26,2773,1885],{"className":2774,"style":1922},[69,70],[26,2776,2778,2798],{"className":2777},[29],[26,2779,2781],{"className":2780},[33],[35,2782,2783],{"xmlns":37},[39,2784,2785,2795],{},[42,2786,2787,2789,2791,2793],{},[45,2788,2637],{"mathvariant":2636},[45,2790,87],{},[45,2792,47],{},[45,2794,94],{},[49,2796,2797],{"encoding":51},"\\angle APB",[26,2799,2801],{"className":2800,"ariaHidden":56},[55],[26,2802,2804,2807,2810,2813,2816],{"className":2803},[60],[26,2805],{"className":2806,"style":2656},[64],[26,2808,2637],{"className":2809},[69],[26,2811,87],{"className":2812},[69,70],[26,2814,47],{"className":2815,"style":71},[69,70],[26,2817,94],{"className":2818,"style":128},[69,70],[26,2820,2821],{},"7, 8",[13,2823,2825],{"id":2824},"field-workflow","Field Workflow",[2827,2828,2829,2833,2922,2926,2961,2965,3101,3105],"steps",{},[1738,2830,2832],{"id":2831},"reconnaissance","Reconnaissance",[18,2834,2835,2836,2890,2891,2919,2920,151],{},"Identify three stable control points (",[26,2837,2839,2860],{"className":2838},[29],[26,2840,2842],{"className":2841},[33],[35,2843,2844],{"xmlns":37},[39,2845,2846,2858],{},[42,2847,2848,2850,2852,2854,2856],{},[45,2849,87],{},[89,2851,91],{"separator":56},[45,2853,94],{},[89,2855,91],{"separator":56},[45,2857,99],{},[49,2859,102],{"encoding":51},[26,2861,2863],{"className":2862,"ariaHidden":56},[55],[26,2864,2866,2869,2872,2875,2878,2881,2884,2887],{"className":2865},[60],[26,2867],{"className":2868,"style":112},[64],[26,2870,87],{"className":2871},[69,70],[26,2873,91],{"className":2874},[119],[26,2876],{"className":2877,"style":124},[123],[26,2879,94],{"className":2880,"style":128},[69,70],[26,2882,91],{"className":2883},[119],[26,2885],{"className":2886,"style":124},[123],[26,2888,99],{"className":2889,"style":138},[69,70],") with known coordinates. Ensure your station ",[26,2892,2894,2907],{"className":2893},[29],[26,2895,2897],{"className":2896},[33],[35,2898,2899],{"xmlns":37},[39,2900,2901,2905],{},[42,2902,2903],{},[45,2904,47],{},[49,2906,47],{"encoding":51},[26,2908,2910],{"className":2909,"ariaHidden":56},[55],[26,2911,2913,2916],{"className":2912},[60],[26,2914],{"className":2915,"style":65},[64],[26,2917,47],{"className":2918,"style":71},[69,70]," is not near the \"Danger Circle\" (see below) ",[26,2921,572],{},[1738,2923,2925],{"id":2924},"angular-observation","Angular Observation",[18,2927,2928,2929,2957,2958,151],{},"Set up the theodolite or total station at point ",[26,2930,2932,2945],{"className":2931},[29],[26,2933,2935],{"className":2934},[33],[35,2936,2937],{"xmlns":37},[39,2938,2939,2943],{},[42,2940,2941],{},[45,2942,47],{},[49,2944,47],{"encoding":51},[26,2946,2948],{"className":2947,"ariaHidden":56},[55],[26,2949,2951,2954],{"className":2950},[60],[26,2952],{"className":2953,"style":65},[64],[26,2955,47],{"className":2956,"style":71},[69,70],". Measure the horizontal angles between the three control points with high precision (multiple sets on both faces) ",[26,2959,2960],{},"2, 9",[1738,2962,2964],{"id":2963},"interior-angle-calculation","Interior Angle Calculation",[18,2966,2967,2968,275,3014,3042,3043,3097,3098,151],{},"Using the known coordinates of ",[26,2969,2971,2990],{"className":2970},[29],[26,2972,2974],{"className":2973},[33],[35,2975,2976],{"xmlns":37},[39,2977,2978,2988],{},[42,2979,2980,2982,2984,2986],{},[45,2981,87],{},[89,2983,91],{"separator":56},[45,2985,94],{},[89,2987,91],{"separator":56},[49,2989,250],{"encoding":51},[26,2991,2993],{"className":2992,"ariaHidden":56},[55],[26,2994,2996,2999,3002,3005,3008,3011],{"className":2995},[60],[26,2997],{"className":2998,"style":112},[64],[26,3000,87],{"className":3001},[69,70],[26,3003,91],{"className":3004},[119],[26,3006],{"className":3007,"style":124},[123],[26,3009,94],{"className":3010,"style":128},[69,70],[26,3012,91],{"className":3013},[119],[26,3015,3017,3030],{"className":3016},[29],[26,3018,3020],{"className":3019},[33],[35,3021,3022],{"xmlns":37},[39,3023,3024,3028],{},[42,3025,3026],{},[45,3027,99],{},[49,3029,99],{"encoding":51},[26,3031,3033],{"className":3032,"ariaHidden":56},[55],[26,3034,3036,3039],{"className":3035},[60],[26,3037],{"className":3038,"style":65},[64],[26,3040,99],{"className":3041,"style":138},[69,70],", calculate the interior angles of the control triangle (",[26,3044,3046,3067],{"className":3045},[29],[26,3047,3049],{"className":3048},[33],[35,3050,3051],{"xmlns":37},[39,3052,3053,3065],{},[42,3054,3055,3057,3059,3061,3063],{},[45,3056,87],{},[89,3058,91],{"separator":56},[45,3060,94],{},[89,3062,91],{"separator":56},[45,3064,99],{},[49,3066,102],{"encoding":51},[26,3068,3070],{"className":3069,"ariaHidden":56},[55],[26,3071,3073,3076,3079,3082,3085,3088,3091,3094],{"className":3072},[60],[26,3074],{"className":3075,"style":112},[64],[26,3077,87],{"className":3078},[69,70],[26,3080,91],{"className":3081},[119],[26,3083],{"className":3084,"style":124},[123],[26,3086,94],{"className":3087,"style":128},[69,70],[26,3089,91],{"className":3090},[119],[26,3092],{"className":3093,"style":124},[123],[26,3095,99],{"className":3096,"style":138},[69,70],") using standard COGO \"Join\" computations ",[26,3099,3100],{},"7, 9",[1738,3102,3104],{"id":3103},"computation","Computation",[18,3106,3107,3108,3136,3137,275,3207,304,3277,151],{},"Calculate the three ",[26,3109,3111,3124],{"className":3110},[29],[26,3112,3114],{"className":3113},[33],[35,3115,3116],{"xmlns":37},[39,3117,3118,3122],{},[42,3119,3120],{},[45,3121,323],{},[49,3123,323],{"encoding":51},[26,3125,3127],{"className":3126,"ariaHidden":56},[55],[26,3128,3130,3133],{"className":3129},[60],[26,3131],{"className":3132,"style":65},[64],[26,3134,323],{"className":3135,"style":138},[69,70]," weights using the cotangent formula, then apply the weighted mean equations to find ",[26,3138,3140,3158],{"className":3139},[29],[26,3141,3143],{"className":3142},[33],[35,3144,3145],{"xmlns":37},[39,3146,3147,3155],{},[42,3148,3149],{},[368,3150,3151,3153],{},[45,3152,372],{},[45,3154,18],{},[49,3156,3157],{"encoding":51},"E_p",[26,3159,3161],{"className":3160,"ariaHidden":56},[55],[26,3162,3164,3167],{"className":3163},[60],[26,3165],{"className":3166,"style":632},[64],[26,3168,3170,3173],{"className":3169},[69],[26,3171,372],{"className":3172,"style":413},[69,70],[26,3174,3176],{"className":3175},[417],[26,3177,3179,3199],{"className":3178},[421,422],[26,3180,3182,3196],{"className":3181},[426],[26,3183,3185],{"className":3184,"style":431},[430],[26,3186,3187,3190],{"style":434},[26,3188],{"className":3189,"style":439},[438],[26,3191,3193],{"className":3192},[443,444,445,446],[26,3194,18],{"className":3195},[69,70,446],[26,3197,454],{"className":3198},[453],[26,3200,3202],{"className":3201},[426],[26,3203,3205],{"className":3204,"style":461},[430],[26,3206],{},[26,3208,3210,3228],{"className":3209},[29],[26,3211,3213],{"className":3212},[33],[35,3214,3215],{"xmlns":37},[39,3216,3217,3225],{},[42,3218,3219],{},[368,3220,3221,3223],{},[45,3222,381],{},[45,3224,18],{},[49,3226,3227],{"encoding":51},"N_p",[26,3229,3231],{"className":3230,"ariaHidden":56},[55],[26,3232,3234,3237],{"className":3233},[60],[26,3235],{"className":3236,"style":632},[64],[26,3238,3240,3243],{"className":3239},[69],[26,3241,381],{"className":3242,"style":476},[69,70],[26,3244,3246],{"className":3245},[417],[26,3247,3249,3269],{"className":3248},[421,422],[26,3250,3252,3266],{"className":3251},[426],[26,3253,3255],{"className":3254,"style":431},[430],[26,3256,3257,3260],{"style":491},[26,3258],{"className":3259,"style":439},[438],[26,3261,3263],{"className":3262},[443,444,445,446],[26,3264,18],{"className":3265},[69,70,446],[26,3267,454],{"className":3268},[453],[26,3270,3272],{"className":3271},[426],[26,3273,3275],{"className":3274,"style":461},[430],[26,3276],{},[26,3278,3279],{},"1, 9",[13,3281,3283],{"id":3282},"step-by-step-example","Step-by-Step Example",[18,3285,3286,3289,3290,2669,3350,2744,3408,3466,3467,3495,3496,2669,3587,2744,3673,304,3759,151],{},[21,3287,3288],{},"Scenario:"," Control triangle points are ",[26,3291,3293,3319],{"className":3292},[29],[26,3294,3296],{"className":3295},[33],[35,3297,3298],{"xmlns":37},[39,3299,3300,3316],{},[42,3301,3302,3304,3306,3309,3311,3314],{},[45,3303,87],{},[89,3305,366],{"stretchy":365},[554,3307,3308],{},"1000",[89,3310,91],{"separator":56},[554,3312,3313],{},"2000",[89,3315,386],{"stretchy":365},[49,3317,3318],{"encoding":51},"A(1000, 2000)",[26,3320,3322],{"className":3321,"ariaHidden":56},[55],[26,3323,3325,3329,3332,3335,3338,3341,3344,3347],{"className":3324},[60],[26,3326],{"className":3327,"style":3328},[64],"height:1em;vertical-align:-0.25em;",[26,3330,87],{"className":3331},[69,70],[26,3333,366],{"className":3334},[406],[26,3336,3308],{"className":3337},[69],[26,3339,91],{"className":3340},[119],[26,3342],{"className":3343,"style":124},[123],[26,3345,3313],{"className":3346},[69],[26,3348,386],{"className":3349},[515],[26,3351,3353,3378],{"className":3352},[29],[26,3354,3356],{"className":3355},[33],[35,3357,3358],{"xmlns":37},[39,3359,3360,3375],{},[42,3361,3362,3364,3366,3369,3371,3373],{},[45,3363,94],{},[89,3365,366],{"stretchy":365},[554,3367,3368],{},"3000",[89,3370,91],{"separator":56},[554,3372,3313],{},[89,3374,386],{"stretchy":365},[49,3376,3377],{"encoding":51},"B(3000, 2000)",[26,3379,3381],{"className":3380,"ariaHidden":56},[55],[26,3382,3384,3387,3390,3393,3396,3399,3402,3405],{"className":3383},[60],[26,3385],{"className":3386,"style":3328},[64],[26,3388,94],{"className":3389,"style":128},[69,70],[26,3391,366],{"className":3392},[406],[26,3394,3368],{"className":3395},[69],[26,3397,91],{"className":3398},[119],[26,3400],{"className":3401,"style":124},[123],[26,3403,3313],{"className":3404},[69],[26,3406,386],{"className":3407},[515],[26,3409,3411,3436],{"className":3410},[29],[26,3412,3414],{"className":3413},[33],[35,3415,3416],{"xmlns":37},[39,3417,3418,3433],{},[42,3419,3420,3422,3424,3426,3428,3431],{},[45,3421,99],{},[89,3423,366],{"stretchy":365},[554,3425,3313],{},[89,3427,91],{"separator":56},[554,3429,3430],{},"3732",[89,3432,386],{"stretchy":365},[49,3434,3435],{"encoding":51},"C(2000, 3732)",[26,3437,3439],{"className":3438,"ariaHidden":56},[55],[26,3440,3442,3445,3448,3451,3454,3457,3460,3463],{"className":3441},[60],[26,3443],{"className":3444,"style":3328},[64],[26,3446,99],{"className":3447,"style":138},[69,70],[26,3449,366],{"className":3450},[406],[26,3452,3313],{"className":3453},[69],[26,3455,91],{"className":3456},[119],[26,3458],{"className":3459,"style":124},[123],[26,3461,3430],{"className":3462},[69],[26,3464,386],{"className":3465},[515],". The measured angles at ",[26,3468,3470,3483],{"className":3469},[29],[26,3471,3473],{"className":3472},[33],[35,3474,3475],{"xmlns":37},[39,3476,3477,3481],{},[42,3478,3479],{},[45,3480,47],{},[49,3482,47],{"encoding":51},[26,3484,3486],{"className":3485,"ariaHidden":56},[55],[26,3487,3489,3492],{"className":3488},[60],[26,3490],{"className":3491,"style":65},[64],[26,3493,47],{"className":3494,"style":71},[69,70]," are ",[26,3497,3499,3524],{"className":3498},[29],[26,3500,3502],{"className":3501},[33],[35,3503,3504],{"xmlns":37},[39,3505,3506,3521],{},[42,3507,3508,3510,3512],{},[45,3509,1875],{},[89,3511,543],{},[3513,3514,3515,3518],"msup",{},[554,3516,3517],{},"110",[89,3519,3520],{},"∘",[49,3522,3523],{"encoding":51},"\\alpha = 110^\\circ",[26,3525,3527,3545],{"className":3526,"ariaHidden":56},[55],[26,3528,3530,3533,3536,3539,3542],{"className":3529},[60],[26,3531],{"className":3532,"style":2617},[64],[26,3534,1875],{"className":3535,"style":1902},[69,70],[26,3537],{"className":3538,"style":676},[123],[26,3540,543],{"className":3541},[680],[26,3543],{"className":3544,"style":676},[123],[26,3546,3548,3552,3556],{"className":3547},[60],[26,3549],{"className":3550,"style":3551},[64],"height:0.6741em;",[26,3553,3555],{"className":3554},[69],"11",[26,3557,3559,3563],{"className":3558},[69],[26,3560,3562],{"className":3561},[69],"0",[26,3564,3566],{"className":3565},[417],[26,3567,3569],{"className":3568},[421],[26,3570,3572],{"className":3571},[426],[26,3573,3575],{"className":3574,"style":3551},[430],[26,3576,3578,3581],{"style":3577},"top:-3.063em;margin-right:0.05em;",[26,3579],{"className":3580,"style":439},[438],[26,3582,3584],{"className":3583},[443,444,445,446],[26,3585,3520],{"className":3586},[773,446],[26,3588,3590,3613],{"className":3589},[29],[26,3591,3593],{"className":3592},[33],[35,3594,3595],{"xmlns":37},[39,3596,3597,3610],{},[42,3598,3599,3601,3603],{},[45,3600,1880],{},[89,3602,543],{},[3513,3604,3605,3608],{},[554,3606,3607],{},"120",[89,3609,3520],{},[49,3611,3612],{"encoding":51},"\\beta = 120^\\circ",[26,3614,3616,3634],{"className":3615,"ariaHidden":56},[55],[26,3617,3619,3622,3625,3628,3631],{"className":3618},[60],[26,3620],{"className":3621,"style":1898},[64],[26,3623,1880],{"className":3624,"style":1912},[69,70],[26,3626],{"className":3627,"style":676},[123],[26,3629,543],{"className":3630},[680],[26,3632],{"className":3633,"style":676},[123],[26,3635,3637,3640,3644],{"className":3636},[60],[26,3638],{"className":3639,"style":3551},[64],[26,3641,3643],{"className":3642},[69],"12",[26,3645,3647,3650],{"className":3646},[69],[26,3648,3562],{"className":3649},[69],[26,3651,3653],{"className":3652},[417],[26,3654,3656],{"className":3655},[421],[26,3657,3659],{"className":3658},[426],[26,3660,3662],{"className":3661,"style":3551},[430],[26,3663,3664,3667],{"style":3577},[26,3665],{"className":3666,"style":439},[438],[26,3668,3670],{"className":3669},[443,444,445,446],[26,3671,3520],{"className":3672},[773,446],[26,3674,3676,3699],{"className":3675},[29],[26,3677,3679],{"className":3678},[33],[35,3680,3681],{"xmlns":37},[39,3682,3683,3696],{},[42,3684,3685,3687,3689],{},[45,3686,1885],{},[89,3688,543],{},[3513,3690,3691,3694],{},[554,3692,3693],{},"130",[89,3695,3520],{},[49,3697,3698],{"encoding":51},"\\gamma = 130^\\circ",[26,3700,3702,3720],{"className":3701,"ariaHidden":56},[55],[26,3703,3705,3708,3711,3714,3717],{"className":3704},[60],[26,3706],{"className":3707,"style":2771},[64],[26,3709,1885],{"className":3710,"style":1922},[69,70],[26,3712],{"className":3713,"style":676},[123],[26,3715,543],{"className":3716},[680],[26,3718],{"className":3719,"style":676},[123],[26,3721,3723,3726,3730],{"className":3722},[60],[26,3724],{"className":3725,"style":3551},[64],[26,3727,3729],{"className":3728},[69],"13",[26,3731,3733,3736],{"className":3732},[69],[26,3734,3562],{"className":3735},[69],[26,3737,3739],{"className":3738},[417],[26,3740,3742],{"className":3741},[421],[26,3743,3745],{"className":3744},[426],[26,3746,3748],{"className":3747,"style":3551},[430],[26,3749,3750,3753],{"style":3577},[26,3751],{"className":3752,"style":439},[438],[26,3754,3756],{"className":3755},[443,444,445,446],[26,3757,3520],{"className":3758},[773,446],[26,3760,3761],{},"10",[3763,3764,3765,3892,4274,4477,4873],"ol",{},[1929,3766,3767,3770,3771,3825,3826,3889,3890,151],{},[21,3768,3769],{},"Triangle Geometry:"," Assume internal angles ",[26,3772,3774,3795],{"className":3773},[29],[26,3775,3777],{"className":3776},[33],[35,3778,3779],{"xmlns":37},[39,3780,3781,3793],{},[42,3782,3783,3785,3787,3789,3791],{},[45,3784,87],{},[89,3786,91],{"separator":56},[45,3788,94],{},[89,3790,91],{"separator":56},[45,3792,99],{},[49,3794,102],{"encoding":51},[26,3796,3798],{"className":3797,"ariaHidden":56},[55],[26,3799,3801,3804,3807,3810,3813,3816,3819,3822],{"className":3800},[60],[26,3802],{"className":3803,"style":112},[64],[26,3805,87],{"className":3806},[69,70],[26,3808,91],{"className":3809},[119],[26,3811],{"className":3812,"style":124},[123],[26,3814,94],{"className":3815,"style":128},[69,70],[26,3817,91],{"className":3818},[119],[26,3820],{"className":3821,"style":124},[123],[26,3823,99],{"className":3824,"style":138},[69,70]," are all ",[26,3827,3829,3848],{"className":3828},[29],[26,3830,3832],{"className":3831},[33],[35,3833,3834],{"xmlns":37},[39,3835,3836,3845],{},[42,3837,3838],{},[3513,3839,3840,3843],{},[554,3841,3842],{},"60",[89,3844,3520],{},[49,3846,3847],{"encoding":51},"60^\\circ",[26,3849,3851],{"className":3850,"ariaHidden":56},[55],[26,3852,3854,3857,3860],{"className":3853},[60],[26,3855],{"className":3856,"style":3551},[64],[26,3858,307],{"className":3859},[69],[26,3861,3863,3866],{"className":3862},[69],[26,3864,3562],{"className":3865},[69],[26,3867,3869],{"className":3868},[417],[26,3870,3872],{"className":3871},[421],[26,3873,3875],{"className":3874},[426],[26,3876,3878],{"className":3877,"style":3551},[430],[26,3879,3880,3883],{"style":3577},[26,3881],{"className":3882,"style":439},[438],[26,3884,3886],{"className":3885},[443,444,445,446],[26,3887,3520],{"className":3888},[773,446]," (equilateral) ",[26,3891,3761],{},[1929,3893,3894,3967],{},[21,3895,3896,3897,520],{},"Calculate ",[26,3898,3900,3918],{"className":3899},[29],[26,3901,3903],{"className":3902},[33],[35,3904,3905],{"xmlns":37},[39,3906,3907,3915],{},[42,3908,3909],{},[368,3910,3911,3913],{},[45,3912,323],{},[554,3914,341],{},[49,3916,3917],{"encoding":51},"K_1",[26,3919,3921],{"className":3920,"ariaHidden":56},[55],[26,3922,3924,3927],{"className":3923},[60],[26,3925],{"className":3926,"style":1986},[64],[26,3928,3930,3933],{"className":3929},[69],[26,3931,323],{"className":3932,"style":138},[69,70],[26,3934,3936],{"className":3935},[417],[26,3937,3939,3959],{"className":3938},[421,422],[26,3940,3942,3956],{"className":3941},[426],[26,3943,3945],{"className":3944,"style":2005},[430],[26,3946,3947,3950],{"style":2008},[26,3948],{"className":3949,"style":439},[438],[26,3951,3953],{"className":3952},[443,444,445,446],[26,3954,341],{"className":3955},[69,446],[26,3957,454],{"className":3958},[453],[26,3960,3962],{"className":3961},[426],[26,3963,3965],{"className":3964,"style":2027},[430],[26,3966],{},[26,3968,3970,4044],{"className":3969},[29],[26,3971,3973],{"className":3972},[33],[35,3974,3975],{"xmlns":37},[39,3976,3977,4041],{},[42,3978,3979,3985,3987,3989,3992,3994,3996,3998,4004,4006,4008,4010,4016,4018,4020,4022,4024,4026,4029,4031,4034,4036,4038],{},[368,3980,3981,3983],{},[45,3982,323],{},[554,3984,341],{},[89,3986,543],{},[554,3988,341],{},[45,3990,3991],{"mathvariant":2636},"\u002F",[89,3993,366],{"stretchy":365},[45,3995,1959],{},[89,3997,1962],{},[3513,3999,4000,4002],{},[554,4001,3842],{},[89,4003,3520],{},[89,4005,1967],{},[45,4007,1959],{},[89,4009,1962],{},[3513,4011,4012,4014],{},[554,4013,3517],{},[89,4015,3520],{},[89,4017,386],{"stretchy":365},[89,4019,543],{},[554,4021,341],{},[45,4023,3991],{"mathvariant":2636},[89,4025,366],{"stretchy":365},[554,4027,4028],{},"0.57735",[89,4030,565],{},[554,4032,4033],{},"0.36397",[89,4035,386],{"stretchy":365},[89,4037,543],{},[554,4039,4040],{},"1.0623",[49,4042,4043],{"encoding":51},"K_1 = 1 \u002F (\\cot 60^\\circ - \\cot 110^\\circ) = 1 \u002F (0.57735 + 0.36397) = 1.0623",[26,4045,4047,4102,4163,4219,4243,4264],{"className":4046,"ariaHidden":56},[55],[26,4048,4050,4053,4093,4096,4099],{"className":4049},[60],[26,4051],{"className":4052,"style":1986},[64],[26,4054,4056,4059],{"className":4055},[69],[26,4057,323],{"className":4058,"style":138},[69,70],[26,4060,4062],{"className":4061},[417],[26,4063,4065,4085],{"className":4064},[421,422],[26,4066,4068,4082],{"className":4067},[426],[26,4069,4071],{"className":4070,"style":2005},[430],[26,4072,4073,4076],{"style":2008},[26,4074],{"className":4075,"style":439},[438],[26,4077,4079],{"className":4078},[443,444,445,446],[26,4080,341],{"className":4081},[69,446],[26,4083,454],{"className":4084},[453],[26,4086,4088],{"className":4087},[426],[26,4089,4091],{"className":4090,"style":2027},[430],[26,4092],{},[26,4094],{"className":4095,"style":676},[123],[26,4097,543],{"className":4098},[680],[26,4100],{"className":4101,"style":676},[123],[26,4103,4105,4108,4112,4115,4118,4121,4124,4153,4157,4160],{"className":4104},[60],[26,4106],{"className":4107,"style":3328},[64],[26,4109,4111],{"className":4110},[69],"1\u002F",[26,4113,366],{"className":4114},[406],[26,4116,1959],{"className":4117},[2079],[26,4119],{"className":4120,"style":124},[123],[26,4122,307],{"className":4123},[69],[26,4125,4127,4130],{"className":4126},[69],[26,4128,3562],{"className":4129},[69],[26,4131,4133],{"className":4132},[417],[26,4134,4136],{"className":4135},[421],[26,4137,4139],{"className":4138},[426],[26,4140,4142],{"className":4141,"style":3551},[430],[26,4143,4144,4147],{"style":3577},[26,4145],{"className":4146,"style":439},[438],[26,4148,4150],{"className":4149},[443,444,445,446],[26,4151,3520],{"className":4152},[773,446],[26,4154],{"className":4155,"style":4156},[123],"margin-right:0.2222em;",[26,4158,1967],{"className":4159},[773],[26,4161],{"className":4162,"style":4156},[123],[26,4164,4166,4169,4172,4175,4178,4207,4210,4213,4216],{"className":4165},[60],[26,4167],{"className":4168,"style":3328},[64],[26,4170,1959],{"className":4171},[2079],[26,4173],{"className":4174,"style":124},[123],[26,4176,3555],{"className":4177},[69],[26,4179,4181,4184],{"className":4180},[69],[26,4182,3562],{"className":4183},[69],[26,4185,4187],{"className":4186},[417],[26,4188,4190],{"className":4189},[421],[26,4191,4193],{"className":4192},[426],[26,4194,4196],{"className":4195,"style":3551},[430],[26,4197,4198,4201],{"style":3577},[26,4199],{"className":4200,"style":439},[438],[26,4202,4204],{"className":4203},[443,444,445,446],[26,4205,3520],{"className":4206},[773,446],[26,4208,386],{"className":4209},[515],[26,4211],{"className":4212,"style":676},[123],[26,4214,543],{"className":4215},[680],[26,4217],{"className":4218,"style":676},[123],[26,4220,4222,4225,4228,4231,4234,4237,4240],{"className":4221},[60],[26,4223],{"className":4224,"style":3328},[64],[26,4226,4111],{"className":4227},[69],[26,4229,366],{"className":4230},[406],[26,4232,4028],{"className":4233},[69],[26,4235],{"className":4236,"style":4156},[123],[26,4238,565],{"className":4239},[773],[26,4241],{"className":4242,"style":4156},[123],[26,4244,4246,4249,4252,4255,4258,4261],{"className":4245},[60],[26,4247],{"className":4248,"style":3328},[64],[26,4250,4033],{"className":4251},[69],[26,4253,386],{"className":4254},[515],[26,4256],{"className":4257,"style":676},[123],[26,4259,543],{"className":4260},[680],[26,4262],{"className":4263,"style":676},[123],[26,4265,4267,4271],{"className":4266},[60],[26,4268],{"className":4269,"style":4270},[64],"height:0.6444em;",[26,4272,4040],{"className":4273},[69],[1929,4275,4276,4419,4420,275,4448,4476],{},[21,4277,3896,4278,4348,4349,520],{},[26,4279,4281,4299],{"className":4280},[29],[26,4282,4284],{"className":4283},[33],[35,4285,4286],{"xmlns":37},[39,4287,4288,4296],{},[42,4289,4290],{},[368,4291,4292,4294],{},[45,4293,323],{},[554,4295,572],{},[49,4297,4298],{"encoding":51},"K_2",[26,4300,4302],{"className":4301,"ariaHidden":56},[55],[26,4303,4305,4308],{"className":4304},[60],[26,4306],{"className":4307,"style":1986},[64],[26,4309,4311,4314],{"className":4310},[69],[26,4312,323],{"className":4313,"style":138},[69,70],[26,4315,4317],{"className":4316},[417],[26,4318,4320,4340],{"className":4319},[421,422],[26,4321,4323,4337],{"className":4322},[426],[26,4324,4326],{"className":4325,"style":2005},[430],[26,4327,4328,4331],{"style":2008},[26,4329],{"className":4330,"style":439},[438],[26,4332,4334],{"className":4333},[443,444,445,446],[26,4335,572],{"className":4336},[69,446],[26,4338,454],{"className":4339},[453],[26,4341,4343],{"className":4342},[426],[26,4344,4346],{"className":4345,"style":2027},[430],[26,4347],{}," & ",[26,4350,4352,4370],{"className":4351},[29],[26,4353,4355],{"className":4354},[33],[35,4356,4357],{"xmlns":37},[39,4358,4359,4367],{},[42,4360,4361],{},[368,4362,4363,4365],{},[45,4364,323],{},[554,4366,588],{},[49,4368,4369],{"encoding":51},"K_3",[26,4371,4373],{"className":4372,"ariaHidden":56},[55],[26,4374,4376,4379],{"className":4375},[60],[26,4377],{"className":4378,"style":1986},[64],[26,4380,4382,4385],{"className":4381},[69],[26,4383,323],{"className":4384,"style":138},[69,70],[26,4386,4388],{"className":4387},[417],[26,4389,4391,4411],{"className":4390},[421,422],[26,4392,4394,4408],{"className":4393},[426],[26,4395,4397],{"className":4396,"style":2005},[430],[26,4398,4399,4402],{"style":2008},[26,4400],{"className":4401,"style":439},[438],[26,4403,4405],{"className":4404},[443,444,445,446],[26,4406,588],{"className":4407},[69,446],[26,4409,454],{"className":4410},[453],[26,4412,4414],{"className":4413},[426],[26,4415,4417],{"className":4416,"style":2027},[430],[26,4418],{}," (Follow the same process using ",[26,4421,4423,4436],{"className":4422},[29],[26,4424,4426],{"className":4425},[33],[35,4427,4428],{"xmlns":37},[39,4429,4430,4434],{},[42,4431,4432],{},[45,4433,1880],{},[49,4435,2686],{"encoding":51},[26,4437,4439],{"className":4438,"ariaHidden":56},[55],[26,4440,4442,4445],{"className":4441},[60],[26,4443],{"className":4444,"style":1898},[64],[26,4446,1880],{"className":4447,"style":1912},[69,70],[26,4449,4451,4464],{"className":4450},[29],[26,4452,4454],{"className":4453},[33],[35,4455,4456],{"xmlns":37},[39,4457,4458,4462],{},[42,4459,4460],{},[45,4461,1885],{},[49,4463,2761],{"encoding":51},[26,4465,4467],{"className":4466,"ariaHidden":56},[55],[26,4468,4470,4473],{"className":4469},[60],[26,4471],{"className":4472,"style":2771},[64],[26,4474,1885],{"className":4475,"style":1922},[69,70],").",[1929,4478,4479,4552,151],{},[21,4480,4481,4482,4551],{},"Final Easting (",[26,4483,4485,4502],{"className":4484},[29],[26,4486,4488],{"className":4487},[33],[35,4489,4490],{"xmlns":37},[39,4491,4492,4500],{},[42,4493,4494],{},[368,4495,4496,4498],{},[45,4497,372],{},[45,4499,18],{},[49,4501,3157],{"encoding":51},[26,4503,4505],{"className":4504,"ariaHidden":56},[55],[26,4506,4508,4511],{"className":4507},[60],[26,4509],{"className":4510,"style":632},[64],[26,4512,4514,4517],{"className":4513},[69],[26,4515,372],{"className":4516,"style":413},[69,70],[26,4518,4520],{"className":4519},[417],[26,4521,4523,4543],{"className":4522},[421,422],[26,4524,4526,4540],{"className":4525},[426],[26,4527,4529],{"className":4528,"style":431},[430],[26,4530,4531,4534],{"style":434},[26,4532],{"className":4533,"style":439},[438],[26,4535,4537],{"className":4536},[443,444,445,446],[26,4538,18],{"className":4539},[69,70,446],[26,4541,454],{"className":4542},[453],[26,4544,4546],{"className":4545},[426],[26,4547,4549],{"className":4548,"style":461},[430],[26,4550],{},"):",[26,4553,4555,4617],{"className":4554},[29],[26,4556,4558],{"className":4557},[33],[35,4559,4560],{"xmlns":37},[39,4561,4562,4614],{},[42,4563,4564,4570,4572,4574,4576,4579,4581,4583,4589,4591,4593,4595,4601,4603,4605,4607,4609,4612],{},[368,4565,4566,4568],{},[45,4567,372],{},[45,4569,18],{},[89,4571,543],{},[89,4573,366],{"stretchy":365},[554,4575,4040],{},[89,4577,4578],{},"×",[554,4580,3308],{},[89,4582,565],{},[368,4584,4585,4587],{},[45,4586,323],{},[554,4588,572],{},[89,4590,4578],{},[554,4592,3368],{},[89,4594,565],{},[368,4596,4597,4599],{},[45,4598,323],{},[554,4600,588],{},[89,4602,4578],{},[554,4604,3313],{},[89,4606,386],{"stretchy":365},[45,4608,3991],{"mathvariant":2636},[89,4610,4611],{},"∑",[45,4613,323],{},[49,4615,4616],{"encoding":51},"E_p = (1.0623 \\times 1000 + K_2 \\times 3000 + K_3 \\times 2000) \u002F \\sum K",[26,4618,4620,4675,4696,4715,4770,4788,4843],{"className":4619,"ariaHidden":56},[55],[26,4621,4623,4626,4666,4669,4672],{"className":4622},[60],[26,4624],{"className":4625,"style":632},[64],[26,4627,4629,4632],{"className":4628},[69],[26,4630,372],{"className":4631,"style":413},[69,70],[26,4633,4635],{"className":4634},[417],[26,4636,4638,4658],{"className":4637},[421,422],[26,4639,4641,4655],{"className":4640},[426],[26,4642,4644],{"className":4643,"style":431},[430],[26,4645,4646,4649],{"style":434},[26,4647],{"className":4648,"style":439},[438],[26,4650,4652],{"className":4651},[443,444,445,446],[26,4653,18],{"className":4654},[69,70,446],[26,4656,454],{"className":4657},[453],[26,4659,4661],{"className":4660},[426],[26,4662,4664],{"className":4663,"style":461},[430],[26,4665],{},[26,4667],{"className":4668,"style":676},[123],[26,4670,543],{"className":4671},[680],[26,4673],{"className":4674,"style":676},[123],[26,4676,4678,4681,4684,4687,4690,4693],{"className":4677},[60],[26,4679],{"className":4680,"style":3328},[64],[26,4682,366],{"className":4683},[406],[26,4685,4040],{"className":4686},[69],[26,4688],{"className":4689,"style":4156},[123],[26,4691,4578],{"className":4692},[773],[26,4694],{"className":4695,"style":4156},[123],[26,4697,4699,4703,4706,4709,4712],{"className":4698},[60],[26,4700],{"className":4701,"style":4702},[64],"height:0.7278em;vertical-align:-0.0833em;",[26,4704,3308],{"className":4705},[69],[26,4707],{"className":4708,"style":4156},[123],[26,4710,565],{"className":4711},[773],[26,4713],{"className":4714,"style":4156},[123],[26,4716,4718,4721,4761,4764,4767],{"className":4717},[60],[26,4719],{"className":4720,"style":1986},[64],[26,4722,4724,4727],{"className":4723},[69],[26,4725,323],{"className":4726,"style":138},[69,70],[26,4728,4730],{"className":4729},[417],[26,4731,4733,4753],{"className":4732},[421,422],[26,4734,4736,4750],{"className":4735},[426],[26,4737,4739],{"className":4738,"style":2005},[430],[26,4740,4741,4744],{"style":2008},[26,4742],{"className":4743,"style":439},[438],[26,4745,4747],{"className":4746},[443,444,445,446],[26,4748,572],{"className":4749},[69,446],[26,4751,454],{"className":4752},[453],[26,4754,4756],{"className":4755},[426],[26,4757,4759],{"className":4758,"style":2027},[430],[26,4760],{},[26,4762],{"className":4763,"style":4156},[123],[26,4765,4578],{"className":4766},[773],[26,4768],{"className":4769,"style":4156},[123],[26,4771,4773,4776,4779,4782,4785],{"className":4772},[60],[26,4774],{"className":4775,"style":4702},[64],[26,4777,3368],{"className":4778},[69],[26,4780],{"className":4781,"style":4156},[123],[26,4783,565],{"className":4784},[773],[26,4786],{"className":4787,"style":4156},[123],[26,4789,4791,4794,4834,4837,4840],{"className":4790},[60],[26,4792],{"className":4793,"style":1986},[64],[26,4795,4797,4800],{"className":4796},[69],[26,4798,323],{"className":4799,"style":138},[69,70],[26,4801,4803],{"className":4802},[417],[26,4804,4806,4826],{"className":4805},[421,422],[26,4807,4809,4823],{"className":4808},[426],[26,4810,4812],{"className":4811,"style":2005},[430],[26,4813,4814,4817],{"style":2008},[26,4815],{"className":4816,"style":439},[438],[26,4818,4820],{"className":4819},[443,444,445,446],[26,4821,588],{"className":4822},[69,446],[26,4824,454],{"className":4825},[453],[26,4827,4829],{"className":4828},[426],[26,4830,4832],{"className":4831,"style":2027},[430],[26,4833],{},[26,4835],{"className":4836,"style":4156},[123],[26,4838,4578],{"className":4839},[773],[26,4841],{"className":4842,"style":4156},[123],[26,4844,4846,4849,4852,4855,4858,4861,4867,4870],{"className":4845},[60],[26,4847],{"className":4848,"style":3328},[64],[26,4850,3313],{"className":4851},[69],[26,4853,386],{"className":4854},[515],[26,4856,3991],{"className":4857},[69],[26,4859],{"className":4860,"style":124},[123],[26,4862,4611],{"className":4863,"style":4866},[2079,4864,4865],"op-symbol","small-op","position:relative;top:0em;",[26,4868],{"className":4869,"style":124},[123],[26,4871,323],{"className":4872,"style":138},[69,70],[1929,4874,4875,4878,4879,4907,4908,151],{},[21,4876,4877],{},"Result:"," This direct calculation yields the ",[26,4880,4882,4895],{"className":4881},[29],[26,4883,4885],{"className":4884},[33],[35,4886,4887],{"xmlns":37},[39,4888,4889,4893],{},[42,4890,4891],{},[45,4892,47],{},[49,4894,47],{"encoding":51},[26,4896,4898],{"className":4897,"ariaHidden":56},[55],[26,4899,4901,4904],{"className":4900},[60],[26,4902],{"className":4903,"style":65},[64],[26,4905,47],{"className":4906,"style":71},[69,70]," coordinates without any distance measurements required ",[26,4909,4910],{},"4, 11",[13,4912,4914],{"id":4913},"formula-breakdown-the-danger-circle","Formula Breakdown: The Danger Circle",[18,4916,4917,4918,4920,4921,4949,4950,275,4996,304,5024,151],{},"The ",[21,4919,165],{}," occurs when the unknown point ",[26,4922,4924,4937],{"className":4923},[29],[26,4925,4927],{"className":4926},[33],[35,4928,4929],{"xmlns":37},[39,4930,4931,4935],{},[42,4932,4933],{},[45,4934,47],{},[49,4936,47],{"encoding":51},[26,4938,4940],{"className":4939,"ariaHidden":56},[55],[26,4941,4943,4946],{"className":4942},[60],[26,4944],{"className":4945,"style":65},[64],[26,4947,47],{"className":4948,"style":71},[69,70]," lies on the circumcircle passing through ",[26,4951,4953,4972],{"className":4952},[29],[26,4954,4956],{"className":4955},[33],[35,4957,4958],{"xmlns":37},[39,4959,4960,4970],{},[42,4961,4962,4964,4966,4968],{},[45,4963,87],{},[89,4965,91],{"separator":56},[45,4967,94],{},[89,4969,91],{"separator":56},[49,4971,250],{"encoding":51},[26,4973,4975],{"className":4974,"ariaHidden":56},[55],[26,4976,4978,4981,4984,4987,4990,4993],{"className":4977},[60],[26,4979],{"className":4980,"style":112},[64],[26,4982,87],{"className":4983},[69,70],[26,4985,91],{"className":4986},[119],[26,4988],{"className":4989,"style":124},[123],[26,4991,94],{"className":4992,"style":128},[69,70],[26,4994,91],{"className":4995},[119],[26,4997,4999,5012],{"className":4998},[29],[26,5000,5002],{"className":5001},[33],[35,5003,5004],{"xmlns":37},[39,5005,5006,5010],{},[42,5007,5008],{},[45,5009,99],{},[49,5011,99],{"encoding":51},[26,5013,5015],{"className":5014,"ariaHidden":56},[55],[26,5016,5018,5021],{"className":5017},[60],[26,5019],{"className":5020,"style":65},[64],[26,5022,99],{"className":5023,"style":138},[69,70],[26,5025,5026],{},"2, 12",[5028,5029,5031],"callout",{"type":5030},"danger",[18,5032,5033,5034,5128,5129,5193,5194,304,5258,5261,5262,5295,5296,151],{},"Mathematically, on this circle, the sum of the angles ",[26,5035,5037,5067],{"className":5036},[29],[26,5038,5040],{"className":5039},[33],[35,5041,5042],{"xmlns":37},[39,5043,5044,5064],{},[42,5045,5046,5048,5050,5052,5054,5056,5060,5062],{},[89,5047,366],{"stretchy":365},[45,5049,1875],{},[89,5051,565],{},[45,5053,1880],{},[89,5055,565],{},[5057,5058,5059],"mtext",{},"Internal Angle ",[45,5061,94],{},[89,5063,386],{"stretchy":365},[49,5065,5066],{"encoding":51},"(\\alpha + \\beta + \\text{Internal Angle } B)",[26,5068,5070,5091,5109],{"className":5069,"ariaHidden":56},[55],[26,5071,5073,5076,5079,5082,5085,5088],{"className":5072},[60],[26,5074],{"className":5075,"style":3328},[64],[26,5077,366],{"className":5078},[406],[26,5080,1875],{"className":5081,"style":1902},[69,70],[26,5083],{"className":5084,"style":4156},[123],[26,5086,565],{"className":5087},[773],[26,5089],{"className":5090,"style":4156},[123],[26,5092,5094,5097,5100,5103,5106],{"className":5093},[60],[26,5095],{"className":5096,"style":1898},[64],[26,5098,1880],{"className":5099,"style":1912},[69,70],[26,5101],{"className":5102,"style":4156},[123],[26,5104,565],{"className":5105},[773],[26,5107],{"className":5108,"style":4156},[123],[26,5110,5112,5115,5122,5125],{"className":5111},[60],[26,5113],{"className":5114,"style":3328},[64],[26,5116,5119],{"className":5117},[69,5118],"text",[26,5120,5059],{"className":5121},[69],[26,5123,94],{"className":5124,"style":128},[69,70],[26,5126,386],{"className":5127},[515]," equals ",[26,5130,5132,5151],{"className":5131},[29],[26,5133,5135],{"className":5134},[33],[35,5136,5137],{"xmlns":37},[39,5138,5139,5148],{},[42,5140,5141],{},[3513,5142,5143,5146],{},[554,5144,5145],{},"180",[89,5147,3520],{},[49,5149,5150],{"encoding":51},"180^\\circ",[26,5152,5154],{"className":5153,"ariaHidden":56},[55],[26,5155,5157,5160,5164],{"className":5156},[60],[26,5158],{"className":5159,"style":3551},[64],[26,5161,5163],{"className":5162},[69],"18",[26,5165,5167,5170],{"className":5166},[69],[26,5168,3562],{"className":5169},[69],[26,5171,5173],{"className":5172},[417],[26,5174,5176],{"className":5175},[421],[26,5177,5179],{"className":5178},[426],[26,5180,5182],{"className":5181,"style":3551},[430],[26,5183,5184,5187],{"style":3577},[26,5185],{"className":5186,"style":439},[438],[26,5188,5190],{"className":5189},[443,444,445,446],[26,5191,3520],{"className":5192},[773,446]," or ",[26,5195,5197,5216],{"className":5196},[29],[26,5198,5200],{"className":5199},[33],[35,5201,5202],{"xmlns":37},[39,5203,5204,5213],{},[42,5205,5206],{},[3513,5207,5208,5211],{},[554,5209,5210],{},"360",[89,5212,3520],{},[49,5214,5215],{"encoding":51},"360^\\circ",[26,5217,5219],{"className":5218,"ariaHidden":56},[55],[26,5220,5222,5225,5229],{"className":5221},[60],[26,5223],{"className":5224,"style":3551},[64],[26,5226,5228],{"className":5227},[69],"36",[26,5230,5232,5235],{"className":5231},[69],[26,5233,3562],{"className":5234},[69],[26,5236,5238],{"className":5237},[417],[26,5239,5241],{"className":5240},[421],[26,5242,5244],{"className":5243},[426],[26,5245,5247],{"className":5246,"style":3551},[430],[26,5248,5249,5252],{"style":3577},[26,5250],{"className":5251,"style":439},[438],[26,5253,5255],{"className":5254},[443,444,445,446],[26,5256,3520],{"className":5257},[773,446],[26,5259,5260],{},"2, 13",". In this configuration, the denominators in the Tienstra weights approach zero, and the coordinates become indeterminate (",[26,5263,5265,5283],{"className":5264},[29],[26,5266,5268],{"className":5267},[33],[35,5269,5270],{"xmlns":37},[39,5271,5272,5280],{},[42,5273,5274,5276,5278],{},[554,5275,3562],{},[45,5277,3991],{"mathvariant":2636},[554,5279,3562],{},[49,5281,5282],{"encoding":51},"0\u002F0",[26,5284,5286],{"className":5285,"ariaHidden":56},[55],[26,5287,5289,5292],{"className":5288},[60],[26,5290],{"className":5291,"style":3328},[64],[26,5293,5282],{"className":5294},[69],"). Your position is mathematically \"lost\" ",[26,5297,169],{},[13,5299,5301],{"id":5300},"practical-tips","Practical Tips",[1926,5303,5304,5418,5427],{},[1929,5305,5306,5309,5310,304,5338,5341,5342,275,5388,304,5416,151],{},[21,5307,5308],{},"Inside is Safest:"," To physically guarantee you avoid the Danger Circle, always site your station ",[26,5311,5313,5326],{"className":5312},[29],[26,5314,5316],{"className":5315},[33],[35,5317,5318],{"xmlns":37},[39,5319,5320,5324],{},[42,5321,5322],{},[45,5323,47],{},[49,5325,47],{"encoding":51},[26,5327,5329],{"className":5328,"ariaHidden":56},[55],[26,5330,5332,5335],{"className":5331},[60],[26,5333],{"className":5334,"style":65},[64],[26,5336,47],{"className":5337,"style":71},[69,70],[21,5339,5340],{},"inside"," the triangle formed by ",[26,5343,5345,5364],{"className":5344},[29],[26,5346,5348],{"className":5347},[33],[35,5349,5350],{"xmlns":37},[39,5351,5352,5362],{},[42,5353,5354,5356,5358,5360],{},[45,5355,87],{},[89,5357,91],{"separator":56},[45,5359,94],{},[89,5361,91],{"separator":56},[49,5363,250],{"encoding":51},[26,5365,5367],{"className":5366,"ariaHidden":56},[55],[26,5368,5370,5373,5376,5379,5382,5385],{"className":5369},[60],[26,5371],{"className":5372,"style":112},[64],[26,5374,87],{"className":5375},[69,70],[26,5377,91],{"className":5378},[119],[26,5380],{"className":5381,"style":124},[123],[26,5383,94],{"className":5384,"style":128},[69,70],[26,5386,91],{"className":5387},[119],[26,5389,5391,5404],{"className":5390},[29],[26,5392,5394],{"className":5393},[33],[35,5395,5396],{"xmlns":37},[39,5397,5398,5402],{},[42,5399,5400],{},[45,5401,99],{},[49,5403,99],{"encoding":51},[26,5405,5407],{"className":5406,"ariaHidden":56},[55],[26,5408,5410,5413],{"className":5409},[60],[26,5411],{"className":5412,"style":65},[64],[26,5414,99],{"className":5415,"style":138},[69,70],[26,5417,572],{},[1929,5419,5420,5423,5424,151],{},[21,5421,5422],{},"4th Point Redundancy:"," Always observe a fourth control point if available. This allows you to compute a second resection as a check and provides a statistical measure of error ",[26,5425,5426],{},"2, 6",[1929,5428,5429,5432,5433,5562,5563,151],{},[21,5430,5431],{},"Check Your Sums:"," Ensure your measured angles ",[26,5434,5436,5466],{"className":5435},[29],[26,5437,5439],{"className":5438},[33],[35,5440,5441],{"xmlns":37},[39,5442,5443,5463],{},[42,5444,5445,5447,5449,5451,5453,5455,5457],{},[45,5446,1875],{},[89,5448,565],{},[45,5450,1880],{},[89,5452,565],{},[45,5454,1885],{},[89,5456,543],{},[3513,5458,5459,5461],{},[554,5460,5210],{},[89,5462,3520],{},[49,5464,5465],{"encoding":51},"\\alpha + \\beta + \\gamma = 360^\\circ",[26,5467,5469,5488,5506,5524],{"className":5468,"ariaHidden":56},[55],[26,5470,5472,5476,5479,5482,5485],{"className":5471},[60],[26,5473],{"className":5474,"style":5475},[64],"height:0.6667em;vertical-align:-0.0833em;",[26,5477,1875],{"className":5478,"style":1902},[69,70],[26,5480],{"className":5481,"style":4156},[123],[26,5483,565],{"className":5484},[773],[26,5486],{"className":5487,"style":4156},[123],[26,5489,5491,5494,5497,5500,5503],{"className":5490},[60],[26,5492],{"className":5493,"style":1898},[64],[26,5495,1880],{"className":5496,"style":1912},[69,70],[26,5498],{"className":5499,"style":4156},[123],[26,5501,565],{"className":5502},[773],[26,5504],{"className":5505,"style":4156},[123],[26,5507,5509,5512,5515,5518,5521],{"className":5508},[60],[26,5510],{"className":5511,"style":2771},[64],[26,5513,1885],{"className":5514,"style":1922},[69,70],[26,5516],{"className":5517,"style":676},[123],[26,5519,543],{"className":5520},[680],[26,5522],{"className":5523,"style":676},[123],[26,5525,5527,5530,5533],{"className":5526},[60],[26,5528],{"className":5529,"style":3551},[64],[26,5531,5228],{"className":5532},[69],[26,5534,5536,5539],{"className":5535},[69],[26,5537,3562],{"className":5538},[69],[26,5540,5542],{"className":5541},[417],[26,5543,5545],{"className":5544},[421],[26,5546,5548],{"className":5547},[426],[26,5549,5551],{"className":5550,"style":3551},[430],[26,5552,5553,5556],{"style":3577},[26,5554],{"className":5555,"style":439},[438],[26,5557,5559],{"className":5558},[443,444,445,446],[26,5560,3520],{"className":5561},[773,446],". Any deviation indicates a gross observation error or a misidentified target ",[26,5564,5565],{},"9",[13,5567,5569],{"id":5568},"common-mistakes","Common Mistakes",[1926,5571,5572,5638],{},[1929,5573,5574,5577,5578,5606,5607,5635,5636,151],{},[21,5575,5576],{},"Sign Confusion:"," When ",[26,5579,5581,5594],{"className":5580},[29],[26,5582,5584],{"className":5583},[33],[35,5585,5586],{"xmlns":37},[39,5587,5588,5592],{},[42,5589,5590],{},[45,5591,47],{},[49,5593,47],{"encoding":51},[26,5595,5597],{"className":5596,"ariaHidden":56},[55],[26,5598,5600,5603],{"className":5599},[60],[26,5601],{"className":5602,"style":65},[64],[26,5604,47],{"className":5605,"style":71},[69,70]," is outside the control triangle, one or more ",[26,5608,5610,5623],{"className":5609},[29],[26,5611,5613],{"className":5612},[33],[35,5614,5615],{"xmlns":37},[39,5616,5617,5621],{},[42,5618,5619],{},[45,5620,323],{},[49,5622,323],{"encoding":51},[26,5624,5626],{"className":5625,"ariaHidden":56},[55],[26,5627,5629,5632],{"className":5628},[60],[26,5630],{"className":5631,"style":65},[64],[26,5633,323],{"className":5634,"style":138},[69,70]," values may become negative. Ensure you maintain the correct algebraic signs throughout the summation ",[26,5637,341],{},[1929,5639,5640,5643,5644,5647,5648,151],{},[21,5641,5642],{},"Near-Circle Geometry:"," Even if you aren't ",[2588,5645,5646],{},"exactly"," on the Danger Circle, being near it results in very small denominators, which massively magnifies small angular measurement errors ",[26,5649,572],{},[13,5651,5653],{"id":5652},"best-practices","Best Practices",[18,5655,5656,5657,5685,5686,304,5689,151],{},"If your reconnaissance shows that you must set up outside the control triangle, perform a quick field check: if the angle at ",[26,5658,5660,5673],{"className":5659},[29],[26,5661,5663],{"className":5662},[33],[35,5664,5665],{"xmlns":37},[39,5666,5667,5671],{},[42,5668,5669],{},[45,5670,47],{},[49,5672,47],{"encoding":51},[26,5674,5676],{"className":5675,"ariaHidden":56},[55],[26,5677,5679,5682],{"className":5678},[60],[26,5680],{"className":5681,"style":65},[64],[26,5683,47],{"className":5684,"style":71},[69,70]," to the two \"outside\" points is roughly equal to the internal angle of the triangle at the \"middle\" point, you are near the danger circle and ",[21,5687,5688],{},"must reposition",[26,5690,572],{},[13,5692,5694],{"id":5693},"faq","FAQ",[5696,5697,5698,5737,5745],"accordion",{},[5699,5700,5702],"accordion-item",{"label":5701},"Why use Tienstra instead of the Cotangent method?",[18,5703,5704,5705,5733,5734,151],{},"The Tienstra formula is more \"elegant\" because it uses the same weights (",[26,5706,5708,5721],{"className":5707},[29],[26,5709,5711],{"className":5710},[33],[35,5712,5713],{"xmlns":37},[39,5714,5715,5719],{},[42,5716,5717],{},[45,5718,323],{},[49,5720,323],{"encoding":51},[26,5722,5724],{"className":5723,"ariaHidden":56},[55],[26,5725,5727,5730],{"className":5726},[60],[26,5728],{"className":5729,"style":65},[64],[26,5731,323],{"className":5732,"style":138},[69,70],") for both Easting and Northing, reducing the potential for calculation errors compared to the traditional cotangent method ",[26,5735,5736],{},"1, 14",[5699,5738,5740],{"label":5739},"Can I use Tienstra with a Total Station?",[18,5741,5742,5743,151],{},"Yes. While most Total Stations have built-in \"Free Station\" routines (which often use Least Squares), understanding Tienstra allows you to manually verify a position if the internal software is suspect ",[26,5744,572],{},[5699,5746,5748],{"label":5747},"Does this work in 3D?",[18,5749,5750,5751,151],{},"Tienstra is a 2D (horizontal) resection method. Vertical position is typically determined separately through trigonometrical levelling after the horizontal position is fixed ",[26,5752,5753],{},"15",[13,5755,5757],{"id":5756},"conclusion","Conclusion",[18,5759,5760],{},"The Tienstra method remains a masterclass in geodetic mathematics, offering a direct path from angular field data to coordinate mapping. By understanding the barycentric principles and respecting the \"trap\" of the Danger Circle, surveyors can utilize this method to establish high-precision site control in even the most challenging environments.",[13,5762,5764],{"id":5763},"references","References",[18,5766,5767,5768,5771,5772,5775,5776,5779],{},"Schofield, W. (2001). ",[2588,5769,5770],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.\nAllan, A. L. (1969). ",[2588,5773,5774],{},"The Tienstra Method of Resection",".\nPorta, J. M., & Thomas, F. (2009). ",[2588,5777,5778],{},"Concise Proof of Tienstra’s Formula",". Journal of Surveying Engineering.",{"title":5781,"searchDepth":5782,"depth":5782,"links":5783},"",2,[5784,5785,5786,5787,5788,5793,5799,5800,5801,5802,5803,5804,5805,5806],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":172,"depth":5782,"text":173},{"id":194,"depth":5782,"text":195},{"id":344,"depth":5782,"text":345,"children":5789},[5790],{"id":1740,"depth":5791,"text":5792},3,"Defining the Weights (KKK)",{"id":2824,"depth":5782,"text":2825,"children":5794},[5795,5796,5797,5798],{"id":2831,"depth":5791,"text":2832},{"id":2924,"depth":5791,"text":2925},{"id":2963,"depth":5791,"text":2964},{"id":3103,"depth":5791,"text":3104},{"id":3282,"depth":5782,"text":3283},{"id":4913,"depth":5782,"text":4914},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5652,"depth":5782,"text":5653},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Technical Deep-Dive","The Tienstra method is one of the most elegant solutions in surveying mathematics — but it carries a geometric trap that has caused field problems for generations of surveyors. Understanding the danger circle is not optional.","md","https:\u002F\u002Fimages.unsplash.com\u002Fphoto-1503387762-592deb58ef4e?w=800&q=80",{"authorCredentials":5812,"authorBio":5813,"commentCount":5814,"tags":5815,"relatedTools":5821,"relatedArticles":5823,"sources":5826},"FInstCES","Principal engineer with 35 years in engineering surveying. Specialist in structural monitoring and high-precision setting out.",23,[5816,5817,5818,5819,5820],"tienstra","resection","COGO","danger-circle","geometry",[5822],"tienstra-resection",[5824,5825],"coordinate-computation","traversing-control-networks",[5827,5831,5835],{"title":5828,"type":5829,"note":5830},"Engineering Surveying (Fifth Edition)","Book","Source for general resection theory and the Circle of Danger.",{"title":5832,"type":5833,"note":5834},"THE TIENSTRA METHOD OF RESECTION","PDF","Primary source for the barycentric formula and mathematical proof.",{"title":5836,"type":5837,"note":5838},"Tienstra formula - Wikipedia","Webpage","Source for the standard coordinate weighting equations.",true,"\u002Fblog\u002Ftienstra-resection-danger-circle","2026-01-15",18,{"title":7,"description":5808},{"loc":5840},"blog\u002F23.tienstra-resection-danger-circle","vfTDzY5VqLXyWF5-KefXoBjAx9vnDBViLbypT6I3pt8",{"id":5848,"title":5849,"author":5850,"body":5851,"category":6395,"description":6396,"extension":5809,"image":6397,"meta":6398,"navigation":5839,"path":6416,"publishedAt":6417,"readTime":6418,"seo":6419,"sitemap":6420,"stem":6421,"__hash__":6422},"blog\u002Fblog\u002F28.inertial-surveying-systems-iss.md","Inertial Surveying Systems (ISS): High-Speed Positioning Without Satellites","Surveyiiing Expert",{"type":10,"value":5852,"toc":6375},[5853,5855,5862,5864,5871,5873,5884,5973,5975,5982,5986,5992,6009,6011,6093,6097,6266,6268,6325,6327,6341,6343,6363,6365,6368,6370],[13,5854,16],{"id":15},[18,5856,5857,5858,5861],{},"While GPS is the dominant positioning tool today, ",[21,5859,5860],{},"Inertial Surveying Systems (ISS)"," offer a unique advantage: they do not require external signals from satellites or base stations. Originally developed for military and aerospace navigation, ISS is used in surveying for high-speed point positioning, road inventory, and projects in areas where GPS signals are blocked .",[13,5863,155],{"id":154},[18,5865,5866,5867,5870],{},"ISS is completely self-contained and computerized, making it virtually free from human error during data capture. It can fix hundreds of points in the time it takes a traditional team to fix one. Its primary drawback is its high cost and the tendency for its precision to drift over time, requiring a specialized field procedure known as a ",[21,5868,5869],{},"ZUPT"," .",[13,5872,195],{"id":194},[18,5874,5875,5876,5879,5880,5883],{},"An ISS consists of a triad of ",[21,5877,5878],{},"accelerometers"," (to measure movement) and a triad of ",[21,5881,5882],{},"gyroscopes"," (to maintain orientation) mounted on a stabilized platform.",[3763,5885,5886,5961,5967],{},[1929,5887,5888,5891,5892,5960],{},[21,5889,5890],{},"Accelerometers:"," Detect changes in velocity along the ",[26,5893,5895,5923],{"className":5894},[29],[26,5896,5898],{"className":5897},[33],[35,5899,5900],{"xmlns":37},[39,5901,5902,5920],{},[42,5903,5904,5907,5909,5912,5914,5917],{},[45,5905,5906],{},"X",[89,5908,91],{"separator":56},[45,5910,5911],{},"Y",[89,5913,91],{"separator":56},[5057,5915,5916],{}," and ",[45,5918,5919],{},"Z",[49,5921,5922],{"encoding":51},"X, Y, \\text{ and } Z",[26,5924,5926],{"className":5925,"ariaHidden":56},[55],[26,5927,5929,5932,5936,5939,5942,5945,5948,5951,5957],{"className":5928},[60],[26,5930],{"className":5931,"style":1898},[64],[26,5933,5906],{"className":5934,"style":5935},[69,70],"margin-right:0.0785em;",[26,5937,91],{"className":5938},[119],[26,5940],{"className":5941,"style":124},[123],[26,5943,5911],{"className":5944,"style":4156},[69,70],[26,5946,91],{"className":5947},[119],[26,5949],{"className":5950,"style":124},[123],[26,5952,5954],{"className":5953},[69,5118],[26,5955,5916],{"className":5956},[69],[26,5958,5919],{"className":5959,"style":138},[69,70]," axes .",[1929,5962,5963,5966],{},[21,5964,5965],{},"Gyroscopes:"," Keep the platform aligned with a specific coordinate system (usually North, East, and the Local Vertical) .",[1929,5968,5969,5972],{},[21,5970,5971],{},"Integration:"," The system's computer integrates acceleration over time to find velocity, and integrates velocity to find the change in distance and position .",[13,5974,345],{"id":344},[18,5976,5977,5978,5981],{},"The system uses ",[21,5979,5980],{},"Kalman Filtering",", a statistical algorithm that continuously estimates the state of the system and corrects for small errors in the sensors based on redundant data .",[13,5983,5985],{"id":5984},"the-zero-velocity-update-zupt","The Zero Velocity Update (ZUPT)",[18,5987,5988,5989,5991],{},"The most critical procedure in ISS is the ",[21,5990,5869],{},". Because sensors are not perfect, small errors accumulate as \"drift.\"",[1926,5993,5994,6003],{},[1929,5995,5996,5999,6000,5870],{},[21,5997,5998],{},"Procedure:"," The vehicle carrying the ISS must stop every ",[21,6001,6002],{},"3–5 minutes",[1929,6004,6005,6008],{},[21,6006,6007],{},"Function:"," During the stop, any movement detected by the system is known to be error (since the vehicle is actually stationary). The computer identifies this drift and resets the velocity to zero, effectively cleaning the data .",[13,6010,2825],{"id":2824},[2827,6012,6013,6017,6065,6069,6072,6076,6079,6083,6086,6090],{},[1738,6014,6016],{"id":6015},"initialization","Initialization",[18,6018,6019,6020,6064],{},"The system is set up at a known control point. It takes roughly ",[26,6021,6023,6043],{"className":6022},[29],[26,6024,6026],{"className":6025},[33],[35,6027,6028],{"xmlns":37},[39,6029,6030,6040],{},[42,6031,6032,6035,6038],{},[554,6033,6034],{},"30",[5057,6036,6037],{},"–",[554,6039,3842],{},[49,6041,6042],{"encoding":51},"30\\text{–}60",[26,6044,6046],{"className":6045,"ariaHidden":56},[55],[26,6047,6049,6052,6055,6061],{"className":6048},[60],[26,6050],{"className":6051,"style":4270},[64],[26,6053,6034],{"className":6054},[69],[26,6056,6058],{"className":6057},[69,5118],[26,6059,6037],{"className":6060},[69],[26,6062,3842],{"className":6063},[69]," minutes to \"align\" itself with the Earth's rotation and local gravity .",[1738,6066,6068],{"id":6067},"movement","Movement",[18,6070,6071],{},"The vehicle travels along the survey route at a uniform time rate to optimize the Kalman filtering .",[1738,6073,6075],{"id":6074},"periodic-zupts","Periodic ZUPTs",[18,6077,6078],{},"Perform a ZUPT every few minutes. Failure to do so will cause the positional error to grow exponentially .",[1738,6080,6082],{"id":6081},"closure","Closure",[18,6084,6085],{},"The survey must close on another known control point. The system then runs back to the beginning to verify the results .",[1738,6087,6089],{"id":6088},"post-processing","Post-Processing",[18,6091,6092],{},"The data is analyzed to distribute residual errors throughout the traverse, similar to a Bowditch adjustment but much more complex .",[13,6094,6096],{"id":6095},"accuracy-and-performance","Accuracy and Performance",[1926,6098,6099,6184,6230],{},[1929,6100,6101,6104,6105,6144,6145,6183],{},[21,6102,6103],{},"Typical Precision:"," Approximately ",[26,6106,6108,6126],{"className":6107},[29],[26,6109,6111],{"className":6110},[33],[35,6112,6113],{"xmlns":37},[39,6114,6115,6123],{},[42,6116,6117,6120],{},[554,6118,6119],{},"200",[5057,6121,6122],{}," mm",[49,6124,6125],{"encoding":51},"200\\text{ mm}",[26,6127,6129],{"className":6128,"ariaHidden":56},[55],[26,6130,6132,6135,6138],{"className":6131},[60],[26,6133],{"className":6134,"style":4270},[64],[26,6136,6119],{"className":6137},[69],[26,6139,6141],{"className":6140},[69,5118],[26,6142,6122],{"className":6143},[69]," in plan and ",[26,6146,6148,6165],{"className":6147},[29],[26,6149,6151],{"className":6150},[33],[35,6152,6153],{"xmlns":37},[39,6154,6155,6162],{},[42,6156,6157,6160],{},[554,6158,6159],{},"100",[5057,6161,6122],{},[49,6163,6164],{"encoding":51},"100\\text{ mm}",[26,6166,6168],{"className":6167,"ariaHidden":56},[55],[26,6169,6171,6174,6177],{"className":6170},[60],[26,6172],{"className":6173,"style":4270},[64],[26,6175,6159],{"className":6176},[69],[26,6178,6180],{"className":6179},[69,5118],[26,6181,6122],{"className":6182},[69]," in elevation .",[1929,6185,6186,6189,6190,6229],{},[21,6187,6188],{},"Enhanced Accuracy:"," By shortening the interval between ZUPTs and reducing the total survey time to less than 2 hours, accuracies in the region of ",[21,6191,6192],{},[26,6193,6195,6211],{"className":6194},[29],[26,6196,6198],{"className":6197},[33],[35,6199,6200],{"xmlns":37},[39,6201,6202,6208],{},[42,6203,6204,6206],{},[554,6205,3761],{},[5057,6207,6122],{},[49,6209,6210],{"encoding":51},"10\\text{ mm}",[26,6212,6214],{"className":6213,"ariaHidden":56},[55],[26,6215,6217,6220,6223],{"className":6216},[60],[26,6218],{"className":6219,"style":4270},[64],[26,6221,3761],{"className":6222},[69],[26,6224,6226],{"className":6225},[69,5118],[26,6227,6122],{"className":6228},[69]," have been achieved .",[1929,6231,6232,6235,6236,6265],{},[21,6233,6234],{},"Speed:"," Point positioning can be up to ",[26,6237,6239,6253],{"className":6238},[29],[26,6240,6242],{"className":6241},[33],[35,6243,6244],{"xmlns":37},[39,6245,6246,6251],{},[42,6247,6248],{},[554,6249,6250],{},"20",[49,6252,6250],{"encoding":51},[26,6254,6256],{"className":6255,"ariaHidden":56},[55],[26,6257,6259,6262],{"className":6258},[60],[26,6260],{"className":6261,"style":4270},[64],[26,6263,6250],{"className":6264},[69]," times quicker than conventional methods .",[13,6267,5301],{"id":5300},[1926,6269,6270,6276],{},[1929,6271,6272,6275],{},[21,6273,6274],{},"Avoid Bumpy Roads:"," Rapid accelerations and sudden direction changes increase orientation errors. Smooth, straight routes are ideal .",[1929,6277,6278,6281,6282,6324],{},[21,6279,6280],{},"Frequent Checks:"," Include known reference points every ",[26,6283,6285,6303],{"className":6284},[29],[26,6286,6288],{"className":6287},[33],[35,6289,6290],{"xmlns":37},[39,6291,6292,6300],{},[42,6293,6294,6296,6298],{},[554,6295,341],{},[5057,6297,6037],{},[554,6299,572],{},[49,6301,6302],{"encoding":51},"1\\text{–}2",[26,6304,6306],{"className":6305,"ariaHidden":56},[55],[26,6307,6309,6312,6315,6321],{"className":6308},[60],[26,6310],{"className":6311,"style":4270},[64],[26,6313,341],{"className":6314},[69],[26,6316,6318],{"className":6317},[69,5118],[26,6319,6037],{"className":6320},[69],[26,6322,572],{"className":6323},[69]," hours to provide external validation of the system's performance .",[13,6326,5569],{"id":5568},[1926,6328,6329,6335],{},[1929,6330,6331,6334],{},[21,6332,6333],{},"Missing a ZUPT:"," Neglecting the 3–5 minute stop is the most common cause of survey failure in ISS.",[1929,6336,6337,6340],{},[21,6338,6339],{},"Temperature Gradients:"," Like gyro-theodolites, ISS components are sensitive to thermal changes. Ensure the system is adequately warmed up and shielded from direct sunlight .",[13,6342,5694],{"id":5693},[5696,6344,6345,6351,6357],{},[5699,6346,6348],{"label":6347},"How does ISS differ from GPS?",[18,6349,6350],{},"GPS requires a line of sight to satellites; ISS is self-contained. GPS errors are generally constant with time; ISS errors increase with time (drift) unless ZUPTs are performed .",[5699,6352,6354],{"label":6353},"Can ISS be used in tunnels?",[18,6355,6356],{},"Yes. This is one of its primary advantages. Once initialized at the tunnel portal, it can maintain its position as it travels deep underground .",[5699,6358,6360],{"label":6359},"What does 'Kalman Filter' do?",[18,6361,6362],{},"It is a mathematical algorithm that combines the current measurement with previous predictions to provide the \"most probable\" current position while accounting for sensor noise .",[13,6364,5757],{"id":5756},[18,6366,6367],{},"Inertial Surveying Systems represent the high-water mark of automated positioning. While expensive, their ability to operate in any weather and without external signals makes them a powerful tool for large-scale infrastructure and rapid topographic detailing.",[13,6369,5764],{"id":5763},[18,6371,5767,6372,6374],{},[2588,6373,5770],{},". 5th ed. Butterworth-Heinemann.",{"title":5781,"searchDepth":5782,"depth":5782,"links":6376},[6377,6378,6379,6380,6381,6382,6389,6390,6391,6392,6393,6394],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":194,"depth":5782,"text":195},{"id":344,"depth":5782,"text":345},{"id":5984,"depth":5782,"text":5985},{"id":2824,"depth":5782,"text":2825,"children":6383},[6384,6385,6386,6387,6388],{"id":6015,"depth":5791,"text":6016},{"id":6067,"depth":5791,"text":6068},{"id":6074,"depth":5791,"text":6075},{"id":6081,"depth":5791,"text":6082},{"id":6088,"depth":5791,"text":6089},{"id":6095,"depth":5782,"text":6096},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Instrumentation","A technical look at Inertial Surveying Systems, including the use of accelerometers, gyroscopes, and the critical Zero Velocity Update (ZUPT) procedure.","\u002Fimages\u002Finertial-surveying.jpg",{"authorCredentials":6399,"commentCount":6400,"tags":6401,"relatedTools":6407,"relatedArticles":6410,"sources":6413},"LS, P.Eng",0,[6402,6403,6404,6405,6406],"ISS","Inertial Navigation","Gyroscopes","Accelerometers","Point Positioning",[6408,6409],"Inertial Platform","Kalman Filter",[6411,6412],"gyro-theodolite-underground-orientation","rtk-gps-real-time-kinematic",[6414],{"title":5828,"type":5829,"note":6415},"Primary source for ISS theory and ZUPT procedures.","\u002Fblog\u002Finertial-surveying-systems-iss","2024-04-15",12,{"title":5849,"description":6396},{"loc":6416},"blog\u002F28.inertial-surveying-systems-iss","unwErSnqBgSQfTIS9T-85LaYj2W2duaD3g4Nc2KWIsM",{"id":6424,"title":6425,"author":5850,"body":6426,"category":7103,"description":7104,"extension":5809,"image":7105,"meta":7106,"navigation":5839,"path":7123,"publishedAt":7124,"readTime":6418,"seo":7125,"sitemap":7126,"stem":7127,"__hash__":7128},"blog\u002Fblog\u002F27.laser-safety-and-management-on-site.md","Laser Safety: Hazard Classes and Management on the Construction Site",{"type":10,"value":6427,"toc":7081},[6428,6430,6433,6435,6438,6440,6446,6450,6454,6676,6680,6683,6687,6690,6924,6928,6931,6945,6947,7016,7018,7032,7034,7048,7050,7070,7072,7075,7077],[13,6429,16],{"id":15},[18,6431,6432],{},"Lasers are indispensable tools for modern setting out, from simple levelling to high-precision tunnel alignment. However, because they produce intense, coherent beams of light, they pose significant risks to eyesight. Surveyors must adhere to strict safety codes (such as BS4803 in the UK) to protect both personnel and the public .",[13,6434,155],{"id":154},[18,6436,6437],{},"A laser beam concentrated by a telescope or even viewed directly can cause permanent blindness in milliseconds. In engineering surveying, the hazard is compounded because beams are often at head height and can be reflected off shiny surfaces (specular reflections) like windows or mirrors .",[13,6439,173],{"id":172},[18,6441,6442,6443,5870],{},"The Health and Safety at Work Act (1974) and the RICS Laser Safety Code dictate how lasers are managed on site. Every site using hazardous lasers must appoint a ",[21,6444,6445],{},"Laser Safety Officer (LSO)",[13,6447,6449],{"id":6448},"theory-of-laser-hazards","Theory of Laser Hazards",[1738,6451,6453],{"id":6452},"_1-beam-intensity","1. Beam Intensity",[18,6455,6456,6457,6496,6497,6535,6536,6607,6608,6675],{},"The power of a commercial survey laser is typically small (around ",[26,6458,6460,6478],{"className":6459},[29],[26,6461,6463],{"className":6462},[33],[35,6464,6465],{"xmlns":37},[39,6466,6467,6475],{},[42,6468,6469,6472],{},[554,6470,6471],{},"5",[5057,6473,6474],{}," mW",[49,6476,6477],{"encoding":51},"5 \\text{ mW}",[26,6479,6481],{"className":6480,"ariaHidden":56},[55],[26,6482,6484,6487,6490],{"className":6483},[60],[26,6485],{"className":6486,"style":65},[64],[26,6488,6471],{"className":6489},[69],[26,6491,6493],{"className":6492},[69,5118],[26,6494,6474],{"className":6495},[69],"), but its intensity is high. A ",[26,6498,6500,6517],{"className":6499},[29],[26,6501,6503],{"className":6502},[33],[35,6504,6505],{"xmlns":37},[39,6506,6507,6514],{},[42,6508,6509,6511],{},[554,6510,572],{},[5057,6512,6513],{}," cm",[49,6515,6516],{"encoding":51},"2 \\text{ cm}",[26,6518,6520],{"className":6519,"ariaHidden":56},[55],[26,6521,6523,6526,6529],{"className":6522},[60],[26,6524],{"className":6525,"style":4270},[64],[26,6527,572],{"className":6528},[69],[26,6530,6532],{"className":6531},[69,5118],[26,6533,6513],{"className":6534},[69]," diameter beam has an intensity of ",[26,6537,6539,6560],{"className":6538},[29],[26,6540,6542],{"className":6541},[33],[35,6543,6544],{"xmlns":37},[39,6545,6546,6557],{},[42,6547,6548,6550],{},[554,6549,3729],{},[3513,6551,6552,6555],{},[5057,6553,6554],{}," mW\u002Fcm",[554,6556,572],{},[49,6558,6559],{"encoding":51},"13 \\text{ mW\u002Fcm}^2",[26,6561,6563],{"className":6562,"ariaHidden":56},[55],[26,6564,6566,6570,6573],{"className":6565},[60],[26,6567],{"className":6568,"style":6569},[64],"height:1.204em;vertical-align:-0.25em;",[26,6571,3729],{"className":6572},[69],[26,6574,6576,6582],{"className":6575},[69],[26,6577,6579],{"className":6578},[69,5118],[26,6580,6554],{"className":6581},[69],[26,6583,6585],{"className":6584},[417],[26,6586,6588],{"className":6587},[421],[26,6589,6591],{"className":6590},[426],[26,6592,6595],{"className":6593,"style":6594},[430],"height:0.954em;",[26,6596,6598,6601],{"style":6597},"top:-3.2029em;margin-right:0.05em;",[26,6599],{"className":6600,"style":439},[438],[26,6602,6604],{"className":6603},[443,444,445,446],[26,6605,572],{"className":6606},[69,446],", which is significantly higher than direct sunlight (",[26,6609,6611,6631],{"className":6610},[29],[26,6612,6614],{"className":6613},[33],[35,6615,6616],{"xmlns":37},[39,6617,6618,6628],{},[42,6619,6620,6622],{},[554,6621,6159],{},[3513,6623,6624,6626],{},[5057,6625,6554],{},[554,6627,572],{},[49,6629,6630],{"encoding":51},"100 \\text{ mW\u002Fcm}^2",[26,6632,6634],{"className":6633,"ariaHidden":56},[55],[26,6635,6637,6640,6643],{"className":6636},[60],[26,6638],{"className":6639,"style":6569},[64],[26,6641,6159],{"className":6642},[69],[26,6644,6646,6652],{"className":6645},[69],[26,6647,6649],{"className":6648},[69,5118],[26,6650,6554],{"className":6651},[69],[26,6653,6655],{"className":6654},[417],[26,6656,6658],{"className":6657},[421],[26,6659,6661],{"className":6660},[426],[26,6662,6664],{"className":6663,"style":6594},[430],[26,6665,6666,6669],{"style":6597},[26,6667],{"className":6668,"style":439},[438],[26,6670,6672],{"className":6671},[443,444,445,446],[26,6673,572],{"className":6674},[69,446],") because the laser does not scatter like ordinary light .",[1738,6677,6679],{"id":6678},"_2-specular-reflection","2. Specular Reflection",[18,6681,6682],{},"This occurs when the beam hits a mirror-like surface. The reflected beam can be just as powerful as the original and may be directed into the eyes of unsuspecting workers .",[13,6684,6686],{"id":6685},"laser-classifications-bs4803","Laser Classifications (BS4803)",[18,6688,6689],{},"Only three classes are generally relevant to on-site work :",[6691,6692,6693,6710],"table",{},[6694,6695,6696],"thead",{},[6697,6698,6699,6704,6707],"tr",{},[6700,6701,6703],"th",{"align":6702},"left","Class",[6700,6705,6706],{"align":6702},"Description",[6700,6708,6709],{"align":6702},"Risk Level",[6711,6712,6713,6764,6842],"tbody",{},[6697,6714,6715,6721,6761],{},[6716,6717,6718],"td",{"align":6702},[21,6719,6720],{},"Class 2",[6716,6722,6723,6724,151],{"align":6702},"Visible radiant power of ",[26,6725,6727,6743],{"className":6726},[29],[26,6728,6730],{"className":6729},[33],[35,6731,6732],{"xmlns":37},[39,6733,6734,6740],{},[42,6735,6736,6738],{},[554,6737,341],{},[5057,6739,6474],{},[49,6741,6742],{"encoding":51},"1 \\text{ mW}",[26,6744,6746],{"className":6745,"ariaHidden":56},[55],[26,6747,6749,6752,6755],{"className":6748},[60],[26,6750],{"className":6751,"style":65},[64],[26,6753,341],{"className":6754},[69],[26,6756,6758],{"className":6757},[69,5118],[26,6759,6474],{"className":6760},[69],[6716,6762,6763],{"align":6702},"Eye protection afforded by the blink-reflex .",[6697,6765,6766,6771,6839],{},[6716,6767,6768],{"align":6702},[21,6769,6770],{},"Class 3A",[6716,6772,6773,6774,6802,6803,151],{"align":6702},"Radiant power of ",[26,6775,6777,6790],{"className":6776},[29],[26,6778,6780],{"className":6779},[33],[35,6781,6782],{"xmlns":37},[39,6783,6784,6788],{},[42,6785,6786],{},[554,6787,341],{},[49,6789,341],{"encoding":51},[26,6791,6793],{"className":6792,"ariaHidden":56},[55],[26,6794,6796,6799],{"className":6795},[60],[26,6797],{"className":6798,"style":4270},[64],[26,6800,341],{"className":6801},[69]," to ",[26,6804,6806,6821],{"className":6805},[29],[26,6807,6809],{"className":6808},[33],[35,6810,6811],{"xmlns":37},[39,6812,6813,6819],{},[42,6814,6815,6817],{},[554,6816,6471],{},[5057,6818,6474],{},[49,6820,6477],{"encoding":51},[26,6822,6824],{"className":6823,"ariaHidden":56},[55],[26,6825,6827,6830,6833],{"className":6826},[60],[26,6828],{"className":6829,"style":65},[64],[26,6831,6471],{"className":6832},[69],[26,6834,6836],{"className":6835},[69,5118],[26,6837,6474],{"className":6838},[69],[6716,6840,6841],{"align":6702},"Safe if blink-reflex is allowed to work .",[6697,6843,6844,6849,6918],{},[6716,6845,6846],{"align":6702},[21,6847,6848],{},"Class 3B",[6716,6850,6851,6852,6802,6880,151],{"align":6702},"Power of ",[26,6853,6855,6868],{"className":6854},[29],[26,6856,6858],{"className":6857},[33],[35,6859,6860],{"xmlns":37},[39,6861,6862,6866],{},[42,6863,6864],{},[554,6865,341],{},[49,6867,341],{"encoding":51},[26,6869,6871],{"className":6870,"ariaHidden":56},[55],[26,6872,6874,6877],{"className":6873},[60],[26,6875],{"className":6876,"style":4270},[64],[26,6878,341],{"className":6879},[69],[26,6881,6883,6900],{"className":6882},[29],[26,6884,6886],{"className":6885},[33],[35,6887,6888],{"xmlns":37},[39,6889,6890,6897],{},[42,6891,6892,6895],{},[554,6893,6894],{},"500",[5057,6896,6474],{},[49,6898,6899],{"encoding":51},"500 \\text{ mW}",[26,6901,6903],{"className":6902,"ariaHidden":56},[55],[26,6904,6906,6909,6912],{"className":6905},[60],[26,6907],{"className":6908,"style":65},[64],[26,6910,6894],{"className":6911},[69],[26,6913,6915],{"className":6914},[69,5118],[26,6916,6474],{"className":6917},[69],[6716,6919,6920,6923],{"align":6702},[21,6921,6922],{},"Highly Dangerous."," Direct viewing or specular reflections can cause injury .",[13,6925,6927],{"id":6926},"the-role-of-the-laser-safety-officer-lso","The Role of the Laser Safety Officer (LSO)",[18,6929,6930],{},"The LSO is a person knowledgeable in the evaluation and control of laser hazards. Their responsibilities include :",[1926,6932,6933,6936,6939,6942],{},[1929,6934,6935],{},"Evaluating the site for potential specular reflections .",[1929,6937,6938],{},"Establishing \"safe viewing distances\" .",[1929,6940,6941],{},"Ensuring only authorized personnel operate the equipment .",[1929,6943,6944],{},"Setting up safety barriers around Class 3B lasers .",[13,6946,2825],{"id":2824},[2827,6948,6949,6953,6956,6960,6963,6967,7009,7013],{},[1738,6950,6952],{"id":6951},"hazard-assessment","Hazard Assessment",[18,6954,6955],{},"Identify the laser class being used and check if any reflective surfaces (glass, polished steel) are in the beam's path .",[1738,6957,6959],{"id":6958},"signage-and-barriers","Signage and Barriers",[18,6961,6962],{},"Erect warning signs at site entrances. For higher-class lasers, create physical barriers with a radius greater than the \"minimum safe viewing distance\" .",[1738,6964,6966],{"id":6965},"height-selection","Height Selection",[18,6968,6969,6970,7008],{},"Wherever possible, position the laser either well above (above ",[26,6971,6973,6990],{"className":6972},[29],[26,6974,6976],{"className":6975},[33],[35,6977,6978],{"xmlns":37},[39,6979,6980,6987],{},[42,6981,6982,6984],{},[554,6983,572],{},[5057,6985,6986],{}," m",[49,6988,6989],{"encoding":51},"2 \\text{ m}",[26,6991,6993],{"className":6992,"ariaHidden":56},[55],[26,6994,6996,6999,7002],{"className":6995},[60],[26,6997],{"className":6998,"style":4270},[64],[26,7000,572],{"className":7001},[69],[26,7003,7005],{"className":7004},[69,5118],[26,7006,6986],{"className":7007},[69],") or well below head height to prevent accidental eye contact .",[1738,7010,7012],{"id":7011},"personal-protective-equipment-ppe","Personal Protective Equipment (PPE)",[18,7014,7015],{},"Issue and ensure the use of laser safety goggles where the class of laser requires it .",[13,7017,5301],{"id":5300},[1926,7019,7020,7026],{},[1929,7021,7022,7025],{},[21,7023,7024],{},"Beam Obstruction:"," If using a laser in a tunnel, ensure it does not \"graze\" the wall, as this increases refraction and pointing error. If the beam is obscured, it may indicate structural movement .",[1929,7027,7028,7031],{},[21,7029,7030],{},"Don't Use Scopes:"," Never view a laser source through a theodolite, level, or binoculars unless the instrument is specifically fitted with an appropriate filter .",[13,7033,5569],{"id":5568},[1926,7035,7036,7042],{},[1929,7037,7038,7041],{},[21,7039,7040],{},"Assuming Blink-Reflex:"," Relying on the blink-reflex for Class 3B lasers is a fatal error; the intensity is too high for the eye to react fast enough .",[1929,7043,7044,7047],{},[21,7045,7046],{},"Dirty Optics:"," Dust on the lens can cause the beam to diverge (scatter), reducing its precision and potentially making it a hazard over a wider area .",[13,7049,5694],{"id":5693},[5696,7051,7052,7058,7064],{},[5699,7053,7055],{"label":7054},"Are Class 2 lasers totally safe?",[18,7056,7057],{},"They are considered safe for accidental viewing because the eye will blink. However, staring into any laser beam intentionally is dangerous .",[5699,7059,7061],{"label":7060},"What is 'Specular Reflection'?",[18,7062,7063],{},"It is a reflection off a mirror-like surface. 'Diffuse Reflection' is off a rough surface (like a concrete wall) and is generally much safer .",[5699,7065,7067],{"label":7066},"Can I use Class 3B lasers in public areas?",[18,7068,7069],{},"Generally, no. Class 3B should only be used in controlled environments with strict safety protocols and an LSO present .",[13,7071,5757],{"id":5756},[18,7073,7074],{},"Lasers are the \"invisible strings\" of the construction site. By adhering to classification standards and appointing a competent LSO, the engineering surveyor can harness the precision of light without compromising the safety of the site team.",[13,7076,5764],{"id":5763},[18,7078,5767,7079,6374],{},[2588,7080,5770],{},{"title":5781,"searchDepth":5782,"depth":5782,"links":7082},[7083,7084,7085,7086,7090,7091,7092,7098,7099,7100,7101,7102],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":172,"depth":5782,"text":173},{"id":6448,"depth":5782,"text":6449,"children":7087},[7088,7089],{"id":6452,"depth":5791,"text":6453},{"id":6678,"depth":5791,"text":6679},{"id":6685,"depth":5782,"text":6686},{"id":6926,"depth":5782,"text":6927},{"id":2824,"depth":5782,"text":2825,"children":7093},[7094,7095,7096,7097],{"id":6951,"depth":5791,"text":6952},{"id":6958,"depth":5791,"text":6959},{"id":6965,"depth":5791,"text":6966},{"id":7011,"depth":5791,"text":7012},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Setting Out","A comprehensive guide to the safety requirements, hazard classifications, and professional responsibilities when using lasers for setting out and alignment.","\u002Fimages\u002Flaser-safety.jpg",{"authorCredentials":7107,"commentCount":6400,"tags":7108,"relatedTools":7114,"relatedArticles":7117,"sources":7120},"LS",[7109,7110,7111,7112,7113],"Lasers","Safety","LSO","Construction Management","Regulations",[7115,7116],"Rotating Laser","Laser Safety Goggles",[7118,7119],"rotating-lasers-level-control","tunnel-alignment-lasers",[7121],{"title":5828,"type":5829,"note":7122},"Primary source for laser classifications (BS4803) and the role of the LSO.","\u002Fblog\u002Flaser-safety-and-management-on-site","2024-04-14",{"title":6425,"description":7104},{"loc":7123},"blog\u002F27.laser-safety-and-management-on-site","Qwt4-zEHw2cekhiSmJnIeCSZJlh9GuGqLDYIx21M-yo",{"id":7130,"title":7131,"author":5850,"body":7132,"category":9822,"description":9823,"extension":5809,"image":9824,"meta":9825,"navigation":5839,"path":9841,"publishedAt":9842,"readTime":6418,"seo":9843,"sitemap":9844,"stem":9845,"__hash__":9846},"blog\u002Fblog\u002F26.dip-and-strike-geological-surveying.md","Dip and Strike: Mastering Sloping Planes in Earthworks",{"type":10,"value":7133,"toc":9805},[7134,7136,7146,7148,7232,7234,7506,7508,7760,9330,9332,9402,9404,9665,9667,9714,9716,9736,9738,9799,9801],[13,7135,16],{"id":15},[18,7137,7138,7139,275,7142,7145],{},"In earthworks and geological engineering, we often deal with tilted planes—such as rock strata or sloping site excavations. To define these planes mathematically, surveyors use the concepts of ",[21,7140,7141],{},"Dip",[21,7143,7144],{},"Strike",". Understanding these is essential for predicting the depth of rock layers or designing the intersection of different site slopes .",[13,7147,195],{"id":194},[1926,7149,7150,7156,7226],{},[1929,7151,7152,7155],{},[21,7153,7154],{},"Full Dip:"," The direction of maximum tilt on a plane .",[1929,7157,7158,7161,7162,7225],{},[21,7159,7160],{},"Strike Line:"," A level line on the plane, which is always at right angles (",[26,7163,7165,7184],{"className":7164},[29],[26,7166,7168],{"className":7167},[33],[35,7169,7170],{"xmlns":37},[39,7171,7172,7181],{},[42,7173,7174],{},[3513,7175,7176,7179],{},[554,7177,7178],{},"90",[89,7180,3520],{},[49,7182,7183],{"encoding":51},"90^\\circ",[26,7185,7187],{"className":7186,"ariaHidden":56},[55],[26,7188,7190,7193,7196],{"className":7189},[60],[26,7191],{"className":7192,"style":3551},[64],[26,7194,5565],{"className":7195},[69],[26,7197,7199,7202],{"className":7198},[69],[26,7200,3562],{"className":7201},[69],[26,7203,7205],{"className":7204},[417],[26,7206,7208],{"className":7207},[421],[26,7209,7211],{"className":7210},[426],[26,7212,7214],{"className":7213,"style":3551},[430],[26,7215,7216,7219],{"style":3577},[26,7217],{"className":7218,"style":439},[438],[26,7220,7222],{"className":7221},[443,444,445,446],[26,7223,3520],{"className":7224},[773,446],") to the direction of full dip .",[1929,7227,7228,7231],{},[21,7229,7230],{},"Apparent Dip:"," Any grade on the plane measured in a direction other than full dip. An apparent dip is always less steep than the full dip .",[13,7233,345],{"id":344},[18,7235,7236,7237,7269,7270,7342,7343,7373,7374],{},"The fundamental relationship between full dip (",[26,7238,7240,7255],{"className":7239},[29],[26,7241,7243],{"className":7242},[33],[35,7244,7245],{"xmlns":37},[39,7246,7247,7252],{},[42,7248,7249],{},[45,7250,7251],{},"θ",[49,7253,7254],{"encoding":51},"\\theta",[26,7256,7258],{"className":7257,"ariaHidden":56},[55],[26,7259,7261,7265],{"className":7260},[60],[26,7262],{"className":7263,"style":7264},[64],"height:0.6944em;",[26,7266,7251],{"className":7267,"style":7268},[69,70],"margin-right:0.0278em;",") and an apparent dip (",[26,7271,7273,7291],{"className":7272},[29],[26,7274,7276],{"className":7275},[33],[35,7277,7278],{"xmlns":37},[39,7279,7280,7288],{},[42,7281,7282],{},[368,7283,7284,7286],{},[45,7285,7251],{},[554,7287,341],{},[49,7289,7290],{"encoding":51},"\\theta_1",[26,7292,7294],{"className":7293,"ariaHidden":56},[55],[26,7295,7297,7301],{"className":7296},[60],[26,7298],{"className":7299,"style":7300},[64],"height:0.8444em;vertical-align:-0.15em;",[26,7302,7304,7307],{"className":7303},[69],[26,7305,7251],{"className":7306,"style":7268},[69,70],[26,7308,7310],{"className":7309},[417],[26,7311,7313,7334],{"className":7312},[421,422],[26,7314,7316,7331],{"className":7315},[426],[26,7317,7319],{"className":7318,"style":2005},[430],[26,7320,7322,7325],{"style":7321},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[26,7323],{"className":7324,"style":439},[438],[26,7326,7328],{"className":7327},[443,444,445,446],[26,7329,341],{"className":7330},[69,446],[26,7332,454],{"className":7333},[453],[26,7335,7337],{"className":7336},[426],[26,7338,7340],{"className":7339,"style":2027},[430],[26,7341],{},") measured at an angle ",[26,7344,7346,7361],{"className":7345},[29],[26,7347,7349],{"className":7348},[33],[35,7350,7351],{"xmlns":37},[39,7352,7353,7358],{},[42,7354,7355],{},[45,7356,7357],{},"ϕ",[49,7359,7360],{"encoding":51},"\\phi",[26,7362,7364],{"className":7363,"ariaHidden":56},[55],[26,7365,7367,7370],{"className":7366},[60],[26,7368],{"className":7369,"style":1898},[64],[26,7371,7357],{"className":7372},[69,70]," to the full dip direction is:\n",[26,7375,7377,7415],{"className":7376},[29],[26,7378,7380],{"className":7379},[33],[35,7381,7382],{"xmlns":37},[39,7383,7384,7412],{},[42,7385,7386,7389,7391,7397,7399,7401,7403,7405,7408,7410],{},[45,7387,7388],{},"tan",[89,7390,1962],{},[368,7392,7393,7395],{},[45,7394,7251],{},[554,7396,341],{},[89,7398,543],{},[45,7400,7388],{},[89,7402,1962],{},[45,7404,7251],{},[45,7406,7407],{},"cos",[89,7409,1962],{},[45,7411,7357],{},[49,7413,7414],{"encoding":51},"\\tan \\theta_1 = \\tan \\theta \\cos \\phi",[26,7416,7418,7479],{"className":7417,"ariaHidden":56},[55],[26,7419,7421,7424,7427,7430,7470,7473,7476],{"className":7420},[60],[26,7422],{"className":7423,"style":7300},[64],[26,7425,7388],{"className":7426},[2079],[26,7428],{"className":7429,"style":124},[123],[26,7431,7433,7436],{"className":7432},[69],[26,7434,7251],{"className":7435,"style":7268},[69,70],[26,7437,7439],{"className":7438},[417],[26,7440,7442,7462],{"className":7441},[421,422],[26,7443,7445,7459],{"className":7444},[426],[26,7446,7448],{"className":7447,"style":2005},[430],[26,7449,7450,7453],{"style":7321},[26,7451],{"className":7452,"style":439},[438],[26,7454,7456],{"className":7455},[443,444,445,446],[26,7457,341],{"className":7458},[69,446],[26,7460,454],{"className":7461},[453],[26,7463,7465],{"className":7464},[426],[26,7466,7468],{"className":7467,"style":2027},[430],[26,7469],{},[26,7471],{"className":7472,"style":676},[123],[26,7474,543],{"className":7475},[680],[26,7477],{"className":7478,"style":676},[123],[26,7480,7482,7485,7488,7491,7494,7497,7500,7503],{"className":7481},[60],[26,7483],{"className":7484,"style":1898},[64],[26,7486,7388],{"className":7487},[2079],[26,7489],{"className":7490,"style":124},[123],[26,7492,7251],{"className":7493,"style":7268},[69,70],[26,7495],{"className":7496,"style":124},[123],[26,7498,7407],{"className":7499},[2079],[26,7501],{"className":7502,"style":124},[123],[26,7504,7357],{"className":7505},[69,70],[13,7507,3283],{"id":3282},[18,7509,7510,7513,7514,7542,7543,7572,7573,7637,7638,2699,7703,7542,7731,7759],{},[21,7511,7512],{},"Problem:"," On a stratum plane, an apparent dip of ",[26,7515,7517,7530],{"className":7516},[29],[26,7518,7520],{"className":7519},[33],[35,7521,7522],{"xmlns":37},[39,7523,7524,7528],{},[42,7525,7526],{},[554,7527,341],{},[49,7529,341],{"encoding":51},[26,7531,7533],{"className":7532,"ariaHidden":56},[55],[26,7534,7536,7539],{"className":7535},[60],[26,7537],{"className":7538,"style":4270},[64],[26,7540,341],{"className":7541},[69]," in ",[26,7544,7546,7560],{"className":7545},[29],[26,7547,7549],{"className":7548},[33],[35,7550,7551],{"xmlns":37},[39,7552,7553,7558],{},[42,7554,7555],{},[554,7556,7557],{},"16",[49,7559,7557],{"encoding":51},[26,7561,7563],{"className":7562,"ariaHidden":56},[55],[26,7564,7566,7569],{"className":7565},[60],[26,7567],{"className":7568,"style":4270},[64],[26,7570,7557],{"className":7571},[69]," bears ",[26,7574,7576,7595],{"className":7575},[29],[26,7577,7579],{"className":7578},[33],[35,7580,7581],{"xmlns":37},[39,7582,7583,7592],{},[42,7584,7585],{},[3513,7586,7587,7590],{},[554,7588,7589],{},"170",[89,7591,3520],{},[49,7593,7594],{"encoding":51},"170^\\circ",[26,7596,7598],{"className":7597,"ariaHidden":56},[55],[26,7599,7601,7604,7608],{"className":7600},[60],[26,7602],{"className":7603,"style":3551},[64],[26,7605,7607],{"className":7606},[69],"17",[26,7609,7611,7614],{"className":7610},[69],[26,7612,3562],{"className":7613},[69],[26,7615,7617],{"className":7616},[417],[26,7618,7620],{"className":7619},[421],[26,7621,7623],{"className":7622},[426],[26,7624,7626],{"className":7625,"style":3551},[430],[26,7627,7628,7631],{"style":3577},[26,7629],{"className":7630,"style":439},[438],[26,7632,7634],{"className":7633},[443,444,445,446],[26,7635,3520],{"className":7636},[773,446],", and another apparent dip in the direction ",[26,7639,7641,7660],{"className":7640},[29],[26,7642,7644],{"className":7643},[33],[35,7645,7646],{"xmlns":37},[39,7647,7648,7657],{},[42,7649,7650],{},[3513,7651,7652,7655],{},[554,7653,7654],{},"194",[89,7656,3520],{},[49,7658,7659],{"encoding":51},"194^\\circ",[26,7661,7663],{"className":7662,"ariaHidden":56},[55],[26,7664,7666,7669,7673],{"className":7665},[60],[26,7667],{"className":7668,"style":3551},[64],[26,7670,7672],{"className":7671},[69],"19",[26,7674,7676,7680],{"className":7675},[69],[26,7677,7679],{"className":7678},[69],"4",[26,7681,7683],{"className":7682},[417],[26,7684,7686],{"className":7685},[421],[26,7687,7689],{"className":7688},[426],[26,7690,7692],{"className":7691,"style":3551},[430],[26,7693,7694,7697],{"style":3577},[26,7695],{"className":7696,"style":439},[438],[26,7698,7700],{"className":7699},[443,444,445,446],[26,7701,3520],{"className":7702},[773,446],[26,7704,7706,7719],{"className":7705},[29],[26,7707,7709],{"className":7708},[33],[35,7710,7711],{"xmlns":37},[39,7712,7713,7717],{},[42,7714,7715],{},[554,7716,341],{},[49,7718,341],{"encoding":51},[26,7720,7722],{"className":7721,"ariaHidden":56},[55],[26,7723,7725,7728],{"className":7724},[60],[26,7726],{"className":7727,"style":4270},[64],[26,7729,341],{"className":7730},[69],[26,7732,7734,7747],{"className":7733},[29],[26,7735,7737],{"className":7736},[33],[35,7738,7739],{"xmlns":37},[39,7740,7741,7745],{},[42,7742,7743],{},[554,7744,3555],{},[49,7746,3555],{"encoding":51},[26,7748,7750],{"className":7749,"ariaHidden":56},[55],[26,7751,7753,7756],{"className":7752},[60],[26,7754],{"className":7755,"style":4270},[64],[26,7757,3555],{"className":7758},[69],". Calculate the direction and rate of full dip .",[3763,7761,7762,8229,8445,8954],{},[1929,7763,7764,7767,7768,7799,7800,8017],{},[21,7765,7766],{},"Set up equations:","\nLet full dip direction be ",[26,7769,7771,7786],{"className":7770},[29],[26,7772,7774],{"className":7773},[33],[35,7775,7776],{"xmlns":37},[39,7777,7778,7783],{},[42,7779,7780],{},[45,7781,7782],{},"δ",[49,7784,7785],{"encoding":51},"\\delta",[26,7787,7789],{"className":7788,"ariaHidden":56},[55],[26,7790,7792,7795],{"className":7791},[60],[26,7793],{"className":7794,"style":7264},[64],[26,7796,7782],{"className":7797,"style":7798},[69,70],"margin-right:0.0379em;",".\n",[26,7801,7803,7854],{"className":7802},[29],[26,7804,7806],{"className":7805},[33],[35,7807,7808],{"xmlns":37},[39,7809,7810,7851],{},[42,7811,7812,7814,7816,7822,7824,7826,7828,7831,7833,7835,7837,7839,7845,7847,7849],{},[45,7813,7388],{},[89,7815,1962],{},[368,7817,7818,7820],{},[5057,7819,7141],{},[554,7821,341],{},[89,7823,543],{},[45,7825,7388],{},[89,7827,1962],{},[5057,7829,7830],{},"Full Dip",[89,7832,4578],{},[45,7834,7407],{},[89,7836,1962],{},[89,7838,366],{"stretchy":365},[3513,7840,7841,7843],{},[554,7842,7589],{},[89,7844,3520],{},[89,7846,1967],{},[45,7848,7782],{},[89,7850,386],{"stretchy":365},[49,7852,7853],{"encoding":51},"\\tan \\text{Dip}_1 = \\tan \\text{Full Dip} \\times \\cos(170^\\circ - \\delta)",[26,7855,7857,7925,7952,8005],{"className":7856,"ariaHidden":56},[55],[26,7858,7860,7864,7867,7870,7916,7919,7922],{"className":7859},[60],[26,7861],{"className":7862,"style":7863},[64],"height:0.9275em;vertical-align:-0.2441em;",[26,7865,7388],{"className":7866},[2079],[26,7868],{"className":7869,"style":124},[123],[26,7871,7873,7879],{"className":7872},[69],[26,7874,7876],{"className":7875},[69,5118],[26,7877,7141],{"className":7878},[69],[26,7880,7882],{"className":7881},[417],[26,7883,7885,7907],{"className":7884},[421,422],[26,7886,7888,7904],{"className":7887},[426],[26,7889,7892],{"className":7890,"style":7891},[430],"height:0.207em;",[26,7893,7895,7898],{"style":7894},"top:-2.4559em;margin-right:0.05em;",[26,7896],{"className":7897,"style":439},[438],[26,7899,7901],{"className":7900},[443,444,445,446],[26,7902,341],{"className":7903},[69,446],[26,7905,454],{"className":7906},[453],[26,7908,7910],{"className":7909},[426],[26,7911,7914],{"className":7912,"style":7913},[430],"height:0.2441em;",[26,7915],{},[26,7917],{"className":7918,"style":676},[123],[26,7920,543],{"className":7921},[680],[26,7923],{"className":7924,"style":676},[123],[26,7926,7928,7931,7934,7937,7943,7946,7949],{"className":7927},[60],[26,7929],{"className":7930,"style":1898},[64],[26,7932,7388],{"className":7933},[2079],[26,7935],{"className":7936,"style":124},[123],[26,7938,7940],{"className":7939},[69,5118],[26,7941,7830],{"className":7942},[69],[26,7944],{"className":7945,"style":4156},[123],[26,7947,4578],{"className":7948},[773],[26,7950],{"className":7951,"style":4156},[123],[26,7953,7955,7958,7961,7964,7967,7996,7999,8002],{"className":7954},[60],[26,7956],{"className":7957,"style":3328},[64],[26,7959,7407],{"className":7960},[2079],[26,7962,366],{"className":7963},[406],[26,7965,7607],{"className":7966},[69],[26,7968,7970,7973],{"className":7969},[69],[26,7971,3562],{"className":7972},[69],[26,7974,7976],{"className":7975},[417],[26,7977,7979],{"className":7978},[421],[26,7980,7982],{"className":7981},[426],[26,7983,7985],{"className":7984,"style":3551},[430],[26,7986,7987,7990],{"style":3577},[26,7988],{"className":7989,"style":439},[438],[26,7991,7993],{"className":7992},[443,444,445,446],[26,7994,3520],{"className":7995},[773,446],[26,7997],{"className":7998,"style":4156},[123],[26,8000,1967],{"className":8001},[773],[26,8003],{"className":8004,"style":4156},[123],[26,8006,8008,8011,8014],{"className":8007},[60],[26,8009],{"className":8010,"style":3328},[64],[26,8012,7782],{"className":8013,"style":7798},[69,70],[26,8015,386],{"className":8016},[515],[26,8018,8020,8070],{"className":8019},[29],[26,8021,8023],{"className":8022},[33],[35,8024,8025],{"xmlns":37},[39,8026,8027,8067],{},[42,8028,8029,8031,8033,8039,8041,8043,8045,8047,8049,8051,8053,8055,8061,8063,8065],{},[45,8030,7388],{},[89,8032,1962],{},[368,8034,8035,8037],{},[5057,8036,7141],{},[554,8038,572],{},[89,8040,543],{},[45,8042,7388],{},[89,8044,1962],{},[5057,8046,7830],{},[89,8048,4578],{},[45,8050,7407],{},[89,8052,1962],{},[89,8054,366],{"stretchy":365},[3513,8056,8057,8059],{},[554,8058,7654],{},[89,8060,3520],{},[89,8062,1967],{},[45,8064,7782],{},[89,8066,386],{"stretchy":365},[49,8068,8069],{"encoding":51},"\\tan \\text{Dip}_2 = \\tan \\text{Full Dip} \\times \\cos(194^\\circ - \\delta)",[26,8071,8073,8137,8164,8217],{"className":8072,"ariaHidden":56},[55],[26,8074,8076,8079,8082,8085,8128,8131,8134],{"className":8075},[60],[26,8077],{"className":8078,"style":7863},[64],[26,8080,7388],{"className":8081},[2079],[26,8083],{"className":8084,"style":124},[123],[26,8086,8088,8094],{"className":8087},[69],[26,8089,8091],{"className":8090},[69,5118],[26,8092,7141],{"className":8093},[69],[26,8095,8097],{"className":8096},[417],[26,8098,8100,8120],{"className":8099},[421,422],[26,8101,8103,8117],{"className":8102},[426],[26,8104,8106],{"className":8105,"style":7891},[430],[26,8107,8108,8111],{"style":7894},[26,8109],{"className":8110,"style":439},[438],[26,8112,8114],{"className":8113},[443,444,445,446],[26,8115,572],{"className":8116},[69,446],[26,8118,454],{"className":8119},[453],[26,8121,8123],{"className":8122},[426],[26,8124,8126],{"className":8125,"style":7913},[430],[26,8127],{},[26,8129],{"className":8130,"style":676},[123],[26,8132,543],{"className":8133},[680],[26,8135],{"className":8136,"style":676},[123],[26,8138,8140,8143,8146,8149,8155,8158,8161],{"className":8139},[60],[26,8141],{"className":8142,"style":1898},[64],[26,8144,7388],{"className":8145},[2079],[26,8147],{"className":8148,"style":124},[123],[26,8150,8152],{"className":8151},[69,5118],[26,8153,7830],{"className":8154},[69],[26,8156],{"className":8157,"style":4156},[123],[26,8159,4578],{"className":8160},[773],[26,8162],{"className":8163,"style":4156},[123],[26,8165,8167,8170,8173,8176,8179,8208,8211,8214],{"className":8166},[60],[26,8168],{"className":8169,"style":3328},[64],[26,8171,7407],{"className":8172},[2079],[26,8174,366],{"className":8175},[406],[26,8177,7672],{"className":8178},[69],[26,8180,8182,8185],{"className":8181},[69],[26,8183,7679],{"className":8184},[69],[26,8186,8188],{"className":8187},[417],[26,8189,8191],{"className":8190},[421],[26,8192,8194],{"className":8193},[426],[26,8195,8197],{"className":8196,"style":3551},[430],[26,8198,8199,8202],{"style":3577},[26,8200],{"className":8201,"style":439},[438],[26,8203,8205],{"className":8204},[443,444,445,446],[26,8206,3520],{"className":8207},[773,446],[26,8209],{"className":8210,"style":4156},[123],[26,8212,1967],{"className":8213},[773],[26,8215],{"className":8216,"style":4156},[123],[26,8218,8220,8223,8226],{"className":8219},[60],[26,8221],{"className":8222,"style":3328},[64],[26,8224,7782],{"className":8225,"style":7798},[69,70],[26,8227,386],{"className":8228},[515],[1929,8230,8231,8234,5870],{},[21,8232,8233],{},"Equate through Full Dip:",[26,8235,8237,8291],{"className":8236},[29],[26,8238,8240],{"className":8239},[33],[35,8241,8242],{"xmlns":37},[39,8243,8244,8288],{},[42,8245,8246,8248,8250,8252,8254,8260,8262,8264,8266,8268,8270,8272,8274,8276,8282,8284,8286],{},[554,8247,7557],{},[45,8249,7407],{},[89,8251,1962],{},[89,8253,366],{"stretchy":365},[3513,8255,8256,8258],{},[554,8257,7589],{},[89,8259,3520],{},[89,8261,1967],{},[45,8263,7782],{},[89,8265,386],{"stretchy":365},[89,8267,543],{},[554,8269,3555],{},[45,8271,7407],{},[89,8273,1962],{},[89,8275,366],{"stretchy":365},[3513,8277,8278,8280],{},[554,8279,7654],{},[89,8281,3520],{},[89,8283,1967],{},[45,8285,7782],{},[89,8287,386],{"stretchy":365},[49,8289,8290],{"encoding":51},"16 \\cos(170^\\circ - \\delta) = 11 \\cos(194^\\circ - \\delta)",[26,8292,8294,8353,8374,8433],{"className":8293,"ariaHidden":56},[55],[26,8295,8297,8300,8303,8306,8309,8312,8315,8344,8347,8350],{"className":8296},[60],[26,8298],{"className":8299,"style":3328},[64],[26,8301,7557],{"className":8302},[69],[26,8304],{"className":8305,"style":124},[123],[26,8307,7407],{"className":8308},[2079],[26,8310,366],{"className":8311},[406],[26,8313,7607],{"className":8314},[69],[26,8316,8318,8321],{"className":8317},[69],[26,8319,3562],{"className":8320},[69],[26,8322,8324],{"className":8323},[417],[26,8325,8327],{"className":8326},[421],[26,8328,8330],{"className":8329},[426],[26,8331,8333],{"className":8332,"style":3551},[430],[26,8334,8335,8338],{"style":3577},[26,8336],{"className":8337,"style":439},[438],[26,8339,8341],{"className":8340},[443,444,445,446],[26,8342,3520],{"className":8343},[773,446],[26,8345],{"className":8346,"style":4156},[123],[26,8348,1967],{"className":8349},[773],[26,8351],{"className":8352,"style":4156},[123],[26,8354,8356,8359,8362,8365,8368,8371],{"className":8355},[60],[26,8357],{"className":8358,"style":3328},[64],[26,8360,7782],{"className":8361,"style":7798},[69,70],[26,8363,386],{"className":8364},[515],[26,8366],{"className":8367,"style":676},[123],[26,8369,543],{"className":8370},[680],[26,8372],{"className":8373,"style":676},[123],[26,8375,8377,8380,8383,8386,8389,8392,8395,8424,8427,8430],{"className":8376},[60],[26,8378],{"className":8379,"style":3328},[64],[26,8381,3555],{"className":8382},[69],[26,8384],{"className":8385,"style":124},[123],[26,8387,7407],{"className":8388},[2079],[26,8390,366],{"className":8391},[406],[26,8393,7672],{"className":8394},[69],[26,8396,8398,8401],{"className":8397},[69],[26,8399,7679],{"className":8400},[69],[26,8402,8404],{"className":8403},[417],[26,8405,8407],{"className":8406},[421],[26,8408,8410],{"className":8409},[426],[26,8411,8413],{"className":8412,"style":3551},[430],[26,8414,8415,8418],{"style":3577},[26,8416],{"className":8417,"style":439},[438],[26,8419,8421],{"className":8420},[443,444,445,446],[26,8422,3520],{"className":8423},[773,446],[26,8425],{"className":8426,"style":4156},[123],[26,8428,1967],{"className":8429},[773],[26,8431],{"className":8432,"style":4156},[123],[26,8434,8436,8439,8442],{"className":8435},[60],[26,8437],{"className":8438,"style":3328},[64],[26,8440,7782],{"className":8441,"style":7798},[69,70],[26,8443,386],{"className":8444},[515],[1929,8446,8447,8478,8479,8649,8650,8819,8820,6802,8853,8886,8887,5870],{},[21,8448,8449,8450,520],{},"Solve for ",[26,8451,8453,8466],{"className":8452},[29],[26,8454,8456],{"className":8455},[33],[35,8457,8458],{"xmlns":37},[39,8459,8460,8464],{},[42,8461,8462],{},[45,8463,7782],{},[49,8465,7785],{"encoding":51},[26,8467,8469],{"className":8468,"ariaHidden":56},[55],[26,8470,8472,8475],{"className":8471},[60],[26,8473],{"className":8474,"style":7264},[64],[26,8476,7782],{"className":8477,"style":7798},[69,70],"\nUsing trigonometric expansion (",[26,8480,8482,8537],{"className":8481},[29],[26,8483,8485],{"className":8484},[33],[35,8486,8487],{"xmlns":37},[39,8488,8489,8534],{},[42,8490,8491,8493,8495,8497,8499,8501,8503,8505,8507,8509,8511,8513,8515,8517,8519,8521,8524,8526,8528,8530,8532],{},[45,8492,7407],{},[89,8494,1962],{},[89,8496,366],{"stretchy":365},[45,8498,87],{},[89,8500,1967],{},[45,8502,94],{},[89,8504,386],{"stretchy":365},[89,8506,543],{},[45,8508,7407],{},[89,8510,1962],{},[45,8512,87],{},[45,8514,7407],{},[89,8516,1962],{},[45,8518,94],{},[89,8520,565],{},[45,8522,8523],{},"sin",[89,8525,1962],{},[45,8527,87],{},[45,8529,8523],{},[89,8531,1962],{},[45,8533,94],{},[49,8535,8536],{"encoding":51},"\\cos(A-B) = \\cos A \\cos B + \\sin A \\sin B",[26,8538,8540,8564,8585,8622],{"className":8539,"ariaHidden":56},[55],[26,8541,8543,8546,8549,8552,8555,8558,8561],{"className":8542},[60],[26,8544],{"className":8545,"style":3328},[64],[26,8547,7407],{"className":8548},[2079],[26,8550,366],{"className":8551},[406],[26,8553,87],{"className":8554},[69,70],[26,8556],{"className":8557,"style":4156},[123],[26,8559,1967],{"className":8560},[773],[26,8562],{"className":8563,"style":4156},[123],[26,8565,8567,8570,8573,8576,8579,8582],{"className":8566},[60],[26,8568],{"className":8569,"style":3328},[64],[26,8571,94],{"className":8572,"style":128},[69,70],[26,8574,386],{"className":8575},[515],[26,8577],{"className":8578,"style":676},[123],[26,8580,543],{"className":8581},[680],[26,8583],{"className":8584,"style":676},[123],[26,8586,8588,8592,8595,8598,8601,8604,8607,8610,8613,8616,8619],{"className":8587},[60],[26,8589],{"className":8590,"style":8591},[64],"height:0.7667em;vertical-align:-0.0833em;",[26,8593,7407],{"className":8594},[2079],[26,8596],{"className":8597,"style":124},[123],[26,8599,87],{"className":8600},[69,70],[26,8602],{"className":8603,"style":124},[123],[26,8605,7407],{"className":8606},[2079],[26,8608],{"className":8609,"style":124},[123],[26,8611,94],{"className":8612,"style":128},[69,70],[26,8614],{"className":8615,"style":4156},[123],[26,8617,565],{"className":8618},[773],[26,8620],{"className":8621,"style":4156},[123],[26,8623,8625,8628,8631,8634,8637,8640,8643,8646],{"className":8624},[60],[26,8626],{"className":8627,"style":65},[64],[26,8629,8523],{"className":8630},[2079],[26,8632],{"className":8633,"style":124},[123],[26,8635,87],{"className":8636},[69,70],[26,8638],{"className":8639,"style":124},[123],[26,8641,8523],{"className":8642},[2079],[26,8644],{"className":8645,"style":124},[123],[26,8647,94],{"className":8648,"style":128},[69,70],"):\n",[26,8651,8653,8704],{"className":8652},[29],[26,8654,8656],{"className":8655},[33],[35,8657,8658],{"xmlns":37},[39,8659,8660,8701],{},[42,8661,8662,8664,8666,8668,8670,8672,8675,8677,8680,8683,8686,8688,8690,8692,8694],{},[45,8663,7388],{},[89,8665,1962],{},[45,8667,7782],{},[89,8669,543],{},[89,8671,1967],{},[554,8673,8674],{},"3.6",[45,8676,3991],{"mathvariant":2636},[554,8678,8679],{},"6.5",[5057,8681,8682],{},"  ",[89,8684,8685],{},"⟹",[5057,8687,8682],{},[45,8689,7782],{},[89,8691,543],{},[89,8693,1967],{},[3513,8695,8696,8699],{},[554,8697,8698],{},"29",[89,8700,3520],{},[49,8702,8703],{"encoding":51},"\\tan \\delta = -3.6 \u002F 6.5 \\implies \\delta = -29^\\circ",[26,8705,8707,8731,8759,8777],{"className":8706,"ariaHidden":56},[55],[26,8708,8710,8713,8716,8719,8722,8725,8728],{"className":8709},[60],[26,8711],{"className":8712,"style":7264},[64],[26,8714,7388],{"className":8715},[2079],[26,8717],{"className":8718,"style":124},[123],[26,8720,7782],{"className":8721,"style":7798},[69,70],[26,8723],{"className":8724,"style":676},[123],[26,8726,543],{"className":8727},[680],[26,8729],{"className":8730,"style":676},[123],[26,8732,8734,8737,8740,8744,8747,8750,8753,8756],{"className":8733},[60],[26,8735],{"className":8736,"style":3328},[64],[26,8738,1967],{"className":8739},[69],[26,8741,8743],{"className":8742},[69],"3.6\u002F6.5",[26,8745],{"className":8746,"style":676},[123],[26,8748],{"className":8749,"style":676},[123],[26,8751,8685],{"className":8752},[680],[26,8754],{"className":8755,"style":676},[123],[26,8757],{"className":8758,"style":676},[123],[26,8760,8762,8765,8768,8771,8774],{"className":8761},[60],[26,8763],{"className":8764,"style":7264},[64],[26,8766,7782],{"className":8767,"style":7798},[69,70],[26,8769],{"className":8770,"style":676},[123],[26,8772,543],{"className":8773},[680],[26,8775],{"className":8776,"style":676},[123],[26,8778,8780,8784,8787,8790],{"className":8779},[60],[26,8781],{"className":8782,"style":8783},[64],"height:0.7574em;vertical-align:-0.0833em;",[26,8785,1967],{"className":8786},[69],[26,8788,572],{"className":8789},[69],[26,8791,8793,8796],{"className":8792},[69],[26,8794,5565],{"className":8795},[69],[26,8797,8799],{"className":8798},[417],[26,8800,8802],{"className":8801},[421],[26,8803,8805],{"className":8804},[426],[26,8806,8808],{"className":8807,"style":3551},[430],[26,8809,8810,8813],{"style":3577},[26,8811],{"className":8812,"style":439},[438],[26,8814,8816],{"className":8815},[443,444,445,446],[26,8817,3520],{"className":8818},[773,446]," .\nAdjusting for direction: Since the grade is increasing from ",[26,8821,8823,8841],{"className":8822},[29],[26,8824,8826],{"className":8825},[33],[35,8827,8828],{"xmlns":37},[39,8829,8830,8838],{},[42,8831,8832,8834,8836],{},[554,8833,341],{},[45,8835,3991],{"mathvariant":2636},[554,8837,7557],{},[49,8839,8840],{"encoding":51},"1\u002F16",[26,8842,8844],{"className":8843,"ariaHidden":56},[55],[26,8845,8847,8850],{"className":8846},[60],[26,8848],{"className":8849,"style":3328},[64],[26,8851,8840],{"className":8852},[69],[26,8854,8856,8874],{"className":8855},[29],[26,8857,8859],{"className":8858},[33],[35,8860,8861],{"xmlns":37},[39,8862,8863,8871],{},[42,8864,8865,8867,8869],{},[554,8866,341],{},[45,8868,3991],{"mathvariant":2636},[554,8870,3555],{},[49,8872,8873],{"encoding":51},"1\u002F11",[26,8875,8877],{"className":8876,"ariaHidden":56},[55],[26,8878,8880,8883],{"className":8879},[60],[26,8881],{"className":8882,"style":3328},[64],[26,8884,8873],{"className":8885},[69],", the full dip must be beyond the second direction.\nResult: ",[21,8888,8889,8890],{},"Full Dip Direction = ",[26,8891,8893,8912],{"className":8892},[29],[26,8894,8896],{"className":8895},[33],[35,8897,8898],{"xmlns":37},[39,8899,8900,8909],{},[42,8901,8902],{},[3513,8903,8904,8907],{},[554,8905,8906],{},"223",[89,8908,3520],{},[49,8910,8911],{"encoding":51},"223^\\circ",[26,8913,8915],{"className":8914,"ariaHidden":56},[55],[26,8916,8918,8921,8925],{"className":8917},[60],[26,8919],{"className":8920,"style":3551},[64],[26,8922,8924],{"className":8923},[69],"22",[26,8926,8928,8931],{"className":8927},[69],[26,8929,588],{"className":8930},[69],[26,8932,8934],{"className":8933},[417],[26,8935,8937],{"className":8936},[421],[26,8938,8940],{"className":8939},[426],[26,8941,8943],{"className":8942,"style":3551},[430],[26,8944,8945,8948],{"style":3577},[26,8946],{"className":8947,"style":439},[438],[26,8949,8951],{"className":8950},[443,444,445,446],[26,8952,3520],{"className":8953},[773,446],[1929,8955,8956,8959,9145,9270,9271,5870],{},[21,8957,8958],{},"Find Rate of Full Dip:",[26,8960,8962,9015],{"className":8961},[29],[26,8963,8965],{"className":8964},[33],[35,8966,8967],{"xmlns":37},[39,8968,8969,9012],{},[42,8970,8971,8973,8975,8977,8979,8981,8983,8985,8988,8990,8992,8994,8996,9002,9004,9010],{},[554,8972,341],{},[45,8974,3991],{"mathvariant":2636},[554,8976,3555],{},[89,8978,543],{},[89,8980,366],{"stretchy":365},[554,8982,341],{},[45,8984,3991],{"mathvariant":2636},[45,8986,8987],{},"x",[89,8989,386],{"stretchy":365},[45,8991,7407],{},[89,8993,1962],{},[89,8995,366],{"stretchy":365},[3513,8997,8998,9000],{},[554,8999,8906],{},[89,9001,3520],{},[89,9003,1967],{},[3513,9005,9006,9008],{},[554,9007,7654],{},[89,9009,3520],{},[89,9011,386],{"stretchy":365},[49,9013,9014],{"encoding":51},"1\u002F11 = (1\u002Fx) \\cos(223^\\circ - 194^\\circ)",[26,9016,9018,9036,9104],{"className":9017,"ariaHidden":56},[55],[26,9019,9021,9024,9027,9030,9033],{"className":9020},[60],[26,9022],{"className":9023,"style":3328},[64],[26,9025,8873],{"className":9026},[69],[26,9028],{"className":9029,"style":676},[123],[26,9031,543],{"className":9032},[680],[26,9034],{"className":9035,"style":676},[123],[26,9037,9039,9042,9045,9048,9051,9054,9057,9060,9063,9066,9095,9098,9101],{"className":9038},[60],[26,9040],{"className":9041,"style":3328},[64],[26,9043,366],{"className":9044},[406],[26,9046,4111],{"className":9047},[69],[26,9049,8987],{"className":9050},[69,70],[26,9052,386],{"className":9053},[515],[26,9055],{"className":9056,"style":124},[123],[26,9058,7407],{"className":9059},[2079],[26,9061,366],{"className":9062},[406],[26,9064,8924],{"className":9065},[69],[26,9067,9069,9072],{"className":9068},[69],[26,9070,588],{"className":9071},[69],[26,9073,9075],{"className":9074},[417],[26,9076,9078],{"className":9077},[421],[26,9079,9081],{"className":9080},[426],[26,9082,9084],{"className":9083,"style":3551},[430],[26,9085,9086,9089],{"style":3577},[26,9087],{"className":9088,"style":439},[438],[26,9090,9092],{"className":9091},[443,444,445,446],[26,9093,3520],{"className":9094},[773,446],[26,9096],{"className":9097,"style":4156},[123],[26,9099,1967],{"className":9100},[773],[26,9102],{"className":9103,"style":4156},[123],[26,9105,9107,9110,9113,9142],{"className":9106},[60],[26,9108],{"className":9109,"style":3328},[64],[26,9111,7672],{"className":9112},[69],[26,9114,9116,9119],{"className":9115},[69],[26,9117,7679],{"className":9118},[69],[26,9120,9122],{"className":9121},[417],[26,9123,9125],{"className":9124},[421],[26,9126,9128],{"className":9127},[426],[26,9129,9131],{"className":9130,"style":3551},[430],[26,9132,9133,9136],{"style":3577},[26,9134],{"className":9135,"style":439},[438],[26,9137,9139],{"className":9138},[443,444,445,446],[26,9140,3520],{"className":9141},[773,446],[26,9143,386],{"className":9144},[515],[26,9146,9148,9181],{"className":9147},[29],[26,9149,9151],{"className":9150},[33],[35,9152,9153],{"xmlns":37},[39,9154,9155,9178],{},[42,9156,9157,9159,9161,9163,9165,9167,9173,9175],{},[45,9158,8987],{},[89,9160,543],{},[554,9162,3555],{},[45,9164,7407],{},[89,9166,1962],{},[3513,9168,9169,9171],{},[554,9170,8698],{},[89,9172,3520],{},[89,9174,543],{},[554,9176,9177],{},"9.6",[49,9179,9180],{"encoding":51},"x = 11 \\cos 29^\\circ = 9.6",[26,9182,9184,9202,9261],{"className":9183,"ariaHidden":56},[55],[26,9185,9187,9190,9193,9196,9199],{"className":9186},[60],[26,9188],{"className":9189,"style":2617},[64],[26,9191,8987],{"className":9192},[69,70],[26,9194],{"className":9195,"style":676},[123],[26,9197,543],{"className":9198},[680],[26,9200],{"className":9201,"style":676},[123],[26,9203,9205,9208,9211,9214,9217,9220,9223,9252,9255,9258],{"className":9204},[60],[26,9206],{"className":9207,"style":3551},[64],[26,9209,3555],{"className":9210},[69],[26,9212],{"className":9213,"style":124},[123],[26,9215,7407],{"className":9216},[2079],[26,9218],{"className":9219,"style":124},[123],[26,9221,572],{"className":9222},[69],[26,9224,9226,9229],{"className":9225},[69],[26,9227,5565],{"className":9228},[69],[26,9230,9232],{"className":9231},[417],[26,9233,9235],{"className":9234},[421],[26,9236,9238],{"className":9237},[426],[26,9239,9241],{"className":9240,"style":3551},[430],[26,9242,9243,9246],{"style":3577},[26,9244],{"className":9245,"style":439},[438],[26,9247,9249],{"className":9248},[443,444,445,446],[26,9250,3520],{"className":9251},[773,446],[26,9253],{"className":9254,"style":676},[123],[26,9256,543],{"className":9257},[680],[26,9259],{"className":9260,"style":676},[123],[26,9262,9264,9267],{"className":9263},[60],[26,9265],{"className":9266,"style":4270},[64],[26,9268,9177],{"className":9269},[69]," .\nResult: ",[21,9272,9273,9274,7542,9302],{},"Rate of Full Dip = ",[26,9275,9277,9290],{"className":9276},[29],[26,9278,9280],{"className":9279},[33],[35,9281,9282],{"xmlns":37},[39,9283,9284,9288],{},[42,9285,9286],{},[554,9287,341],{},[49,9289,341],{"encoding":51},[26,9291,9293],{"className":9292,"ariaHidden":56},[55],[26,9294,9296,9299],{"className":9295},[60],[26,9297],{"className":9298,"style":4270},[64],[26,9300,341],{"className":9301},[69],[26,9303,9305,9318],{"className":9304},[29],[26,9306,9308],{"className":9307},[33],[35,9309,9310],{"xmlns":37},[39,9311,9312,9316],{},[42,9313,9314],{},[554,9315,9177],{},[49,9317,9177],{"encoding":51},[26,9319,9321],{"className":9320,"ariaHidden":56},[55],[26,9322,9324,9327],{"className":9323},[60],[26,9325],{"className":9326,"style":4270},[64],[26,9328,9177],{"className":9329},[69],[13,9331,2825],{"id":2824},[2827,9333,9334,9338,9341,9345,9348,9352,9355,9359],{},[1738,9335,9337],{"id":9336},"observe-apparent-dips","Observe Apparent Dips",[18,9339,9340],{},"Identify two lines on the plane where the gradient (apparent dip) and the compass bearing can be measured .",[1738,9342,9344],{"id":9343},"sketch-the-problem","Sketch the Problem",[18,9346,9347],{},"Plot the two bearings. Note that the full dip direction must lie \"between\" the two observed directions if they are on either side of the peak, or \"beyond\" them if the grade is strictly increasing .",[1738,9349,9351],{"id":9350},"calculate-direction","Calculate Direction",[18,9353,9354],{},"Use the tangent formula to find the bearing of the strike line or full dip .",[1738,9356,9358],{"id":9357},"calculate-rate","Calculate Rate",[18,9360,9361,9362,9401],{},"Apply the ",[26,9363,9365,9383],{"className":9364},[29],[26,9366,9368],{"className":9367},[33],[35,9369,9370],{"xmlns":37},[39,9371,9372,9380],{},[42,9373,9374,9376,9378],{},[45,9375,7407],{},[89,9377,1962],{},[45,9379,7357],{},[49,9381,9382],{"encoding":51},"\\cos \\phi",[26,9384,9386],{"className":9385,"ariaHidden":56},[55],[26,9387,9389,9392,9395,9398],{"className":9388},[60],[26,9390],{"className":9391,"style":1898},[64],[26,9393,7407],{"className":9394},[2079],[26,9396],{"className":9397,"style":124},[123],[26,9399,7357],{"className":9400},[69,70]," correction to find the maximum gradient .",[13,9403,5301],{"id":5300},[1926,9405,9406,9602],{},[1929,9407,9408,9411,9412,9473,9474,275,9538,5870],{},[21,9409,9410],{},"Strike Line Check:"," If you find the direction of full dip is ",[26,9413,9415,9432],{"className":9414},[29],[26,9416,9418],{"className":9417},[33],[35,9419,9420],{"xmlns":37},[39,9421,9422,9430],{},[42,9423,9424],{},[3513,9425,9426,9428],{},[554,9427,5145],{},[89,9429,3520],{},[49,9431,5150],{"encoding":51},[26,9433,9435],{"className":9434,"ariaHidden":56},[55],[26,9436,9438,9441,9444],{"className":9437},[60],[26,9439],{"className":9440,"style":3551},[64],[26,9442,5163],{"className":9443},[69],[26,9445,9447,9450],{"className":9446},[69],[26,9448,3562],{"className":9449},[69],[26,9451,9453],{"className":9452},[417],[26,9454,9456],{"className":9455},[421],[26,9457,9459],{"className":9458},[426],[26,9460,9462],{"className":9461,"style":3551},[430],[26,9463,9464,9467],{"style":3577},[26,9465],{"className":9466,"style":439},[438],[26,9468,9470],{"className":9469},[443,444,445,446],[26,9471,3520],{"className":9472},[773,446],", then the strike lines (where the level is constant) must run exactly ",[26,9475,9477,9496],{"className":9476},[29],[26,9478,9480],{"className":9479},[33],[35,9481,9482],{"xmlns":37},[39,9483,9484,9493],{},[42,9485,9486],{},[3513,9487,9488,9491],{},[554,9489,9490],{},"090",[89,9492,3520],{},[49,9494,9495],{"encoding":51},"090^\\circ",[26,9497,9499],{"className":9498,"ariaHidden":56},[55],[26,9500,9502,9505,9509],{"className":9501},[60],[26,9503],{"className":9504,"style":3551},[64],[26,9506,9508],{"className":9507},[69],"09",[26,9510,9512,9515],{"className":9511},[69],[26,9513,3562],{"className":9514},[69],[26,9516,9518],{"className":9517},[417],[26,9519,9521],{"className":9520},[421],[26,9522,9524],{"className":9523},[426],[26,9525,9527],{"className":9526,"style":3551},[430],[26,9528,9529,9532],{"style":3577},[26,9530],{"className":9531,"style":439},[438],[26,9533,9535],{"className":9534},[443,444,445,446],[26,9536,3520],{"className":9537},[773,446],[26,9539,9541,9560],{"className":9540},[29],[26,9542,9544],{"className":9543},[33],[35,9545,9546],{"xmlns":37},[39,9547,9548,9557],{},[42,9549,9550],{},[3513,9551,9552,9555],{},[554,9553,9554],{},"270",[89,9556,3520],{},[49,9558,9559],{"encoding":51},"270^\\circ",[26,9561,9563],{"className":9562,"ariaHidden":56},[55],[26,9564,9566,9569,9573],{"className":9565},[60],[26,9567],{"className":9568,"style":3551},[64],[26,9570,9572],{"className":9571},[69],"27",[26,9574,9576,9579],{"className":9575},[69],[26,9577,3562],{"className":9578},[69],[26,9580,9582],{"className":9581},[417],[26,9583,9585],{"className":9584},[421],[26,9586,9588],{"className":9587},[426],[26,9589,9591],{"className":9590,"style":3551},[430],[26,9592,9593,9596],{"style":3577},[26,9594],{"className":9595,"style":439},[438],[26,9597,9599],{"className":9598},[443,444,445,446],[26,9600,3520],{"className":9601},[773,446],[1929,9603,9604,9607,9608,9664],{},[21,9605,9606],{},"Reciprocal Ratios:"," In these formulas, it is often easier to work with gradients as decimals (",[26,9609,9611,9634],{"className":9610},[29],[26,9612,9614],{"className":9613},[33],[35,9615,9616],{"xmlns":37},[39,9617,9618,9631],{},[42,9619,9620,9622,9624,9626,9628],{},[554,9621,341],{},[45,9623,3991],{"mathvariant":2636},[554,9625,7557],{},[89,9627,543],{},[554,9629,9630],{},"0.0625",[49,9632,9633],{"encoding":51},"1\u002F16 = 0.0625",[26,9635,9637,9655],{"className":9636,"ariaHidden":56},[55],[26,9638,9640,9643,9646,9649,9652],{"className":9639},[60],[26,9641],{"className":9642,"style":3328},[64],[26,9644,8840],{"className":9645},[69],[26,9647],{"className":9648,"style":676},[123],[26,9650,543],{"className":9651},[680],[26,9653],{"className":9654,"style":676},[123],[26,9656,9658,9661],{"className":9657},[60],[26,9659],{"className":9660,"style":4270},[64],[26,9662,9630],{"className":9663},[69],") to avoid confusion .",[13,9666,5569],{"id":5568},[1926,9668,9669,9675],{},[1929,9670,9671,9674],{},[21,9672,9673],{},"Sign Errors:"," Forgetting that the full dip must be the steepest possible line. If your calculated full dip is flatter than your observed apparent dip, your math is wrong .",[1929,9676,9677,9680,9681,9709,9710,9713],{},[21,9678,9679],{},"Angular Difference:"," The angle ",[26,9682,9684,9697],{"className":9683},[29],[26,9685,9687],{"className":9686},[33],[35,9688,9689],{"xmlns":37},[39,9690,9691,9695],{},[42,9692,9693],{},[45,9694,7357],{},[49,9696,7360],{"encoding":51},[26,9698,9700],{"className":9699,"ariaHidden":56},[55],[26,9701,9703,9706],{"className":9702},[60],[26,9704],{"className":9705,"style":1898},[64],[26,9707,7357],{"className":9708},[69,70]," is the ",[2588,9711,9712],{},"difference"," between the full dip direction and the apparent dip direction, not the bearing itself .",[13,9715,5694],{"id":5693},[5696,9717,9718,9724,9730],{},[5699,9719,9721],{"label":9720},"What is a 'Strike Line' used for?",[18,9722,9723],{},"In site grading, a strike line is used to set out \"contours\" on a plane. Any two points on a strike line have the exact same elevation .",[5699,9725,9727],{"label":9726},"Can I find full dip with only one measurement?",[18,9728,9729],{},"No. You need at least two apparent dip measurements in different directions to uniquely define the 3D plane .",[5699,9731,9733],{"label":9732},"How is this used in volumes?",[18,9734,9735],{},"By knowing the dip and strike of a rock stratum, you can calculate the \"wedge\" of rock that needs to be excavated before you even start digging .",[13,9737,5757],{"id":5756},[18,9739,9740,9741,9798],{},"Dip and strike calculations allow the surveyor to model complex 3D surfaces from just a few field observations. By mastering the ",[26,9742,9744,9768],{"className":9743},[29],[26,9745,9747],{"className":9746},[33],[35,9748,9749],{"xmlns":37},[39,9750,9751,9765],{},[42,9752,9753,9755,9757,9759,9761,9763],{},[45,9754,7388],{},[89,9756,1962],{},[45,9758,7251],{},[45,9760,7407],{},[89,9762,1962],{},[45,9764,7357],{},[49,9766,9767],{"encoding":51},"\\tan \\theta \\cos \\phi",[26,9769,9771],{"className":9770,"ariaHidden":56},[55],[26,9772,9774,9777,9780,9783,9786,9789,9792,9795],{"className":9773},[60],[26,9775],{"className":9776,"style":1898},[64],[26,9778,7388],{"className":9779},[2079],[26,9781],{"className":9782,"style":124},[123],[26,9784,7251],{"className":9785,"style":7268},[69,70],[26,9787],{"className":9788,"style":124},[123],[26,9790,7407],{"className":9791},[2079],[26,9793],{"className":9794,"style":124},[123],[26,9796,7357],{"className":9797},[69,70]," formula, you can provide engineers with critical data on subsurface rock layers and optimize the design of site slopes.",[13,9800,5764],{"id":5763},[18,9802,5767,9803,6374],{},[2588,9804,5770],{},{"title":5781,"searchDepth":5782,"depth":5782,"links":9806},[9807,9808,9809,9810,9811,9817,9818,9819,9820,9821],{"id":15,"depth":5782,"text":16},{"id":194,"depth":5782,"text":195},{"id":344,"depth":5782,"text":345},{"id":3282,"depth":5782,"text":3283},{"id":2824,"depth":5782,"text":2825,"children":9812},[9813,9814,9815,9816],{"id":9336,"depth":5791,"text":9337},{"id":9343,"depth":5791,"text":9344},{"id":9350,"depth":5791,"text":9351},{"id":9357,"depth":5791,"text":9358},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Earthworks","A technical guide to calculating the direction and rate of full dip from apparent dip observations on tilted stratum planes.","\u002Fimages\u002Fdip-strike.jpg",{"authorCredentials":9826,"commentCount":6400,"tags":9827,"relatedTools":9832,"relatedArticles":9835,"sources":9838},"P.Eng",[9828,9822,9829,9830,9831],"Dip and Strike","Geology","Math","Site Grading",[9833,9834],"Scientific Calculator","Geological Compass",[9836,9837],"mass-haul-diagrams-earthwork-economics","prismoidal-vs-end-area-volumes",[9839],{"title":5828,"type":5829,"note":9840},"Primary source for the mathematical relationship between full and apparent dip.","\u002Fblog\u002Fdip-and-strike-geological-surveying","2024-04-13",{"title":7131,"description":9823},{"loc":9841},"blog\u002F26.dip-and-strike-geological-surveying","aeqVAcmdiQ6JrInRB2dXlN8ViW9yXz5SeCOL8e_cu2c",{"id":9848,"title":9849,"author":5850,"body":9850,"category":7103,"description":10429,"extension":5809,"image":10430,"meta":10431,"navigation":5839,"path":10443,"publishedAt":10444,"readTime":10445,"seo":10446,"sitemap":10447,"stem":10448,"__hash__":10449},"blog\u002Fblog\u002F25.machine-guidance-rtk-gps-construction.md","Machine Guidance: Integrating RTK GPS with Construction Plant",{"type":10,"value":9851,"toc":10410},[9852,9854,9861,9863,9870,9872,9875,9877,9948,9950,9987,9989,10281,10283,10304,10306,10320,10322,10399,10401,10404,10406],[13,9853,16],{"id":15},[18,9855,9856,9857,9860],{},"One of the most transformative shifts in modern construction is the move from manual setting-out to ",[21,9858,9859],{},"Machine Guidance",". By mounting RTK GPS receivers directly onto earthmoving plant—such as bulldozers, graders, and excavators—operators can view the design surface in real-time on a cabin-mounted display. This eliminates the need for thousands of timber stakes and batter boards on a site .",[13,9862,155],{"id":154},[18,9864,9865,9866,9869],{},"Traditional earthworks require a surveyor to constantly replace pegs destroyed by machinery. Machine guidance provides continuous, 3D positioning of the machine's blade or bucket relative to the ",[21,9867,9868],{},"Digital Ground Model (DGM)",". This increases productivity, reduces the risk of over-excavation, and significantly improves safety by keeping survey personnel away from heavy machinery .",[13,9871,173],{"id":172},[18,9873,9874],{},"Machine guidance relies on a high-speed data link. An RTK base station on site transmits corrections to a receiver on the machine. Sensors on the machine’s chassis and blade (tilt sensors, encoders) then determine the exact position of the cutting edge relative to the GPS antenna .",[13,9876,195],{"id":194},[18,9878,9879,9880,9947],{},"The system works by comparing the machine's current coordinates ",[26,9881,9883,9910],{"className":9882},[29],[26,9884,9886],{"className":9885},[33],[35,9887,9888],{"xmlns":37},[39,9889,9890,9907],{},[42,9891,9892,9894,9896,9898,9900,9902,9905],{},[89,9893,366],{"stretchy":365},[45,9895,372],{},[89,9897,91],{"separator":56},[45,9899,381],{},[89,9901,91],{"separator":56},[45,9903,9904],{},"H",[89,9906,386],{"stretchy":365},[49,9908,9909],{"encoding":51},"(E, N, H)",[26,9911,9913],{"className":9912,"ariaHidden":56},[55],[26,9914,9916,9919,9922,9925,9928,9931,9934,9937,9940,9944],{"className":9915},[60],[26,9917],{"className":9918,"style":3328},[64],[26,9920,366],{"className":9921},[406],[26,9923,372],{"className":9924,"style":413},[69,70],[26,9926,91],{"className":9927},[119],[26,9929],{"className":9930,"style":124},[123],[26,9932,381],{"className":9933,"style":476},[69,70],[26,9935,91],{"className":9936},[119],[26,9938],{"className":9939,"style":124},[123],[26,9941,9904],{"className":9942,"style":9943},[69,70],"margin-right:0.0813em;",[26,9945,386],{"className":9946},[515]," with the design coordinates stored in the on-board DGM. The difference represents the \"cut\" or \"fill\" required at that exact spot .",[13,9949,2825],{"id":2824},[2827,9951,9952,9956,9959,9963,9966,9970,9973,9977,9980,9984],{},[1738,9953,9955],{"id":9954},"prepare-the-dgm","Prepare the DGM",[18,9957,9958],{},"The surveyor creates a digital design surface (triangular facets or regular grid) representing the final formation levels .",[1738,9960,9962],{"id":9961},"set-up-rtk-base","Set Up RTK Base",[18,9964,9965],{},"Establish a fixed base station on a known control point with a clear view of the sky and a robust radio link to the machinery .",[1738,9967,9969],{"id":9968},"calibration","Calibration",[18,9971,9972],{},"Calibrate the machine’s blade position. This involves taking GPS readings while the blade is on a known survey point to establish the vertical offset between the antenna and the cutting edge .",[1738,9974,9976],{"id":9975},"real-time-operation","Real-Time Operation",[18,9978,9979],{},"The operator follows the cabin screen, which indicates how much to lower or raise the blade. In \"Automatic\" mode, the system directly controls the machine's hydraulics to maintain the design grade .",[1738,9981,9983],{"id":9982},"as-built-verification","As-Built Verification",[18,9985,9986],{},"As the machine finishes a pass, the system records the final coordinates, providing an instantaneous \"as-built\" survey of the formation .",[13,9988,345],{"id":344},[18,9990,9991,9992,8649,10027,10239,10240,10273,10274,10277,10278,5870],{},"The system calculates the vertical height difference (",[26,9993,9995,10012],{"className":9994},[29],[26,9996,9998],{"className":9997},[33],[35,9999,10000],{"xmlns":37},[39,10001,10002,10009],{},[42,10003,10004,10007],{},[45,10005,10006],{"mathvariant":2636},"Δ",[45,10008,9904],{},[49,10010,10011],{"encoding":51},"\\Delta H",[26,10013,10015],{"className":10014,"ariaHidden":56},[55],[26,10016,10018,10021,10024],{"className":10017},[60],[26,10019],{"className":10020,"style":65},[64],[26,10022,10006],{"className":10023},[69],[26,10025,9904],{"className":10026,"style":9943},[69,70],[26,10028,10030,10073],{"className":10029},[29],[26,10031,10033],{"className":10032},[33],[35,10034,10035],{"xmlns":37},[39,10036,10037,10070],{},[42,10038,10039,10041,10043,10045,10052,10054,10056,10059,10061,10063],{},[45,10040,10006],{"mathvariant":2636},[45,10042,9904],{},[89,10044,543],{},[368,10046,10047,10049],{},[45,10048,9904],{},[5057,10050,10051],{},"GPS",[89,10053,1967],{},[89,10055,366],{"stretchy":365},[5057,10057,10058],{},"Antenna Offset",[89,10060,386],{"stretchy":365},[89,10062,1967],{},[368,10064,10065,10067],{},[45,10066,9904],{},[5057,10068,10069],{},"Design",[49,10071,10072],{"encoding":51},"\\Delta H = H_{\\text{GPS}} - (\\text{Antenna Offset}) - H_{\\text{Design}}",[26,10074,10076,10097,10160,10187],{"className":10075,"ariaHidden":56},[55],[26,10077,10079,10082,10085,10088,10091,10094],{"className":10078},[60],[26,10080],{"className":10081,"style":65},[64],[26,10083,10006],{"className":10084},[69],[26,10086,9904],{"className":10087,"style":9943},[69,70],[26,10089],{"className":10090,"style":676},[123],[26,10092,543],{"className":10093},[680],[26,10095],{"className":10096,"style":676},[123],[26,10098,10100,10103,10151,10154,10157],{"className":10099},[60],[26,10101],{"className":10102,"style":1986},[64],[26,10104,10106,10109],{"className":10105},[69],[26,10107,9904],{"className":10108,"style":9943},[69,70],[26,10110,10112],{"className":10111},[417],[26,10113,10115,10143],{"className":10114},[421,422],[26,10116,10118,10140],{"className":10117},[426],[26,10119,10122],{"className":10120,"style":10121},[430],"height:0.3283em;",[26,10123,10125,10128],{"style":10124},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[26,10126],{"className":10127,"style":439},[438],[26,10129,10131],{"className":10130},[443,444,445,446],[26,10132,10134],{"className":10133},[69,446],[26,10135,10137],{"className":10136},[69,5118,446],[26,10138,10051],{"className":10139},[69,446],[26,10141,454],{"className":10142},[453],[26,10144,10146],{"className":10145},[426],[26,10147,10149],{"className":10148,"style":2027},[430],[26,10150],{},[26,10152],{"className":10153,"style":4156},[123],[26,10155,1967],{"className":10156},[773],[26,10158],{"className":10159,"style":4156},[123],[26,10161,10163,10166,10169,10175,10178,10181,10184],{"className":10162},[60],[26,10164],{"className":10165,"style":3328},[64],[26,10167,366],{"className":10168},[406],[26,10170,10172],{"className":10171},[69,5118],[26,10173,10058],{"className":10174},[69],[26,10176,386],{"className":10177},[515],[26,10179],{"className":10180,"style":4156},[123],[26,10182,1967],{"className":10183},[773],[26,10185],{"className":10186,"style":4156},[123],[26,10188,10190,10193],{"className":10189},[60],[26,10191],{"className":10192,"style":632},[64],[26,10194,10196,10199],{"className":10195},[69],[26,10197,9904],{"className":10198,"style":9943},[69,70],[26,10200,10202],{"className":10201},[417],[26,10203,10205,10231],{"className":10204},[421,422],[26,10206,10208,10228],{"className":10207},[426],[26,10209,10211],{"className":10210,"style":10121},[430],[26,10212,10213,10216],{"style":10124},[26,10214],{"className":10215,"style":439},[438],[26,10217,10219],{"className":10218},[443,444,445,446],[26,10220,10222],{"className":10221},[69,446],[26,10223,10225],{"className":10224},[69,5118,446],[26,10226,10069],{"className":10227},[69,446],[26,10229,454],{"className":10230},[453],[26,10232,10234],{"className":10233},[426],[26,10235,10237],{"className":10236,"style":461},[430],[26,10238],{},"\nIf ",[26,10241,10243,10258],{"className":10242},[29],[26,10244,10246],{"className":10245},[33],[35,10247,10248],{"xmlns":37},[39,10249,10250,10256],{},[42,10251,10252,10254],{},[45,10253,10006],{"mathvariant":2636},[45,10255,9904],{},[49,10257,10011],{"encoding":51},[26,10259,10261],{"className":10260,"ariaHidden":56},[55],[26,10262,10264,10267,10270],{"className":10263},[60],[26,10265],{"className":10266,"style":65},[64],[26,10268,10006],{"className":10269},[69],[26,10271,9904],{"className":10272,"style":9943},[69,70]," is positive, it's a ",[21,10275,10276],{},"Cut","; if negative, it's a ",[21,10279,10280],{},"Fill",[13,10282,5301],{"id":5300},[1926,10284,10285,10298],{},[1929,10286,10287,10290,10291,275,10294,10297],{},[21,10288,10289],{},"Dual Antennas:"," Use dual GPS antennas on the machine to determine not just position, but also the precise ",[21,10292,10293],{},"heading",[21,10295,10296],{},"cross-slope"," of the blade .",[1929,10299,10300,10303],{},[21,10301,10302],{},"Base Stability:"," The base station must be perfectly stable. Any settlement or movement of the base tripod will cause every machine on the site to excavate to the wrong level .",[13,10305,5569],{"id":5568},[1926,10307,10308,10314],{},[1929,10309,10310,10313],{},[21,10311,10312],{},"Datum Mismatch:"," Ensuring the design DGM and the GPS base station are on the exact same local coordinate system and vertical datum is the most common point of failure .",[1929,10315,10316,10319],{},[21,10317,10318],{},"Signal Masking:"," Large buildings or dense tree cover can block GPS signals. In these areas, the system may revert to \"Float\" mode, which lacks the precision required for final grading .",[13,10321,5694],{"id":5693},[5696,10323,10324,10330,10393],{},[5699,10325,10327],{"label":10326},"Can machine guidance work at night?",[18,10328,10329],{},"Yes. Since the system uses satellite signals and internal design models, it can operate in total darkness or heavy fog with the same precision as a clear day .",[5699,10331,10333],{"label":10332},"What accuracy can be achieved?",[18,10334,10335,10336,10392],{},"With high-quality RTK, vertical accuracies of ",[26,10337,10339,10362],{"className":10338},[29],[26,10340,10342],{"className":10341},[33],[35,10343,10344],{"xmlns":37},[39,10345,10346,10359],{},[42,10347,10348,10351,10353,10355,10357],{},[89,10349,10350],{},"±",[554,10352,6250],{},[5057,10354,6037],{},[554,10356,6034],{},[5057,10358,6122],{},[49,10360,10361],{"encoding":51},"\\pm 20\\text{–}30\\text{ mm}",[26,10363,10365],{"className":10364,"ariaHidden":56},[55],[26,10366,10368,10371,10374,10377,10383,10386],{"className":10367},[60],[26,10369],{"className":10370,"style":4702},[64],[26,10372,10350],{"className":10373},[69],[26,10375,6250],{"className":10376},[69],[26,10378,10380],{"className":10379},[69,5118],[26,10381,6037],{"className":10382},[69],[26,10384,6034],{"className":10385},[69],[26,10387,10389],{"className":10388},[69,5118],[26,10390,6122],{"className":10391},[69]," are common, which is sufficient for most bulk earthworks and sub-base layers .",[5699,10394,10396],{"label":10395},"Does it replace the surveyor?",[18,10397,10398],{},"No. The surveyor is still required to establish the site control, create the DGM, and perform final audits to ensure the machine was calibrated correctly .",[13,10400,5757],{"id":5756},[18,10402,10403],{},"Machine guidance represents the ultimate integration of surveying data and mechanical execution. By putting the design directly into the operator's hands, the engineering surveyor moves from a role of \"peg-pusher\" to a \"data manager,\" ensuring the project is built faster and more accurately than ever before.",[13,10405,5764],{"id":5763},[18,10407,5767,10408,6374],{},[2588,10409,5770],{},{"title":5781,"searchDepth":5782,"depth":5782,"links":10411},[10412,10413,10414,10415,10416,10423,10424,10425,10426,10427,10428],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":172,"depth":5782,"text":173},{"id":194,"depth":5782,"text":195},{"id":2824,"depth":5782,"text":2825,"children":10417},[10418,10419,10420,10421,10422],{"id":9954,"depth":5791,"text":9955},{"id":9961,"depth":5791,"text":9962},{"id":9968,"depth":5791,"text":9969},{"id":9975,"depth":5791,"text":9976},{"id":9982,"depth":5791,"text":9983},{"id":344,"depth":5782,"text":345},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"A technical look at how RTK GPS and digital ground models are used to automate earthmoving machinery, eliminating the need for traditional setting-out stakes.","\u002Fimages\u002Fmachine-guidance.jpg",{"authorCredentials":9826,"commentCount":6400,"tags":10432,"relatedTools":10435,"relatedArticles":10439,"sources":10440},[9859,10051,10433,9822,10434],"RTK","Automation",[10436,10437,10438],"RTK GPS","Digital Ground Model","Bulldozer Control",[6412,9836],[10441],{"title":5828,"type":5829,"note":10442},"Primary source for the application of GPS in machine control and automation.","\u002Fblog\u002Fmachine-guidance-rtk-gps-construction","2024-04-12",14,{"title":9849,"description":10429},{"loc":10443},"blog\u002F25.machine-guidance-rtk-gps-construction","AiPTHEKy8ZTVOapNLblA895AuLMQ9sy_8ILxh2Zr2fc",{"id":10451,"title":10452,"author":5850,"body":10453,"category":15720,"description":15721,"extension":5809,"image":15722,"meta":15723,"navigation":5839,"path":15739,"publishedAt":15740,"readTime":15741,"seo":15742,"sitemap":15743,"stem":15744,"__hash__":15745},"blog\u002Fblog\u002F24.spherical-excess-large-networks.md","Spherical Excess: Adjusting Large-Scale Control Networks",{"type":10,"value":10454,"toc":15698},[10455,10457,10727,10729,10860,10862,10868,10870,10874,11454,11665,11669,11827,11829,12558,14533,14535,14964,14966,15179,15181,15448,15450,15621,15623,15692,15694],[13,10456,16],{"id":15},[18,10458,10459,10460,10521,10522,10592,10593,10654,10655,5870],{},"In small-scale engineering surveys, we treat the Earth as a flat plane, and the sum of the angles in a triangle equals exactly ",[26,10461,10463,10480],{"className":10462},[29],[26,10464,10466],{"className":10465},[33],[35,10467,10468],{"xmlns":37},[39,10469,10470,10478],{},[42,10471,10472],{},[3513,10473,10474,10476],{},[554,10475,5145],{},[89,10477,3520],{},[49,10479,5150],{"encoding":51},[26,10481,10483],{"className":10482,"ariaHidden":56},[55],[26,10484,10486,10489,10492],{"className":10485},[60],[26,10487],{"className":10488,"style":3551},[64],[26,10490,5163],{"className":10491},[69],[26,10493,10495,10498],{"className":10494},[69],[26,10496,3562],{"className":10497},[69],[26,10499,10501],{"className":10500},[417],[26,10502,10504],{"className":10503},[421],[26,10505,10507],{"className":10506},[426],[26,10508,10510],{"className":10509,"style":3551},[430],[26,10511,10512,10515],{"style":3577},[26,10513],{"className":10514,"style":439},[438],[26,10516,10518],{"className":10517},[443,444,445,446],[26,10519,3520],{"className":10520},[773,446],". However, as the area of the survey grows—typically exceeding ",[26,10523,10525,10546],{"className":10524},[29],[26,10526,10528],{"className":10527},[33],[35,10529,10530],{"xmlns":37},[39,10531,10532,10543],{},[42,10533,10534,10536],{},[554,10535,6119],{},[3513,10537,10538,10541],{},[5057,10539,10540],{}," km",[554,10542,572],{},[49,10544,10545],{"encoding":51},"200 \\text{ km}^2",[26,10547,10549],{"className":10548,"ariaHidden":56},[55],[26,10550,10552,10556,10559],{"className":10551},[60],[26,10553],{"className":10554,"style":10555},[64],"height:0.8984em;",[26,10557,6119],{"className":10558},[69],[26,10560,10562,10568],{"className":10561},[69],[26,10563,10565],{"className":10564},[69,5118],[26,10566,10540],{"className":10567},[69],[26,10569,10571],{"className":10570},[417],[26,10572,10574],{"className":10573},[421],[26,10575,10577],{"className":10576},[426],[26,10578,10580],{"className":10579,"style":10555},[430],[26,10581,10583,10586],{"style":10582},"top:-3.1473em;margin-right:0.05em;",[26,10584],{"className":10585,"style":439},[438],[26,10587,10589],{"className":10588},[443,444,445,446],[26,10590,572],{"className":10591},[69,446],"—the Earth’s curvature becomes significant. The angles of a triangle observed on the curved surface of the ellipsoid will sum to more than ",[26,10594,10596,10613],{"className":10595},[29],[26,10597,10599],{"className":10598},[33],[35,10600,10601],{"xmlns":37},[39,10602,10603,10611],{},[42,10604,10605],{},[3513,10606,10607,10609],{},[554,10608,5145],{},[89,10610,3520],{},[49,10612,5150],{"encoding":51},[26,10614,10616],{"className":10615,"ariaHidden":56},[55],[26,10617,10619,10622,10625],{"className":10618},[60],[26,10620],{"className":10621,"style":3551},[64],[26,10623,5163],{"className":10624},[69],[26,10626,10628,10631],{"className":10627},[69],[26,10629,3562],{"className":10630},[69],[26,10632,10634],{"className":10633},[417],[26,10635,10637],{"className":10636},[421],[26,10638,10640],{"className":10639},[426],[26,10641,10643],{"className":10642,"style":3551},[430],[26,10644,10645,10648],{"style":3577},[26,10646],{"className":10647,"style":439},[438],[26,10649,10651],{"className":10650},[443,444,445,446],[26,10652,3520],{"className":10653},[773,446],". This additional amount is known as ",[21,10656,10657,10658,386],{},"Spherical Excess (",[26,10659,10661,10684],{"className":10660},[29],[26,10662,10664],{"className":10663},[33],[35,10665,10666],{"xmlns":37},[39,10667,10668,10681],{},[42,10669,10670],{},[3513,10671,10672,10674],{},[45,10673,372],{},[42,10675,10676,10679],{},[89,10677,10678],{"mathvariant":2636},"′",[89,10680,10678],{"mathvariant":2636},[49,10682,10683],{"encoding":51},"E''",[26,10685,10687],{"className":10686,"ariaHidden":56},[55],[26,10688,10690,10694],{"className":10689},[60],[26,10691],{"className":10692,"style":10693},[64],"height:0.7519em;",[26,10695,10697,10700],{"className":10696},[69],[26,10698,372],{"className":10699,"style":413},[69,70],[26,10701,10703],{"className":10702},[417],[26,10704,10706],{"className":10705},[421],[26,10707,10709],{"className":10708},[426],[26,10710,10712],{"className":10711,"style":10693},[430],[26,10713,10714,10717],{"style":3577},[26,10715],{"className":10716,"style":439},[438],[26,10718,10720],{"className":10719},[443,444,445,446],[26,10721,10723],{"className":10722},[69,446],[26,10724,10726],{"className":10725},[69,446],"′′",[13,10728,155],{"id":154},[18,10730,10731,10732,10793,10794,10859],{},"For primary triangulation or major infrastructure projects spanning dozens of kilometers, ignoring spherical excess introduces a systematic error into the network. If the angles are forced to close to ",[26,10733,10735,10752],{"className":10734},[29],[26,10736,10738],{"className":10737},[33],[35,10739,10740],{"xmlns":37},[39,10741,10742,10750],{},[42,10743,10744],{},[3513,10745,10746,10748],{},[554,10747,5145],{},[89,10749,3520],{},[49,10751,5150],{"encoding":51},[26,10753,10755],{"className":10754,"ariaHidden":56},[55],[26,10756,10758,10761,10764],{"className":10757},[60],[26,10759],{"className":10760,"style":3551},[64],[26,10762,5163],{"className":10763},[69],[26,10765,10767,10770],{"className":10766},[69],[26,10768,3562],{"className":10769},[69],[26,10771,10773],{"className":10772},[417],[26,10774,10776],{"className":10775},[421],[26,10777,10779],{"className":10778},[426],[26,10780,10782],{"className":10781,"style":3551},[430],[26,10783,10784,10787],{"style":3577},[26,10785],{"className":10786,"style":439},[438],[26,10788,10790],{"className":10789},[443,444,445,446],[26,10791,3520],{"className":10792},[773,446]," without accounting for ",[26,10795,10797,10818],{"className":10796},[29],[26,10798,10800],{"className":10799},[33],[35,10801,10802],{"xmlns":37},[39,10803,10804,10816],{},[42,10805,10806],{},[3513,10807,10808,10810],{},[45,10809,372],{},[42,10811,10812,10814],{},[89,10813,10678],{"mathvariant":2636},[89,10815,10678],{"mathvariant":2636},[49,10817,10683],{"encoding":51},[26,10819,10821],{"className":10820,"ariaHidden":56},[55],[26,10822,10824,10827],{"className":10823},[60],[26,10825],{"className":10826,"style":10693},[64],[26,10828,10830,10833],{"className":10829},[69],[26,10831,372],{"className":10832,"style":413},[69,70],[26,10834,10836],{"className":10835},[417],[26,10837,10839],{"className":10838},[421],[26,10840,10842],{"className":10841},[426],[26,10843,10845],{"className":10844,"style":10693},[430],[26,10846,10847,10850],{"style":3577},[26,10848],{"className":10849,"style":439},[438],[26,10851,10853],{"className":10852},[443,444,445,446],[26,10854,10856],{"className":10855},[69,446],[26,10857,10726],{"className":10858},[69,446],", the resulting coordinates will be distorted, leading to misclosures in long-distance traverses or tunnel breakthroughs .",[13,10861,195],{"id":194},[18,10863,10864,10867],{},[21,10865,10866],{},"Legendre’s Theorem"," states that for a spherical triangle whose sides are very small compared to the radius of the sphere, the area is practically the same as a plane triangle with the same side lengths. To adjust the triangle, the spherical excess is calculated and then distributed among the three observed angles, effectively reducing them to \"plane\" angles for standard trigonometric computation .",[13,10869,345],{"id":344},[1738,10871,10873],{"id":10872},"_1-the-spherical-excess-formula","1. The Spherical Excess Formula",[18,10875,10876,10877,10942,10943,11453],{},"The excess (",[26,10878,10880,10901],{"className":10879},[29],[26,10881,10883],{"className":10882},[33],[35,10884,10885],{"xmlns":37},[39,10886,10887,10899],{},[42,10888,10889],{},[3513,10890,10891,10893],{},[45,10892,372],{},[42,10894,10895,10897],{},[89,10896,10678],{"mathvariant":2636},[89,10898,10678],{"mathvariant":2636},[49,10900,10683],{"encoding":51},[26,10902,10904],{"className":10903,"ariaHidden":56},[55],[26,10905,10907,10910],{"className":10906},[60],[26,10908],{"className":10909,"style":10693},[64],[26,10911,10913,10916],{"className":10912},[69],[26,10914,372],{"className":10915,"style":413},[69,70],[26,10917,10919],{"className":10918},[417],[26,10920,10922],{"className":10921},[421],[26,10923,10925],{"className":10924},[426],[26,10926,10928],{"className":10927,"style":10693},[430],[26,10929,10930,10933],{"style":3577},[26,10931],{"className":10932,"style":439},[438],[26,10934,10936],{"className":10935},[443,444,445,446],[26,10937,10939],{"className":10938},[69,446],[26,10940,10726],{"className":10941},[69,446],") in seconds of arc is proportional to the area of the triangle:\n",[26,10944,10946,11041],{"className":10945},[29],[26,10947,10949],{"className":10948},[33],[35,10950,10951],{"xmlns":37},[39,10952,10953,11038],{},[42,10954,10955,10965,10967,10995,10998,11028,11030,11033,11035],{},[3513,10956,10957,10959],{},[45,10958,372],{},[42,10960,10961,10963],{},[89,10962,10678],{"mathvariant":2636},[89,10964,10678],{"mathvariant":2636},[89,10966,543],{},[545,10968,10969,10972],{},[5057,10970,10971],{},"Area of Triangle",[42,10973,10974,10981,10983,10985],{},[3513,10975,10976,10979],{},[45,10977,10978],{},"R",[554,10980,572],{},[45,10982,8523],{},[89,10984,1962],{},[3513,10986,10987,10989],{},[554,10988,341],{},[42,10990,10991,10993],{},[89,10992,10678],{"mathvariant":2636},[89,10994,10678],{"mathvariant":2636},[89,10996,10997],{},"≈",[545,10999,11000,11018],{},[42,11001,11002,11004,11006,11008,11010],{},[45,11003,562],{},[45,11005,579],{},[45,11007,8523],{},[89,11009,1962],{},[11011,11012,11013,11015],"mover",{"accent":56},[45,11014,99],{},[89,11016,11017],{},"^",[42,11019,11020,11022],{},[554,11021,572],{},[3513,11023,11024,11026],{},[45,11025,10978],{},[554,11027,572],{},[89,11029,4578],{},[554,11031,11032],{},"206",[89,11034,91],{"separator":56},[554,11036,11037],{},"265",[49,11039,11040],{"encoding":51},"E'' = \\frac{\\text{Area of Triangle}}{R^2 \\sin 1''} \\approx \\frac{ab \\sin \\hat{C}}{2R^2} \\times 206,265",[26,11042,11044,11091,11264,11434],{"className":11043,"ariaHidden":56},[55],[26,11045,11047,11050,11082,11085,11088],{"className":11046},[60],[26,11048],{"className":11049,"style":10693},[64],[26,11051,11053,11056],{"className":11052},[69],[26,11054,372],{"className":11055,"style":413},[69,70],[26,11057,11059],{"className":11058},[417],[26,11060,11062],{"className":11061},[421],[26,11063,11065],{"className":11064},[426],[26,11066,11068],{"className":11067,"style":10693},[430],[26,11069,11070,11073],{"style":3577},[26,11071],{"className":11072,"style":439},[438],[26,11074,11076],{"className":11075},[443,444,445,446],[26,11077,11079],{"className":11078},[69,446],[26,11080,10726],{"className":11081},[69,446],[26,11083],{"className":11084,"style":676},[123],[26,11086,543],{"className":11087},[680],[26,11089],{"className":11090,"style":676},[123],[26,11092,11094,11098,11255,11258,11261],{"className":11093},[60],[26,11095],{"className":11096,"style":11097},[64],"height:1.2772em;vertical-align:-0.345em;",[26,11099,11101,11104,11252],{"className":11100},[69],[26,11102],{"className":11103},[406,697],[26,11105,11107],{"className":11106},[545],[26,11108,11110,11243],{"className":11109},[421,422],[26,11111,11113,11240],{"className":11112},[426],[26,11114,11117,11214,11222],{"className":11115,"style":11116},[430],"height:0.9322em;",[26,11118,11119,11122],{"style":713},[26,11120],{"className":11121,"style":717},[438],[26,11123,11125],{"className":11124},[443,444,445,446],[26,11126,11128,11160,11163,11178,11181],{"className":11127},[69,446],[26,11129,11131,11135],{"className":11130},[69,446],[26,11132,10978],{"className":11133,"style":11134},[69,70,446],"margin-right:0.0077em;",[26,11136,11138],{"className":11137},[417],[26,11139,11141],{"className":11140},[421],[26,11142,11144],{"className":11143},[426],[26,11145,11148],{"className":11146,"style":11147},[430],"height:0.7463em;",[26,11149,11151,11154],{"style":11150},"top:-2.786em;margin-right:0.0714em;",[26,11152],{"className":11153,"style":749},[438],[26,11155,11157],{"className":11156},[443,753,754,446],[26,11158,572],{"className":11159},[69,446],[26,11161],{"className":11162,"style":2094},[123,446],[26,11164,11166,11170,11174],{"className":11165},[2079,446],[26,11167,11169],{"className":11168},[446],"s",[26,11171,11173],{"className":11172},[446],"i",[26,11175,11177],{"className":11176},[446],"n",[26,11179],{"className":11180,"style":2094},[123,446],[26,11182,11184,11187],{"className":11183},[69,446],[26,11185,341],{"className":11186},[69,446],[26,11188,11190],{"className":11189},[417],[26,11191,11193],{"className":11192},[421],[26,11194,11196],{"className":11195},[426],[26,11197,11200],{"className":11198,"style":11199},[430],"height:0.6828em;",[26,11201,11202,11205],{"style":11150},[26,11203],{"className":11204,"style":749},[438],[26,11206,11208],{"className":11207},[443,753,754,446],[26,11209,11211],{"className":11210},[69,446],[26,11212,10726],{"className":11213},[69,446],[26,11215,11216,11219],{"style":859},[26,11217],{"className":11218,"style":717},[438],[26,11220],{"className":11221,"style":867},[866],[26,11223,11225,11228],{"style":11224},"top:-3.4461em;",[26,11226],{"className":11227,"style":717},[438],[26,11229,11231],{"className":11230},[443,444,445,446],[26,11232,11234],{"className":11233},[69,446],[26,11235,11237],{"className":11236},[69,5118,446],[26,11238,10971],{"className":11239},[69,446],[26,11241,454],{"className":11242},[453],[26,11244,11246],{"className":11245},[426],[26,11247,11250],{"className":11248,"style":11249},[430],"height:0.345em;",[26,11251],{},[26,11253],{"className":11254},[515,697],[26,11256],{"className":11257,"style":676},[123],[26,11259,10997],{"className":11260},[680],[26,11262],{"className":11263,"style":676},[123],[26,11265,11267,11271,11425,11428,11431],{"className":11266},[60],[26,11268],{"className":11269,"style":11270},[64],"height:1.4017em;vertical-align:-0.345em;",[26,11272,11274,11277,11422],{"className":11273},[69],[26,11275],{"className":11276},[406,697],[26,11278,11280],{"className":11279},[545],[26,11281,11283,11414],{"className":11282},[421,422],[26,11284,11286,11411],{"className":11285},[426],[26,11287,11290,11333,11341],{"className":11288,"style":11289},[430],"height:1.0567em;",[26,11291,11292,11295],{"style":713},[26,11293],{"className":11294,"style":717},[438],[26,11296,11298],{"className":11297},[443,444,445,446],[26,11299,11301,11304],{"className":11300},[69,446],[26,11302,572],{"className":11303},[69,446],[26,11305,11307,11310],{"className":11306},[69,446],[26,11308,10978],{"className":11309,"style":11134},[69,70,446],[26,11311,11313],{"className":11312},[417],[26,11314,11316],{"className":11315},[421],[26,11317,11319],{"className":11318},[426],[26,11320,11322],{"className":11321,"style":11147},[430],[26,11323,11324,11327],{"style":11150},[26,11325],{"className":11326,"style":749},[438],[26,11328,11330],{"className":11329},[443,753,754,446],[26,11331,572],{"className":11332},[69,446],[26,11334,11335,11338],{"style":859},[26,11336],{"className":11337,"style":717},[438],[26,11339],{"className":11340,"style":867},[866],[26,11342,11343,11346],{"style":2129},[26,11344],{"className":11345,"style":717},[438],[26,11347,11349],{"className":11348},[443,444,445,446],[26,11350,11352,11356,11359,11371,11374],{"className":11351},[69,446],[26,11353,11355],{"className":11354},[69,70,446],"ab",[26,11357],{"className":11358,"style":2094},[123,446],[26,11360,11362,11365,11368],{"className":11361},[2079,446],[26,11363,11169],{"className":11364},[446],[26,11366,11173],{"className":11367},[446],[26,11369,11177],{"className":11370},[446],[26,11372],{"className":11373,"style":2094},[123,446],[26,11375,11378],{"className":11376},[69,11377,446],"accent",[26,11379,11381],{"className":11380},[421],[26,11382,11384],{"className":11383},[426],[26,11385,11388,11397],{"className":11386,"style":11387},[430],"height:0.9468em;",[26,11389,11391,11394],{"style":11390},"top:-2.7em;",[26,11392],{"className":11393,"style":439},[438],[26,11395,99],{"className":11396,"style":138},[69,70,446],[26,11398,11400,11403],{"style":11399},"top:-2.9523em;",[26,11401],{"className":11402,"style":439},[438],[26,11404,11408],{"className":11405,"style":11407},[11406],"accent-body","left:-0.1667em;",[26,11409,11017],{"className":11410},[69,446],[26,11412,454],{"className":11413},[453],[26,11415,11417],{"className":11416},[426],[26,11418,11420],{"className":11419,"style":11249},[430],[26,11421],{},[26,11423],{"className":11424},[515,697],[26,11426],{"className":11427,"style":4156},[123],[26,11429,4578],{"className":11430},[773],[26,11432],{"className":11433,"style":4156},[123],[26,11435,11437,11441,11444,11447,11450],{"className":11436},[60],[26,11438],{"className":11439,"style":11440},[64],"height:0.8389em;vertical-align:-0.1944em;",[26,11442,11032],{"className":11443},[69],[26,11445,91],{"className":11446},[119],[26,11448],{"className":11449,"style":124},[123],[26,11451,11037],{"className":11452},[69],"\nWhere:",[1926,11455,11456,11501,11567],{},[1929,11457,11458,11500],{},[26,11459,11461,11479],{"className":11460},[29],[26,11462,11464],{"className":11463},[33],[35,11465,11466],{"xmlns":37},[39,11467,11468,11476],{},[42,11469,11470,11472,11474],{},[45,11471,562],{},[89,11473,91],{"separator":56},[45,11475,579],{},[49,11477,11478],{"encoding":51},"a, b",[26,11480,11482],{"className":11481,"ariaHidden":56},[55],[26,11483,11485,11488,11491,11494,11497],{"className":11484},[60],[26,11486],{"className":11487,"style":1898},[64],[26,11489,562],{"className":11490},[69,70],[26,11492,91],{"className":11493},[119],[26,11495],{"className":11496,"style":124},[123],[26,11498,579],{"className":11499},[69,70]," = Lengths of two sides of the triangle .",[1929,11502,11503,11566],{},[26,11504,11506,11524],{"className":11505},[29],[26,11507,11509],{"className":11508},[33],[35,11510,11511],{"xmlns":37},[39,11512,11513,11521],{},[42,11514,11515],{},[11011,11516,11517,11519],{"accent":56},[45,11518,99],{},[89,11520,11017],{},[49,11522,11523],{"encoding":51},"\\hat{C}",[26,11525,11527],{"className":11526,"ariaHidden":56},[55],[26,11528,11530,11533],{"className":11529},[60],[26,11531],{"className":11532,"style":11387},[64],[26,11534,11536],{"className":11535},[69,11377],[26,11537,11539],{"className":11538},[421],[26,11540,11542],{"className":11541},[426],[26,11543,11545,11554],{"className":11544,"style":11387},[430],[26,11546,11548,11551],{"style":11547},"top:-3em;",[26,11549],{"className":11550,"style":717},[438],[26,11552,99],{"className":11553,"style":138},[69,70],[26,11555,11557,11560],{"style":11556},"top:-3.2523em;",[26,11558],{"className":11559,"style":717},[438],[26,11561,11563],{"className":11562,"style":11407},[11406],[26,11564,11017],{"className":11565},[69]," = The included observed angle .",[1929,11568,11569,11597,11598,11664],{},[26,11570,11572,11585],{"className":11571},[29],[26,11573,11575],{"className":11574},[33],[35,11576,11577],{"xmlns":37},[39,11578,11579,11583],{},[42,11580,11581],{},[45,11582,10978],{},[49,11584,10978],{"encoding":51},[26,11586,11588],{"className":11587,"ariaHidden":56},[55],[26,11589,11591,11594],{"className":11590},[60],[26,11592],{"className":11593,"style":65},[64],[26,11595,10978],{"className":11596,"style":11134},[69,70]," = The mean radius of the Earth (",[26,11599,11601,11624],{"className":11600},[29],[26,11602,11604],{"className":11603},[33],[35,11605,11606],{"xmlns":37},[39,11607,11608,11621],{},[42,11609,11610,11612,11614,11616,11619],{},[89,11611,10997],{},[554,11613,307],{},[89,11615,91],{"separator":56},[554,11617,11618],{},"370",[5057,11620,10540],{},[49,11622,11623],{"encoding":51},"\\approx 6,370 \\text{ km}",[26,11625,11627,11640],{"className":11626,"ariaHidden":56},[55],[26,11628,11630,11634,11637],{"className":11629},[60],[26,11631],{"className":11632,"style":11633},[64],"height:0.4831em;",[26,11635,10997],{"className":11636},[680],[26,11638],{"className":11639,"style":676},[123],[26,11641,11643,11646,11649,11652,11655,11658],{"className":11642},[60],[26,11644],{"className":11645,"style":1898},[64],[26,11647,307],{"className":11648},[69],[26,11650,91],{"className":11651},[119],[26,11653],{"className":11654,"style":124},[123],[26,11656,11618],{"className":11657},[69],[26,11659,11661],{"className":11660},[69,5118],[26,11662,10540],{"className":11663},[69],") .",[1738,11666,11668],{"id":11667},"_2-angular-closure-requirement","2. Angular Closure Requirement",[18,11670,11671,11672],{},"In a major triangulation, the sum of the adjusted spheroidal angles should equal:\n",[26,11673,11675,11712],{"className":11674},[29],[26,11676,11678],{"className":11677},[33],[35,11679,11680],{"xmlns":37},[39,11681,11682,11709],{},[42,11683,11684,11686,11689,11691,11697,11699],{},[89,11685,4611],{},[5057,11687,11688],{},"Angles",[89,11690,543],{},[3513,11692,11693,11695],{},[554,11694,5145],{},[89,11696,3520],{},[89,11698,565],{},[3513,11700,11701,11703],{},[45,11702,372],{},[42,11704,11705,11707],{},[89,11706,10678],{"mathvariant":2636},[89,11708,10678],{"mathvariant":2636},[49,11710,11711],{"encoding":51},"\\sum \\text{Angles} = 180^\\circ + E''",[26,11713,11715,11742,11789],{"className":11714,"ariaHidden":56},[55],[26,11716,11718,11721,11724,11727,11733,11736,11739],{"className":11717},[60],[26,11719],{"className":11720,"style":3328},[64],[26,11722,4611],{"className":11723,"style":4866},[2079,4864,4865],[26,11725],{"className":11726,"style":124},[123],[26,11728,11730],{"className":11729},[69,5118],[26,11731,11688],{"className":11732},[69],[26,11734],{"className":11735,"style":676},[123],[26,11737,543],{"className":11738},[680],[26,11740],{"className":11741,"style":676},[123],[26,11743,11745,11748,11751,11780,11783,11786],{"className":11744},[60],[26,11746],{"className":11747,"style":8783},[64],[26,11749,5163],{"className":11750},[69],[26,11752,11754,11757],{"className":11753},[69],[26,11755,3562],{"className":11756},[69],[26,11758,11760],{"className":11759},[417],[26,11761,11763],{"className":11762},[421],[26,11764,11766],{"className":11765},[426],[26,11767,11769],{"className":11768,"style":3551},[430],[26,11770,11771,11774],{"style":3577},[26,11772],{"className":11773,"style":439},[438],[26,11775,11777],{"className":11776},[443,444,445,446],[26,11778,3520],{"className":11779},[773,446],[26,11781],{"className":11782,"style":4156},[123],[26,11784,565],{"className":11785},[773],[26,11787],{"className":11788,"style":4156},[123],[26,11790,11792,11795],{"className":11791},[60],[26,11793],{"className":11794,"style":10693},[64],[26,11796,11798,11801],{"className":11797},[69],[26,11799,372],{"className":11800,"style":413},[69,70],[26,11802,11804],{"className":11803},[417],[26,11805,11807],{"className":11806},[421],[26,11808,11810],{"className":11809},[426],[26,11811,11813],{"className":11812,"style":10693},[430],[26,11814,11815,11818],{"style":3577},[26,11816],{"className":11817,"style":439},[438],[26,11819,11821],{"className":11820},[443,444,445,446],[26,11822,11824],{"className":11823},[69,446],[26,11825,10726],{"className":11826},[69,446],[13,11828,3283],{"id":3282},[18,11830,11831,11833,11834,275,11880,11908,11909,11943,11944,11982,11983,12034,12035,2669,12209,2669,12383,12557],{},[21,11832,7512],{}," In a major triangulation, the mean values of the angles ",[26,11835,11837,11856],{"className":11836},[29],[26,11838,11840],{"className":11839},[33],[35,11841,11842],{"xmlns":37},[39,11843,11844,11854],{},[42,11845,11846,11848,11850,11852],{},[45,11847,87],{},[89,11849,91],{"separator":56},[45,11851,94],{},[89,11853,91],{"separator":56},[49,11855,250],{"encoding":51},[26,11857,11859],{"className":11858,"ariaHidden":56},[55],[26,11860,11862,11865,11868,11871,11874,11877],{"className":11861},[60],[26,11863],{"className":11864,"style":112},[64],[26,11866,87],{"className":11867},[69,70],[26,11869,91],{"className":11870},[119],[26,11872],{"className":11873,"style":124},[123],[26,11875,94],{"className":11876,"style":128},[69,70],[26,11878,91],{"className":11879},[119],[26,11881,11883,11896],{"className":11882},[29],[26,11884,11886],{"className":11885},[33],[35,11887,11888],{"xmlns":37},[39,11889,11890,11894],{},[42,11891,11892],{},[45,11893,99],{},[49,11895,99],{"encoding":51},[26,11897,11899],{"className":11898,"ariaHidden":56},[55],[26,11900,11902,11905],{"className":11901},[60],[26,11903],{"className":11904,"style":65},[64],[26,11906,99],{"className":11907,"style":138},[69,70]," were measured. The length of side ",[26,11910,11912,11928],{"className":11911},[29],[26,11913,11915],{"className":11914},[33],[35,11916,11917],{"xmlns":37},[39,11918,11919,11925],{},[42,11920,11921,11923],{},[45,11922,94],{},[45,11924,99],{},[49,11926,11927],{"encoding":51},"BC",[26,11929,11931],{"className":11930,"ariaHidden":56},[55],[26,11932,11934,11937,11940],{"className":11933},[60],[26,11935],{"className":11936,"style":65},[64],[26,11938,94],{"className":11939,"style":128},[69,70],[26,11941,99],{"className":11942,"style":138},[69,70]," was ",[26,11945,11947,11964],{"className":11946},[29],[26,11948,11950],{"className":11949},[33],[35,11951,11952],{"xmlns":37},[39,11953,11954,11961],{},[42,11955,11956,11959],{},[554,11957,11958],{},"37.5",[5057,11960,10540],{},[49,11962,11963],{"encoding":51},"37.5 \\text{ km}",[26,11965,11967],{"className":11966,"ariaHidden":56},[55],[26,11968,11970,11973,11976],{"className":11969},[60],[26,11971],{"className":11972,"style":7264},[64],[26,11974,11958],{"className":11975},[69],[26,11977,11979],{"className":11978},[69,5118],[26,11980,10540],{"className":11981},[69]," and the radius of the Earth is ",[26,11984,11986,12007],{"className":11985},[29],[26,11987,11989],{"className":11988},[33],[35,11990,11991],{"xmlns":37},[39,11992,11993,12004],{},[42,11994,11995,11997,11999,12002],{},[554,11996,307],{},[89,11998,91],{"separator":56},[554,12000,12001],{},"267",[5057,12003,10540],{},[49,12005,12006],{"encoding":51},"6,267 \\text{ km}",[26,12008,12010],{"className":12009,"ariaHidden":56},[55],[26,12011,12013,12016,12019,12022,12025,12028],{"className":12012},[60],[26,12014],{"className":12015,"style":1898},[64],[26,12017,307],{"className":12018},[69],[26,12020,91],{"className":12021},[119],[26,12023],{"className":12024,"style":124},[123],[26,12026,12001],{"className":12027},[69],[26,12029,12031],{"className":12030},[69,5118],[26,12032,10540],{"className":12033},[69],". Observed angles: ",[26,12036,12038,12079],{"className":12037},[29],[26,12039,12041],{"className":12040},[33],[35,12042,12043],{"xmlns":37},[39,12044,12045,12076],{},[42,12046,12047,12049,12051,12058,12065],{},[45,12048,87],{},[89,12050,543],{},[3513,12052,12053,12056],{},[554,12054,12055],{},"50",[89,12057,3520],{},[3513,12059,12060,12062],{},[554,12061,8924],{},[89,12063,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},"0em",[3513,12066,12067,12070],{},[554,12068,12069],{},"32.5",[42,12071,12072,12074],{},[89,12073,10678],{"mathvariant":2636},[89,12075,10678],{"mathvariant":2636},[49,12077,12078],{"encoding":51},"A=50^\\circ 22' 32.5''",[26,12080,12082,12100],{"className":12081,"ariaHidden":56},[55],[26,12083,12085,12088,12091,12094,12097],{"className":12084},[60],[26,12086],{"className":12087,"style":65},[64],[26,12089,87],{"className":12090},[69,70],[26,12092],{"className":12093,"style":676},[123],[26,12095,543],{"className":12096},[680],[26,12098],{"className":12099,"style":676},[123],[26,12101,12103,12106,12109,12138,12141,12173,12177],{"className":12102},[60],[26,12104],{"className":12105,"style":10693},[64],[26,12107,6471],{"className":12108},[69],[26,12110,12112,12115],{"className":12111},[69],[26,12113,3562],{"className":12114},[69],[26,12116,12118],{"className":12117},[417],[26,12119,12121],{"className":12120},[421],[26,12122,12124],{"className":12123},[426],[26,12125,12127],{"className":12126,"style":3551},[430],[26,12128,12129,12132],{"style":3577},[26,12130],{"className":12131,"style":439},[438],[26,12133,12135],{"className":12134},[443,444,445,446],[26,12136,3520],{"className":12137},[773,446],[26,12139,572],{"className":12140},[69],[26,12142,12144,12147],{"className":12143},[69],[26,12145,572],{"className":12146},[69],[26,12148,12150],{"className":12149},[417],[26,12151,12153],{"className":12152},[421],[26,12154,12156],{"className":12155},[426],[26,12157,12159],{"className":12158,"style":10693},[430],[26,12160,12161,12164],{"style":3577},[26,12162],{"className":12163,"style":439},[438],[26,12165,12167],{"className":12166},[443,444,445,446],[26,12168,12170],{"className":12169},[69,446],[26,12171,10678],{"className":12172},[69,446],[26,12174,12176],{"className":12175},[69],"32.",[26,12178,12180,12183],{"className":12179},[69],[26,12181,6471],{"className":12182},[69],[26,12184,12186],{"className":12185},[417],[26,12187,12189],{"className":12188},[421],[26,12190,12192],{"className":12191},[426],[26,12193,12195],{"className":12194,"style":10693},[430],[26,12196,12197,12200],{"style":3577},[26,12198],{"className":12199,"style":439},[438],[26,12201,12203],{"className":12202},[443,444,445,446],[26,12204,12206],{"className":12205},[69,446],[26,12207,10726],{"className":12208},[69,446],[26,12210,12212,12253],{"className":12211},[29],[26,12213,12215],{"className":12214},[33],[35,12216,12217],{"xmlns":37},[39,12218,12219,12250],{},[42,12220,12221,12223,12225,12232,12239],{},[45,12222,94],{},[89,12224,543],{},[3513,12226,12227,12230],{},[554,12228,12229],{},"65",[89,12231,3520],{},[3513,12233,12234,12237],{},[554,12235,12236],{},"40",[89,12238,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,12240,12241,12244],{},[554,12242,12243],{},"47.5",[42,12245,12246,12248],{},[89,12247,10678],{"mathvariant":2636},[89,12249,10678],{"mathvariant":2636},[49,12251,12252],{"encoding":51},"B=65^\\circ 40' 47.5''",[26,12254,12256,12274],{"className":12255,"ariaHidden":56},[55],[26,12257,12259,12262,12265,12268,12271],{"className":12258},[60],[26,12260],{"className":12261,"style":65},[64],[26,12263,94],{"className":12264,"style":128},[69,70],[26,12266],{"className":12267,"style":676},[123],[26,12269,543],{"className":12270},[680],[26,12272],{"className":12273,"style":676},[123],[26,12275,12277,12280,12283,12312,12315,12347,12351],{"className":12276},[60],[26,12278],{"className":12279,"style":10693},[64],[26,12281,307],{"className":12282},[69],[26,12284,12286,12289],{"className":12285},[69],[26,12287,6471],{"className":12288},[69],[26,12290,12292],{"className":12291},[417],[26,12293,12295],{"className":12294},[421],[26,12296,12298],{"className":12297},[426],[26,12299,12301],{"className":12300,"style":3551},[430],[26,12302,12303,12306],{"style":3577},[26,12304],{"className":12305,"style":439},[438],[26,12307,12309],{"className":12308},[443,444,445,446],[26,12310,3520],{"className":12311},[773,446],[26,12313,7679],{"className":12314},[69],[26,12316,12318,12321],{"className":12317},[69],[26,12319,3562],{"className":12320},[69],[26,12322,12324],{"className":12323},[417],[26,12325,12327],{"className":12326},[421],[26,12328,12330],{"className":12329},[426],[26,12331,12333],{"className":12332,"style":10693},[430],[26,12334,12335,12338],{"style":3577},[26,12336],{"className":12337,"style":439},[438],[26,12339,12341],{"className":12340},[443,444,445,446],[26,12342,12344],{"className":12343},[69,446],[26,12345,10678],{"className":12346},[69,446],[26,12348,12350],{"className":12349},[69],"47.",[26,12352,12354,12357],{"className":12353},[69],[26,12355,6471],{"className":12356},[69],[26,12358,12360],{"className":12359},[417],[26,12361,12363],{"className":12362},[421],[26,12364,12366],{"className":12365},[426],[26,12367,12369],{"className":12368,"style":10693},[430],[26,12370,12371,12374],{"style":3577},[26,12372],{"className":12373,"style":439},[438],[26,12375,12377],{"className":12376},[443,444,445,446],[26,12378,12380],{"className":12379},[69,446],[26,12381,10726],{"className":12382},[69,446],[26,12384,12386,12427],{"className":12385},[29],[26,12387,12389],{"className":12388},[33],[35,12390,12391],{"xmlns":37},[39,12392,12393,12424],{},[42,12394,12395,12397,12399,12406,12413],{},[45,12396,99],{},[89,12398,543],{},[3513,12400,12401,12404],{},[554,12402,12403],{},"63",[89,12405,3520],{},[3513,12407,12408,12411],{},[554,12409,12410],{},"56",[89,12412,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,12414,12415,12418],{},[554,12416,12417],{},"46.5",[42,12419,12420,12422],{},[89,12421,10678],{"mathvariant":2636},[89,12423,10678],{"mathvariant":2636},[49,12425,12426],{"encoding":51},"C=63^\\circ 56' 46.5''",[26,12428,12430,12448],{"className":12429,"ariaHidden":56},[55],[26,12431,12433,12436,12439,12442,12445],{"className":12432},[60],[26,12434],{"className":12435,"style":65},[64],[26,12437,99],{"className":12438,"style":138},[69,70],[26,12440],{"className":12441,"style":676},[123],[26,12443,543],{"className":12444},[680],[26,12446],{"className":12447,"style":676},[123],[26,12449,12451,12454,12457,12486,12489,12521,12525],{"className":12450},[60],[26,12452],{"className":12453,"style":10693},[64],[26,12455,307],{"className":12456},[69],[26,12458,12460,12463],{"className":12459},[69],[26,12461,588],{"className":12462},[69],[26,12464,12466],{"className":12465},[417],[26,12467,12469],{"className":12468},[421],[26,12470,12472],{"className":12471},[426],[26,12473,12475],{"className":12474,"style":3551},[430],[26,12476,12477,12480],{"style":3577},[26,12478],{"className":12479,"style":439},[438],[26,12481,12483],{"className":12482},[443,444,445,446],[26,12484,3520],{"className":12485},[773,446],[26,12487,6471],{"className":12488},[69],[26,12490,12492,12495],{"className":12491},[69],[26,12493,307],{"className":12494},[69],[26,12496,12498],{"className":12497},[417],[26,12499,12501],{"className":12500},[421],[26,12502,12504],{"className":12503},[426],[26,12505,12507],{"className":12506,"style":10693},[430],[26,12508,12509,12512],{"style":3577},[26,12510],{"className":12511,"style":439},[438],[26,12513,12515],{"className":12514},[443,444,445,446],[26,12516,12518],{"className":12517},[69,446],[26,12519,10678],{"className":12520},[69,446],[26,12522,12524],{"className":12523},[69],"46.",[26,12526,12528,12531],{"className":12527},[69],[26,12529,6471],{"className":12530},[69],[26,12532,12534],{"className":12533},[417],[26,12535,12537],{"className":12536},[421],[26,12538,12540],{"className":12539},[426],[26,12541,12543],{"className":12542,"style":10693},[430],[26,12544,12545,12548],{"style":3577},[26,12546],{"className":12547,"style":439},[438],[26,12549,12551],{"className":12550},[443,444,445,446],[26,12552,12554],{"className":12553},[69,446],[26,12555,10726],{"className":12556},[69,446],". Calculate the spherical excess and adjusted angles .",[3763,12559,12560,13230,13591,13943],{},[1929,12561,12562,12630,12631,12659,12660,12752,12753,13099,13100,5870],{},[21,12563,12564,12565,4551],{},"Calculate Spherical Excess (",[26,12566,12568,12589],{"className":12567},[29],[26,12569,12571],{"className":12570},[33],[35,12572,12573],{"xmlns":37},[39,12574,12575,12587],{},[42,12576,12577],{},[3513,12578,12579,12581],{},[45,12580,372],{},[42,12582,12583,12585],{},[89,12584,10678],{"mathvariant":2636},[89,12586,10678],{"mathvariant":2636},[49,12588,10683],{"encoding":51},[26,12590,12592],{"className":12591,"ariaHidden":56},[55],[26,12593,12595,12598],{"className":12594},[60],[26,12596],{"className":12597,"style":10693},[64],[26,12599,12601,12604],{"className":12600},[69],[26,12602,372],{"className":12603,"style":413},[69,70],[26,12605,12607],{"className":12606},[417],[26,12608,12610],{"className":12609},[421],[26,12611,12613],{"className":12612},[426],[26,12614,12616],{"className":12615,"style":10693},[430],[26,12617,12618,12621],{"style":3577},[26,12619],{"className":12620,"style":439},[438],[26,12622,12624],{"className":12623},[443,444,445,446],[26,12625,12627],{"className":12626},[69,446],[26,12628,10726],{"className":12629},[69,446],"\nUsing the sine rule to find side ",[26,12632,12634,12647],{"className":12633},[29],[26,12635,12637],{"className":12636},[33],[35,12638,12639],{"xmlns":37},[39,12640,12641,12645],{},[42,12642,12643],{},[45,12644,579],{},[49,12646,579],{"encoding":51},[26,12648,12650],{"className":12649,"ariaHidden":56},[55],[26,12651,12653,12656],{"className":12652},[60],[26,12654],{"className":12655,"style":7264},[64],[26,12657,579],{"className":12658},[69,70],": ",[26,12661,12663,12695],{"className":12662},[29],[26,12664,12666],{"className":12665},[33],[35,12667,12668],{"xmlns":37},[39,12669,12670,12692],{},[42,12671,12672,12674,12676,12678,12680,12682,12684,12686,12688,12690],{},[45,12673,579],{},[89,12675,543],{},[45,12677,562],{},[45,12679,8523],{},[89,12681,1962],{},[45,12683,94],{},[45,12685,3991],{"mathvariant":2636},[45,12687,8523],{},[89,12689,1962],{},[45,12691,87],{},[49,12693,12694],{"encoding":51},"b = a \\sin B \u002F \\sin A",[26,12696,12698,12716],{"className":12697,"ariaHidden":56},[55],[26,12699,12701,12704,12707,12710,12713],{"className":12700},[60],[26,12702],{"className":12703,"style":7264},[64],[26,12705,579],{"className":12706},[69,70],[26,12708],{"className":12709,"style":676},[123],[26,12711,543],{"className":12712},[680],[26,12714],{"className":12715,"style":676},[123],[26,12717,12719,12722,12725,12728,12731,12734,12737,12740,12743,12746,12749],{"className":12718},[60],[26,12720],{"className":12721,"style":3328},[64],[26,12723,562],{"className":12724},[69,70],[26,12726],{"className":12727,"style":124},[123],[26,12729,8523],{"className":12730},[2079],[26,12732],{"className":12733,"style":124},[123],[26,12735,94],{"className":12736,"style":128},[69,70],[26,12738,3991],{"className":12739},[69],[26,12741],{"className":12742,"style":124},[123],[26,12744,8523],{"className":12745},[2079],[26,12747],{"className":12748,"style":124},[123],[26,12750,87],{"className":12751},[69,70]," .\n",[26,12754,12756,12826],{"className":12755},[29],[26,12757,12759],{"className":12758},[33],[35,12760,12761],{"xmlns":37},[39,12762,12763,12823],{},[42,12764,12765,12775,12777,12815,12817,12819,12821],{},[3513,12766,12767,12769],{},[45,12768,372],{},[42,12770,12771,12773],{},[89,12772,10678],{"mathvariant":2636},[89,12774,10678],{"mathvariant":2636},[89,12776,543],{},[545,12778,12779,12799],{},[42,12780,12781,12787,12789,12791,12793,12795,12797],{},[3513,12782,12783,12785],{},[45,12784,562],{},[554,12786,572],{},[45,12788,8523],{},[89,12790,1962],{},[45,12792,94],{},[45,12794,8523],{},[89,12796,1962],{},[45,12798,99],{},[42,12800,12801,12803,12809,12811,12813],{},[554,12802,572],{},[3513,12804,12805,12807],{},[45,12806,10978],{},[554,12808,572],{},[45,12810,8523],{},[89,12812,1962],{},[45,12814,87],{},[89,12816,4578],{},[554,12818,11032],{},[89,12820,91],{"separator":56},[554,12822,11037],{},[49,12824,12825],{"encoding":51},"E'' = \\frac{a^2 \\sin B \\sin C}{2R^2 \\sin A} \\times 206,265",[26,12827,12829,12876,13081],{"className":12828,"ariaHidden":56},[55],[26,12830,12832,12835,12867,12870,12873],{"className":12831},[60],[26,12833],{"className":12834,"style":10693},[64],[26,12836,12838,12841],{"className":12837},[69],[26,12839,372],{"className":12840,"style":413},[69,70],[26,12842,12844],{"className":12843},[417],[26,12845,12847],{"className":12846},[421],[26,12848,12850],{"className":12849},[426],[26,12851,12853],{"className":12852,"style":10693},[430],[26,12854,12855,12858],{"style":3577},[26,12856],{"className":12857,"style":439},[438],[26,12859,12861],{"className":12860},[443,444,445,446],[26,12862,12864],{"className":12863},[69,446],[26,12865,10726],{"className":12866},[69,446],[26,12868],{"className":12869,"style":676},[123],[26,12871,543],{"className":12872},[680],[26,12874],{"className":12875,"style":676},[123],[26,12877,12879,12883,13072,13075,13078],{"className":12878},[60],[26,12880],{"className":12881,"style":12882},[64],"height:1.3629em;vertical-align:-0.345em;",[26,12884,12886,12889,13069],{"className":12885},[69],[26,12887],{"className":12888},[406,697],[26,12890,12892],{"className":12891},[545],[26,12893,12895,13061],{"className":12894},[421,422],[26,12896,12898,13058],{"className":12897},[426],[26,12899,12902,12966,12974],{"className":12900,"style":12901},[430],"height:1.0179em;",[26,12903,12904,12907],{"style":713},[26,12905],{"className":12906,"style":717},[438],[26,12908,12910],{"className":12909},[443,444,445,446],[26,12911,12913,12916,12945,12948,12960,12963],{"className":12912},[69,446],[26,12914,572],{"className":12915},[69,446],[26,12917,12919,12922],{"className":12918},[69,446],[26,12920,10978],{"className":12921,"style":11134},[69,70,446],[26,12923,12925],{"className":12924},[417],[26,12926,12928],{"className":12927},[421],[26,12929,12931],{"className":12930},[426],[26,12932,12934],{"className":12933,"style":11147},[430],[26,12935,12936,12939],{"style":11150},[26,12937],{"className":12938,"style":749},[438],[26,12940,12942],{"className":12941},[443,753,754,446],[26,12943,572],{"className":12944},[69,446],[26,12946],{"className":12947,"style":2094},[123,446],[26,12949,12951,12954,12957],{"className":12950},[2079,446],[26,12952,11169],{"className":12953},[446],[26,12955,11173],{"className":12956},[446],[26,12958,11177],{"className":12959},[446],[26,12961],{"className":12962,"style":2094},[123,446],[26,12964,87],{"className":12965},[69,70,446],[26,12967,12968,12971],{"style":859},[26,12969],{"className":12970,"style":717},[438],[26,12972],{"className":12973,"style":867},[866],[26,12975,12976,12979],{"style":2129},[26,12977],{"className":12978,"style":717},[438],[26,12980,12982],{"className":12981},[443,444,445,446],[26,12983,12985,13016,13019,13031,13034,13037,13040,13052,13055],{"className":12984},[69,446],[26,12986,12988,12991],{"className":12987},[69,446],[26,12989,562],{"className":12990},[69,70,446],[26,12992,12994],{"className":12993},[417],[26,12995,12997],{"className":12996},[421],[26,12998,13000],{"className":12999},[426],[26,13001,13004],{"className":13002,"style":13003},[430],"height:0.8913em;",[26,13005,13007,13010],{"style":13006},"top:-2.931em;margin-right:0.0714em;",[26,13008],{"className":13009,"style":749},[438],[26,13011,13013],{"className":13012},[443,753,754,446],[26,13014,572],{"className":13015},[69,446],[26,13017],{"className":13018,"style":2094},[123,446],[26,13020,13022,13025,13028],{"className":13021},[2079,446],[26,13023,11169],{"className":13024},[446],[26,13026,11173],{"className":13027},[446],[26,13029,11177],{"className":13030},[446],[26,13032],{"className":13033,"style":2094},[123,446],[26,13035,94],{"className":13036,"style":128},[69,70,446],[26,13038],{"className":13039,"style":2094},[123,446],[26,13041,13043,13046,13049],{"className":13042},[2079,446],[26,13044,11169],{"className":13045},[446],[26,13047,11173],{"className":13048},[446],[26,13050,11177],{"className":13051},[446],[26,13053],{"className":13054,"style":2094},[123,446],[26,13056,99],{"className":13057,"style":138},[69,70,446],[26,13059,454],{"className":13060},[453],[26,13062,13064],{"className":13063},[426],[26,13065,13067],{"className":13066,"style":11249},[430],[26,13068],{},[26,13070],{"className":13071},[515,697],[26,13073],{"className":13074,"style":4156},[123],[26,13076,4578],{"className":13077},[773],[26,13079],{"className":13080,"style":4156},[123],[26,13082,13084,13087,13090,13093,13096],{"className":13083},[60],[26,13085],{"className":13086,"style":11440},[64],[26,13088,11032],{"className":13089},[69],[26,13091,91],{"className":13092},[119],[26,13094],{"className":13095,"style":124},[123],[26,13097,11037],{"className":13098},[69],"\nSubstituting the values: ",[26,13101,13103,13138],{"className":13102},[29],[26,13104,13106],{"className":13105},[33],[35,13107,13108],{"xmlns":37},[39,13109,13110,13135],{},[42,13111,13112,13122,13124],{},[3513,13113,13114,13116],{},[45,13115,372],{},[42,13117,13118,13120],{},[89,13119,10678],{"mathvariant":2636},[89,13121,10678],{"mathvariant":2636},[89,13123,10997],{},[3513,13125,13126,13129],{},[554,13127,13128],{},"3.9",[42,13130,13131,13133],{},[89,13132,10678],{"mathvariant":2636},[89,13134,10678],{"mathvariant":2636},[49,13136,13137],{"encoding":51},"E'' \\approx 3.9''",[26,13139,13141,13188],{"className":13140,"ariaHidden":56},[55],[26,13142,13144,13147,13179,13182,13185],{"className":13143},[60],[26,13145],{"className":13146,"style":10693},[64],[26,13148,13150,13153],{"className":13149},[69],[26,13151,372],{"className":13152,"style":413},[69,70],[26,13154,13156],{"className":13155},[417],[26,13157,13159],{"className":13158},[421],[26,13160,13162],{"className":13161},[426],[26,13163,13165],{"className":13164,"style":10693},[430],[26,13166,13167,13170],{"style":3577},[26,13168],{"className":13169,"style":439},[438],[26,13171,13173],{"className":13172},[443,444,445,446],[26,13174,13176],{"className":13175},[69,446],[26,13177,10726],{"className":13178},[69,446],[26,13180],{"className":13181,"style":676},[123],[26,13183,10997],{"className":13184},[680],[26,13186],{"className":13187,"style":676},[123],[26,13189,13191,13194,13198],{"className":13190},[60],[26,13192],{"className":13193,"style":10693},[64],[26,13195,13197],{"className":13196},[69],"3.",[26,13199,13201,13204],{"className":13200},[69],[26,13202,5565],{"className":13203},[69],[26,13205,13207],{"className":13206},[417],[26,13208,13210],{"className":13209},[421],[26,13211,13213],{"className":13212},[426],[26,13214,13216],{"className":13215,"style":10693},[430],[26,13217,13218,13221],{"style":3577},[26,13219],{"className":13220,"style":439},[438],[26,13222,13224],{"className":13223},[443,444,445,446],[26,13225,13227],{"className":13226},[69,446],[26,13228,10726],{"className":13229},[69,446],[1929,13231,13232,13235,13236,5870],{},[21,13233,13234],{},"Target Sum:","\nTarget = ",[26,13237,13239,13311],{"className":13238},[29],[26,13240,13242],{"className":13241},[33],[35,13243,13244],{"xmlns":37},[39,13245,13246,13308],{},[42,13247,13248,13254,13261,13271,13273,13283,13285,13291,13297],{},[3513,13249,13250,13252],{},[554,13251,5145],{},[89,13253,3520],{},[3513,13255,13256,13259],{},[554,13257,13258],{},"00",[89,13260,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,13262,13263,13265],{},[554,13264,13258],{},[42,13266,13267,13269],{},[89,13268,10678],{"mathvariant":2636},[89,13270,10678],{"mathvariant":2636},[89,13272,565],{},[3513,13274,13275,13277],{},[554,13276,13128],{},[42,13278,13279,13281],{},[89,13280,10678],{"mathvariant":2636},[89,13282,10678],{"mathvariant":2636},[89,13284,543],{},[3513,13286,13287,13289],{},[554,13288,5145],{},[89,13290,3520],{},[3513,13292,13293,13295],{},[554,13294,13258],{},[89,13296,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,13298,13299,13302],{},[554,13300,13301],{},"03.9",[42,13303,13304,13306],{},[89,13305,10678],{"mathvariant":2636},[89,13307,10678],{"mathvariant":2636},[49,13309,13310],{"encoding":51},"180^\\circ 00' 00'' + 3.9'' = 180^\\circ 00' 03.9''",[26,13312,13314,13432,13482],{"className":13313,"ariaHidden":56},[55],[26,13315,13317,13321,13324,13353,13356,13388,13391,13423,13426,13429],{"className":13316},[60],[26,13318],{"className":13319,"style":13320},[64],"height:0.8352em;vertical-align:-0.0833em;",[26,13322,5163],{"className":13323},[69],[26,13325,13327,13330],{"className":13326},[69],[26,13328,3562],{"className":13329},[69],[26,13331,13333],{"className":13332},[417],[26,13334,13336],{"className":13335},[421],[26,13337,13339],{"className":13338},[426],[26,13340,13342],{"className":13341,"style":3551},[430],[26,13343,13344,13347],{"style":3577},[26,13345],{"className":13346,"style":439},[438],[26,13348,13350],{"className":13349},[443,444,445,446],[26,13351,3520],{"className":13352},[773,446],[26,13354,3562],{"className":13355},[69],[26,13357,13359,13362],{"className":13358},[69],[26,13360,3562],{"className":13361},[69],[26,13363,13365],{"className":13364},[417],[26,13366,13368],{"className":13367},[421],[26,13369,13371],{"className":13370},[426],[26,13372,13374],{"className":13373,"style":10693},[430],[26,13375,13376,13379],{"style":3577},[26,13377],{"className":13378,"style":439},[438],[26,13380,13382],{"className":13381},[443,444,445,446],[26,13383,13385],{"className":13384},[69,446],[26,13386,10678],{"className":13387},[69,446],[26,13389,3562],{"className":13390},[69],[26,13392,13394,13397],{"className":13393},[69],[26,13395,3562],{"className":13396},[69],[26,13398,13400],{"className":13399},[417],[26,13401,13403],{"className":13402},[421],[26,13404,13406],{"className":13405},[426],[26,13407,13409],{"className":13408,"style":10693},[430],[26,13410,13411,13414],{"style":3577},[26,13412],{"className":13413,"style":439},[438],[26,13415,13417],{"className":13416},[443,444,445,446],[26,13418,13420],{"className":13419},[69,446],[26,13421,10726],{"className":13422},[69,446],[26,13424],{"className":13425,"style":4156},[123],[26,13427,565],{"className":13428},[773],[26,13430],{"className":13431,"style":4156},[123],[26,13433,13435,13438,13441,13473,13476,13479],{"className":13434},[60],[26,13436],{"className":13437,"style":10693},[64],[26,13439,13197],{"className":13440},[69],[26,13442,13444,13447],{"className":13443},[69],[26,13445,5565],{"className":13446},[69],[26,13448,13450],{"className":13449},[417],[26,13451,13453],{"className":13452},[421],[26,13454,13456],{"className":13455},[426],[26,13457,13459],{"className":13458,"style":10693},[430],[26,13460,13461,13464],{"style":3577},[26,13462],{"className":13463,"style":439},[438],[26,13465,13467],{"className":13466},[443,444,445,446],[26,13468,13470],{"className":13469},[69,446],[26,13471,10726],{"className":13472},[69,446],[26,13474],{"className":13475,"style":676},[123],[26,13477,543],{"className":13478},[680],[26,13480],{"className":13481,"style":676},[123],[26,13483,13485,13488,13491,13520,13523,13555,13559],{"className":13484},[60],[26,13486],{"className":13487,"style":10693},[64],[26,13489,5163],{"className":13490},[69],[26,13492,13494,13497],{"className":13493},[69],[26,13495,3562],{"className":13496},[69],[26,13498,13500],{"className":13499},[417],[26,13501,13503],{"className":13502},[421],[26,13504,13506],{"className":13505},[426],[26,13507,13509],{"className":13508,"style":3551},[430],[26,13510,13511,13514],{"style":3577},[26,13512],{"className":13513,"style":439},[438],[26,13515,13517],{"className":13516},[443,444,445,446],[26,13518,3520],{"className":13519},[773,446],[26,13521,3562],{"className":13522},[69],[26,13524,13526,13529],{"className":13525},[69],[26,13527,3562],{"className":13528},[69],[26,13530,13532],{"className":13531},[417],[26,13533,13535],{"className":13534},[421],[26,13536,13538],{"className":13537},[426],[26,13539,13541],{"className":13540,"style":10693},[430],[26,13542,13543,13546],{"style":3577},[26,13544],{"className":13545,"style":439},[438],[26,13547,13549],{"className":13548},[443,444,445,446],[26,13550,13552],{"className":13551},[69,446],[26,13553,10678],{"className":13554},[69,446],[26,13556,13558],{"className":13557},[69],"03.",[26,13560,13562,13565],{"className":13561},[69],[26,13563,5565],{"className":13564},[69],[26,13566,13568],{"className":13567},[417],[26,13569,13571],{"className":13570},[421],[26,13572,13574],{"className":13573},[426],[26,13575,13577],{"className":13576,"style":10693},[430],[26,13578,13579,13582],{"style":3577},[26,13580],{"className":13581,"style":439},[438],[26,13583,13585],{"className":13584},[443,444,445,446],[26,13586,13588],{"className":13587},[69,446],[26,13589,10726],{"className":13590},[69,446],[1929,13592,13593,13596,13597,13747,13748,5870],{},[21,13594,13595],{},"Find Misclosure:","\nObserved Sum = ",[26,13598,13600,13635],{"className":13599},[29],[26,13601,13603],{"className":13602},[33],[35,13604,13605],{"xmlns":37},[39,13606,13607,13632],{},[42,13608,13609,13615,13621],{},[3513,13610,13611,13613],{},[554,13612,5145],{},[89,13614,3520],{},[3513,13616,13617,13619],{},[554,13618,13258],{},[89,13620,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,13622,13623,13626],{},[554,13624,13625],{},"06.5",[42,13627,13628,13630],{},[89,13629,10678],{"mathvariant":2636},[89,13631,10678],{"mathvariant":2636},[49,13633,13634],{"encoding":51},"180^\\circ 00' 06.5''",[26,13636,13638],{"className":13637,"ariaHidden":56},[55],[26,13639,13641,13644,13647,13676,13679,13711,13715],{"className":13640},[60],[26,13642],{"className":13643,"style":10693},[64],[26,13645,5163],{"className":13646},[69],[26,13648,13650,13653],{"className":13649},[69],[26,13651,3562],{"className":13652},[69],[26,13654,13656],{"className":13655},[417],[26,13657,13659],{"className":13658},[421],[26,13660,13662],{"className":13661},[426],[26,13663,13665],{"className":13664,"style":3551},[430],[26,13666,13667,13670],{"style":3577},[26,13668],{"className":13669,"style":439},[438],[26,13671,13673],{"className":13672},[443,444,445,446],[26,13674,3520],{"className":13675},[773,446],[26,13677,3562],{"className":13678},[69],[26,13680,13682,13685],{"className":13681},[69],[26,13683,3562],{"className":13684},[69],[26,13686,13688],{"className":13687},[417],[26,13689,13691],{"className":13690},[421],[26,13692,13694],{"className":13693},[426],[26,13695,13697],{"className":13696,"style":10693},[430],[26,13698,13699,13702],{"style":3577},[26,13700],{"className":13701,"style":439},[438],[26,13703,13705],{"className":13704},[443,444,445,446],[26,13706,13708],{"className":13707},[69,446],[26,13709,10678],{"className":13710},[69,446],[26,13712,13714],{"className":13713},[69],"06.",[26,13716,13718,13721],{"className":13717},[69],[26,13719,6471],{"className":13720},[69],[26,13722,13724],{"className":13723},[417],[26,13725,13727],{"className":13726},[421],[26,13728,13730],{"className":13729},[426],[26,13731,13733],{"className":13732,"style":10693},[430],[26,13734,13735,13738],{"style":3577},[26,13736],{"className":13737,"style":439},[438],[26,13739,13741],{"className":13740},[443,444,445,446],[26,13742,13744],{"className":13743},[69,446],[26,13745,10726],{"className":13746},[69,446],".\nError = ",[26,13749,13751,13798],{"className":13750},[29],[26,13752,13754],{"className":13753},[33],[35,13755,13756],{"xmlns":37},[39,13757,13758,13795],{},[42,13759,13760,13770,13772,13782,13784],{},[3513,13761,13762,13764],{},[554,13763,13625],{},[42,13765,13766,13768],{},[89,13767,10678],{"mathvariant":2636},[89,13769,10678],{"mathvariant":2636},[89,13771,1967],{},[3513,13773,13774,13776],{},[554,13775,13301],{},[42,13777,13778,13780],{},[89,13779,10678],{"mathvariant":2636},[89,13781,10678],{"mathvariant":2636},[89,13783,543],{},[3513,13785,13786,13789],{},[554,13787,13788],{},"2.6",[42,13790,13791,13793],{},[89,13792,10678],{"mathvariant":2636},[89,13794,10678],{"mathvariant":2636},[49,13796,13797],{"encoding":51},"06.5'' - 03.9'' = 2.6''",[26,13799,13801,13851,13901],{"className":13800,"ariaHidden":56},[55],[26,13802,13804,13807,13810,13842,13845,13848],{"className":13803},[60],[26,13805],{"className":13806,"style":13320},[64],[26,13808,13714],{"className":13809},[69],[26,13811,13813,13816],{"className":13812},[69],[26,13814,6471],{"className":13815},[69],[26,13817,13819],{"className":13818},[417],[26,13820,13822],{"className":13821},[421],[26,13823,13825],{"className":13824},[426],[26,13826,13828],{"className":13827,"style":10693},[430],[26,13829,13830,13833],{"style":3577},[26,13831],{"className":13832,"style":439},[438],[26,13834,13836],{"className":13835},[443,444,445,446],[26,13837,13839],{"className":13838},[69,446],[26,13840,10726],{"className":13841},[69,446],[26,13843],{"className":13844,"style":4156},[123],[26,13846,1967],{"className":13847},[773],[26,13849],{"className":13850,"style":4156},[123],[26,13852,13854,13857,13860,13892,13895,13898],{"className":13853},[60],[26,13855],{"className":13856,"style":10693},[64],[26,13858,13558],{"className":13859},[69],[26,13861,13863,13866],{"className":13862},[69],[26,13864,5565],{"className":13865},[69],[26,13867,13869],{"className":13868},[417],[26,13870,13872],{"className":13871},[421],[26,13873,13875],{"className":13874},[426],[26,13876,13878],{"className":13877,"style":10693},[430],[26,13879,13880,13883],{"style":3577},[26,13881],{"className":13882,"style":439},[438],[26,13884,13886],{"className":13885},[443,444,445,446],[26,13887,13889],{"className":13888},[69,446],[26,13890,10726],{"className":13891},[69,446],[26,13893],{"className":13894,"style":676},[123],[26,13896,543],{"className":13897},[680],[26,13899],{"className":13900,"style":676},[123],[26,13902,13904,13907,13911],{"className":13903},[60],[26,13905],{"className":13906,"style":10693},[64],[26,13908,13910],{"className":13909},[69],"2.",[26,13912,13914,13917],{"className":13913},[69],[26,13915,307],{"className":13916},[69],[26,13918,13920],{"className":13919},[417],[26,13921,13923],{"className":13922},[421],[26,13924,13926],{"className":13925},[426],[26,13927,13929],{"className":13928,"style":10693},[430],[26,13930,13931,13934],{"style":3577},[26,13932],{"className":13933,"style":439},[438],[26,13935,13937],{"className":13936},[443,444,445,446],[26,13938,13940],{"className":13939},[69,446],[26,13941,10726],{"className":13942},[69,446],[1929,13944,13945,13948,13949,13978,13979,14015,14016,14189,14190,14189,14361,5870],{},[21,13946,13947],{},"Distribute Error:","\nIf angles have different weights (",[26,13950,13952,13966],{"className":13951},[29],[26,13953,13955],{"className":13954},[33],[35,13956,13957],{"xmlns":37},[39,13958,13959,13964],{},[42,13960,13961],{},[45,13962,13963],{},"W",[49,13965,13963],{"encoding":51},[26,13967,13969],{"className":13968,"ariaHidden":56},[55],[26,13970,13972,13975],{"className":13971},[60],[26,13973],{"className":13974,"style":65},[64],[26,13976,13963],{"className":13977,"style":71},[69,70],"), the correction is applied proportional to the reciprocal weight (",[26,13980,13982,14000],{"className":13981},[29],[26,13983,13985],{"className":13984},[33],[35,13986,13987],{"xmlns":37},[39,13988,13989,13997],{},[42,13990,13991,13993,13995],{},[554,13992,341],{},[45,13994,3991],{"mathvariant":2636},[45,13996,13963],{},[49,13998,13999],{"encoding":51},"1\u002FW",[26,14001,14003],{"className":14002,"ariaHidden":56},[55],[26,14004,14006,14009,14012],{"className":14005},[60],[26,14007],{"className":14008,"style":3328},[64],[26,14010,4111],{"className":14011},[69],[26,14013,13963],{"className":14014,"style":71},[69,70],") .\nCorrected ",[26,14017,14019,14058],{"className":14018},[29],[26,14020,14022],{"className":14021},[33],[35,14023,14024],{"xmlns":37},[39,14025,14026,14055],{},[42,14027,14028,14030,14032,14038,14044],{},[45,14029,87],{},[89,14031,543],{},[3513,14033,14034,14036],{},[554,14035,12055],{},[89,14037,3520],{},[3513,14039,14040,14042],{},[554,14041,8924],{},[89,14043,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,14045,14046,14049],{},[554,14047,14048],{},"31.8",[42,14050,14051,14053],{},[89,14052,10678],{"mathvariant":2636},[89,14054,10678],{"mathvariant":2636},[49,14056,14057],{"encoding":51},"A = 50^\\circ 22' 31.8''",[26,14059,14061,14079],{"className":14060,"ariaHidden":56},[55],[26,14062,14064,14067,14070,14073,14076],{"className":14063},[60],[26,14065],{"className":14066,"style":65},[64],[26,14068,87],{"className":14069},[69,70],[26,14071],{"className":14072,"style":676},[123],[26,14074,543],{"className":14075},[680],[26,14077],{"className":14078,"style":676},[123],[26,14080,14082,14085,14088,14117,14120,14152,14156],{"className":14081},[60],[26,14083],{"className":14084,"style":10693},[64],[26,14086,6471],{"className":14087},[69],[26,14089,14091,14094],{"className":14090},[69],[26,14092,3562],{"className":14093},[69],[26,14095,14097],{"className":14096},[417],[26,14098,14100],{"className":14099},[421],[26,14101,14103],{"className":14102},[426],[26,14104,14106],{"className":14105,"style":3551},[430],[26,14107,14108,14111],{"style":3577},[26,14109],{"className":14110,"style":439},[438],[26,14112,14114],{"className":14113},[443,444,445,446],[26,14115,3520],{"className":14116},[773,446],[26,14118,572],{"className":14119},[69],[26,14121,14123,14126],{"className":14122},[69],[26,14124,572],{"className":14125},[69],[26,14127,14129],{"className":14128},[417],[26,14130,14132],{"className":14131},[421],[26,14133,14135],{"className":14134},[426],[26,14136,14138],{"className":14137,"style":10693},[430],[26,14139,14140,14143],{"style":3577},[26,14141],{"className":14142,"style":439},[438],[26,14144,14146],{"className":14145},[443,444,445,446],[26,14147,14149],{"className":14148},[69,446],[26,14150,10678],{"className":14151},[69,446],[26,14153,14155],{"className":14154},[69],"31.",[26,14157,14159,14163],{"className":14158},[69],[26,14160,14162],{"className":14161},[69],"8",[26,14164,14166],{"className":14165},[417],[26,14167,14169],{"className":14168},[421],[26,14170,14172],{"className":14171},[426],[26,14173,14175],{"className":14174,"style":10693},[430],[26,14176,14177,14180],{"style":3577},[26,14178],{"className":14179,"style":439},[438],[26,14181,14183],{"className":14182},[443,444,445,446],[26,14184,14186],{"className":14185},[69,446],[26,14187,10726],{"className":14188},[69,446],"\nCorrected ",[26,14191,14193,14232],{"className":14192},[29],[26,14194,14196],{"className":14195},[33],[35,14197,14198],{"xmlns":37},[39,14199,14200,14229],{},[42,14201,14202,14204,14206,14212,14218],{},[45,14203,94],{},[89,14205,543],{},[3513,14207,14208,14210],{},[554,14209,12229],{},[89,14211,3520],{},[3513,14213,14214,14216],{},[554,14215,12236],{},[89,14217,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,14219,14220,14223],{},[554,14221,14222],{},"46.2",[42,14224,14225,14227],{},[89,14226,10678],{"mathvariant":2636},[89,14228,10678],{"mathvariant":2636},[49,14230,14231],{"encoding":51},"B = 65^\\circ 40' 46.2''",[26,14233,14235,14253],{"className":14234,"ariaHidden":56},[55],[26,14236,14238,14241,14244,14247,14250],{"className":14237},[60],[26,14239],{"className":14240,"style":65},[64],[26,14242,94],{"className":14243,"style":128},[69,70],[26,14245],{"className":14246,"style":676},[123],[26,14248,543],{"className":14249},[680],[26,14251],{"className":14252,"style":676},[123],[26,14254,14256,14259,14262,14291,14294,14326,14329],{"className":14255},[60],[26,14257],{"className":14258,"style":10693},[64],[26,14260,307],{"className":14261},[69],[26,14263,14265,14268],{"className":14264},[69],[26,14266,6471],{"className":14267},[69],[26,14269,14271],{"className":14270},[417],[26,14272,14274],{"className":14273},[421],[26,14275,14277],{"className":14276},[426],[26,14278,14280],{"className":14279,"style":3551},[430],[26,14281,14282,14285],{"style":3577},[26,14283],{"className":14284,"style":439},[438],[26,14286,14288],{"className":14287},[443,444,445,446],[26,14289,3520],{"className":14290},[773,446],[26,14292,7679],{"className":14293},[69],[26,14295,14297,14300],{"className":14296},[69],[26,14298,3562],{"className":14299},[69],[26,14301,14303],{"className":14302},[417],[26,14304,14306],{"className":14305},[421],[26,14307,14309],{"className":14308},[426],[26,14310,14312],{"className":14311,"style":10693},[430],[26,14313,14314,14317],{"style":3577},[26,14315],{"className":14316,"style":439},[438],[26,14318,14320],{"className":14319},[443,444,445,446],[26,14321,14323],{"className":14322},[69,446],[26,14324,10678],{"className":14325},[69,446],[26,14327,12524],{"className":14328},[69],[26,14330,14332,14335],{"className":14331},[69],[26,14333,572],{"className":14334},[69],[26,14336,14338],{"className":14337},[417],[26,14339,14341],{"className":14340},[421],[26,14342,14344],{"className":14343},[426],[26,14345,14347],{"className":14346,"style":10693},[430],[26,14348,14349,14352],{"style":3577},[26,14350],{"className":14351,"style":439},[438],[26,14353,14355],{"className":14354},[443,444,445,446],[26,14356,14358],{"className":14357},[69,446],[26,14359,10726],{"className":14360},[69,446],[26,14362,14364,14403],{"className":14363},[29],[26,14365,14367],{"className":14366},[33],[35,14368,14369],{"xmlns":37},[39,14370,14371,14400],{},[42,14372,14373,14375,14377,14383,14389],{},[45,14374,99],{},[89,14376,543],{},[3513,14378,14379,14381],{},[554,14380,12403],{},[89,14382,3520],{},[3513,14384,14385,14387],{},[554,14386,12410],{},[89,14388,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,14390,14391,14394],{},[554,14392,14393],{},"45.9",[42,14395,14396,14398],{},[89,14397,10678],{"mathvariant":2636},[89,14399,10678],{"mathvariant":2636},[49,14401,14402],{"encoding":51},"C = 63^\\circ 56' 45.9''",[26,14404,14406,14424],{"className":14405,"ariaHidden":56},[55],[26,14407,14409,14412,14415,14418,14421],{"className":14408},[60],[26,14410],{"className":14411,"style":65},[64],[26,14413,99],{"className":14414,"style":138},[69,70],[26,14416],{"className":14417,"style":676},[123],[26,14419,543],{"className":14420},[680],[26,14422],{"className":14423,"style":676},[123],[26,14425,14427,14430,14433,14462,14465,14497,14501],{"className":14426},[60],[26,14428],{"className":14429,"style":10693},[64],[26,14431,307],{"className":14432},[69],[26,14434,14436,14439],{"className":14435},[69],[26,14437,588],{"className":14438},[69],[26,14440,14442],{"className":14441},[417],[26,14443,14445],{"className":14444},[421],[26,14446,14448],{"className":14447},[426],[26,14449,14451],{"className":14450,"style":3551},[430],[26,14452,14453,14456],{"style":3577},[26,14454],{"className":14455,"style":439},[438],[26,14457,14459],{"className":14458},[443,444,445,446],[26,14460,3520],{"className":14461},[773,446],[26,14463,6471],{"className":14464},[69],[26,14466,14468,14471],{"className":14467},[69],[26,14469,307],{"className":14470},[69],[26,14472,14474],{"className":14473},[417],[26,14475,14477],{"className":14476},[421],[26,14478,14480],{"className":14479},[426],[26,14481,14483],{"className":14482,"style":10693},[430],[26,14484,14485,14488],{"style":3577},[26,14486],{"className":14487,"style":439},[438],[26,14489,14491],{"className":14490},[443,444,445,446],[26,14492,14494],{"className":14493},[69,446],[26,14495,10678],{"className":14496},[69,446],[26,14498,14500],{"className":14499},[69],"45.",[26,14502,14504,14507],{"className":14503},[69],[26,14505,5565],{"className":14506},[69],[26,14508,14510],{"className":14509},[417],[26,14511,14513],{"className":14512},[421],[26,14514,14516],{"className":14515},[426],[26,14517,14519],{"className":14518,"style":10693},[430],[26,14520,14521,14524],{"style":3577},[26,14522],{"className":14523,"style":439},[438],[26,14525,14527],{"className":14526},[443,444,445,446],[26,14528,14530],{"className":14529},[69,446],[26,14531,10726],{"className":14532},[69,446],[13,14534,2825],{"id":2824},[2827,14536,14537,14539,14666,14670,14740,14744,14747,14751,14819,14823],{},[1738,14538,2832],{"id":2831},[18,14540,14541,14542,275,14604,11664],{},"Establish a network where triangles are well-conditioned (angles between ",[26,14543,14545,14563],{"className":14544},[29],[26,14546,14548],{"className":14547},[33],[35,14549,14550],{"xmlns":37},[39,14551,14552,14560],{},[42,14553,14554],{},[3513,14555,14556,14558],{},[554,14557,6034],{},[89,14559,3520],{},[49,14561,14562],{"encoding":51},"30^\\circ",[26,14564,14566],{"className":14565,"ariaHidden":56},[55],[26,14567,14569,14572,14575],{"className":14568},[60],[26,14570],{"className":14571,"style":3551},[64],[26,14573,588],{"className":14574},[69],[26,14576,14578,14581],{"className":14577},[69],[26,14579,3562],{"className":14580},[69],[26,14582,14584],{"className":14583},[417],[26,14585,14587],{"className":14586},[421],[26,14588,14590],{"className":14589},[426],[26,14591,14593],{"className":14592,"style":3551},[430],[26,14594,14595,14598],{"style":3577},[26,14596],{"className":14597,"style":439},[438],[26,14599,14601],{"className":14600},[443,444,445,446],[26,14602,3520],{"className":14603},[773,446],[26,14605,14607,14625],{"className":14606},[29],[26,14608,14610],{"className":14609},[33],[35,14611,14612],{"xmlns":37},[39,14613,14614,14622],{},[42,14615,14616],{},[3513,14617,14618,14620],{},[554,14619,3607],{},[89,14621,3520],{},[49,14623,14624],{"encoding":51},"120^\\circ",[26,14626,14628],{"className":14627,"ariaHidden":56},[55],[26,14629,14631,14634,14637],{"className":14630},[60],[26,14632],{"className":14633,"style":3551},[64],[26,14635,3643],{"className":14636},[69],[26,14638,14640,14643],{"className":14639},[69],[26,14641,3562],{"className":14642},[69],[26,14644,14646],{"className":14645},[417],[26,14647,14649],{"className":14648},[421],[26,14650,14652],{"className":14651},[426],[26,14653,14655],{"className":14654,"style":3551},[430],[26,14656,14657,14660],{"style":3577},[26,14658],{"className":14659,"style":439},[438],[26,14661,14663],{"className":14662},[443,444,445,446],[26,14664,3520],{"className":14665},[773,446],[1738,14667,14669],{"id":14668},"precise-angular-observation","Precise Angular Observation",[18,14671,14672,14673,14739],{},"Measure all three angles of each triangle using a ",[26,14674,14676,14698],{"className":14675},[29],[26,14677,14679],{"className":14678},[33],[35,14680,14681],{"xmlns":37},[39,14682,14683,14695],{},[42,14684,14685],{},[3513,14686,14687,14689],{},[554,14688,341],{},[42,14690,14691,14693],{},[89,14692,10678],{"mathvariant":2636},[89,14694,10678],{"mathvariant":2636},[49,14696,14697],{"encoding":51},"1''",[26,14699,14701],{"className":14700,"ariaHidden":56},[55],[26,14702,14704,14707],{"className":14703},[60],[26,14705],{"className":14706,"style":10693},[64],[26,14708,14710,14713],{"className":14709},[69],[26,14711,341],{"className":14712},[69],[26,14714,14716],{"className":14715},[417],[26,14717,14719],{"className":14718},[421],[26,14720,14722],{"className":14721},[426],[26,14723,14725],{"className":14724,"style":10693},[430],[26,14726,14727,14730],{"style":3577},[26,14728],{"className":14729,"style":439},[438],[26,14731,14733],{"className":14732},[443,444,445,446],[26,14734,14736],{"className":14735},[69,446],[26,14737,10726],{"className":14738},[69,446]," theodolite, completing multiple sets on both faces to minimize instrumental error .",[1738,14741,14743],{"id":14742},"baseline-measurement","Baseline Measurement",[18,14745,14746],{},"Measure at least one side of the network (the baseline) with high-precision EDM or GPS, reducing the length to the ellipsoid .",[1738,14748,14750],{"id":14749},"compute-excess","Compute Excess",[18,14752,3896,14753,14818],{},[26,14754,14756,14777],{"className":14755},[29],[26,14757,14759],{"className":14758},[33],[35,14760,14761],{"xmlns":37},[39,14762,14763,14775],{},[42,14764,14765],{},[3513,14766,14767,14769],{},[45,14768,372],{},[42,14770,14771,14773],{},[89,14772,10678],{"mathvariant":2636},[89,14774,10678],{"mathvariant":2636},[49,14776,10683],{"encoding":51},[26,14778,14780],{"className":14779,"ariaHidden":56},[55],[26,14781,14783,14786],{"className":14782},[60],[26,14784],{"className":14785,"style":10693},[64],[26,14787,14789,14792],{"className":14788},[69],[26,14790,372],{"className":14791,"style":413},[69,70],[26,14793,14795],{"className":14794},[417],[26,14796,14798],{"className":14797},[421],[26,14799,14801],{"className":14800},[426],[26,14802,14804],{"className":14803,"style":10693},[430],[26,14805,14806,14809],{"style":3577},[26,14807],{"className":14808,"style":439},[438],[26,14810,14812],{"className":14811},[443,444,445,446],[26,14813,14815],{"className":14814},[69,446],[26,14816,10726],{"className":14817},[69,446]," for every triangle in the network using the approximate area .",[1738,14820,14822],{"id":14821},"distribution","Distribution",[18,14824,14825,14826,14901,14902,14963],{},"Apply the angular misclosure correction, ensuring the sum of the plane angles (Observed - ",[26,14827,14829,14855],{"className":14828},[29],[26,14830,14832],{"className":14831},[33],[35,14833,14834],{"xmlns":37},[39,14835,14836,14852],{},[42,14837,14838,14848,14850],{},[3513,14839,14840,14842],{},[45,14841,372],{},[42,14843,14844,14846],{},[89,14845,10678],{"mathvariant":2636},[89,14847,10678],{"mathvariant":2636},[45,14849,3991],{"mathvariant":2636},[554,14851,588],{},[49,14853,14854],{"encoding":51},"E''\u002F3",[26,14856,14858],{"className":14857,"ariaHidden":56},[55],[26,14859,14861,14865,14897],{"className":14860},[60],[26,14862],{"className":14863,"style":14864},[64],"height:1.0019em;vertical-align:-0.25em;",[26,14866,14868,14871],{"className":14867},[69],[26,14869,372],{"className":14870,"style":413},[69,70],[26,14872,14874],{"className":14873},[417],[26,14875,14877],{"className":14876},[421],[26,14878,14880],{"className":14879},[426],[26,14881,14883],{"className":14882,"style":10693},[430],[26,14884,14885,14888],{"style":3577},[26,14886],{"className":14887,"style":439},[438],[26,14889,14891],{"className":14890},[443,444,445,446],[26,14892,14894],{"className":14893},[69,446],[26,14895,10726],{"className":14896},[69,446],[26,14898,14900],{"className":14899},[69],"\u002F3",") equals ",[26,14903,14905,14922],{"className":14904},[29],[26,14906,14908],{"className":14907},[33],[35,14909,14910],{"xmlns":37},[39,14911,14912,14920],{},[42,14913,14914],{},[3513,14915,14916,14918],{},[554,14917,5145],{},[89,14919,3520],{},[49,14921,5150],{"encoding":51},[26,14923,14925],{"className":14924,"ariaHidden":56},[55],[26,14926,14928,14931,14934],{"className":14927},[60],[26,14929],{"className":14930,"style":3551},[64],[26,14932,5163],{"className":14933},[69],[26,14935,14937,14940],{"className":14936},[69],[26,14938,3562],{"className":14939},[69],[26,14941,14943],{"className":14942},[417],[26,14944,14946],{"className":14945},[421],[26,14947,14949],{"className":14948},[426],[26,14950,14952],{"className":14951,"style":3551},[430],[26,14953,14954,14957],{"style":3577},[26,14955],{"className":14956,"style":439},[438],[26,14958,14960],{"className":14959},[443,444,445,446],[26,14961,3520],{"className":14962},[773,446]," for further coordinate calculation .",[13,14965,5301],{"id":5300},[1926,14967,14968,15107],{},[1929,14969,14970,14973,14974,15040,15041,15106],{},[21,14971,14972],{},"The 1-Second Rule:"," For a triangle with an area of ",[26,14975,14977,14996],{"className":14976},[29],[26,14978,14980],{"className":14979},[33],[35,14981,14982],{"xmlns":37},[39,14983,14984,14994],{},[42,14985,14986,14988],{},[554,14987,6119],{},[3513,14989,14990,14992],{},[5057,14991,10540],{},[554,14993,572],{},[49,14995,10545],{"encoding":51},[26,14997,14999],{"className":14998,"ariaHidden":56},[55],[26,15000,15002,15005,15008],{"className":15001},[60],[26,15003],{"className":15004,"style":10555},[64],[26,15006,6119],{"className":15007},[69],[26,15009,15011,15017],{"className":15010},[69],[26,15012,15014],{"className":15013},[69,5118],[26,15015,10540],{"className":15016},[69],[26,15018,15020],{"className":15019},[417],[26,15021,15023],{"className":15022},[421],[26,15024,15026],{"className":15025},[426],[26,15027,15029],{"className":15028,"style":10555},[430],[26,15030,15031,15034],{"style":10582},[26,15032],{"className":15033,"style":439},[438],[26,15035,15037],{"className":15036},[443,444,445,446],[26,15038,572],{"className":15039},[69,446],", the spherical excess is approximately ",[26,15042,15044,15065],{"className":15043},[29],[26,15045,15047],{"className":15046},[33],[35,15048,15049],{"xmlns":37},[39,15050,15051,15063],{},[42,15052,15053],{},[3513,15054,15055,15057],{},[554,15056,341],{},[42,15058,15059,15061],{},[89,15060,10678],{"mathvariant":2636},[89,15062,10678],{"mathvariant":2636},[49,15064,14697],{"encoding":51},[26,15066,15068],{"className":15067,"ariaHidden":56},[55],[26,15069,15071,15074],{"className":15070},[60],[26,15072],{"className":15073,"style":10693},[64],[26,15075,15077,15080],{"className":15076},[69],[26,15078,341],{"className":15079},[69],[26,15081,15083],{"className":15082},[417],[26,15084,15086],{"className":15085},[421],[26,15087,15089],{"className":15088},[426],[26,15090,15092],{"className":15091,"style":10693},[430],[26,15093,15094,15097],{"style":3577},[26,15095],{"className":15096,"style":439},[438],[26,15098,15100],{"className":15099},[443,444,445,446],[26,15101,15103],{"className":15102},[69,446],[26,15104,10726],{"className":15105},[69,446],". If your triangles are smaller than this, the effect is often negligible for tertiary work .",[1929,15108,15109,15112,15113,15178],{},[21,15110,15111],{},"R is Variable:"," Use the mean radius of curvature for the latitude of the survey site for higher precision in the ",[26,15114,15116,15137],{"className":15115},[29],[26,15117,15119],{"className":15118},[33],[35,15120,15121],{"xmlns":37},[39,15122,15123,15135],{},[42,15124,15125],{},[3513,15126,15127,15129],{},[45,15128,372],{},[42,15130,15131,15133],{},[89,15132,10678],{"mathvariant":2636},[89,15134,10678],{"mathvariant":2636},[49,15136,10683],{"encoding":51},[26,15138,15140],{"className":15139,"ariaHidden":56},[55],[26,15141,15143,15146],{"className":15142},[60],[26,15144],{"className":15145,"style":10693},[64],[26,15147,15149,15152],{"className":15148},[69],[26,15150,372],{"className":15151,"style":413},[69,70],[26,15153,15155],{"className":15154},[417],[26,15156,15158],{"className":15157},[421],[26,15159,15161],{"className":15160},[426],[26,15162,15164],{"className":15163,"style":10693},[430],[26,15165,15166,15169],{"style":3577},[26,15167],{"className":15168,"style":439},[438],[26,15170,15172],{"className":15171},[443,444,445,446],[26,15173,15175],{"className":15174},[69,446],[26,15176,10726],{"className":15177},[69,446]," calculation .",[13,15180,5569],{"id":5568},[1926,15182,15183,15442],{},[1929,15184,15185,15188,15189,15286,15287,15346,15347,275,15380,11664],{},[21,15186,15187],{},"Forgetting Units:"," When using the formula ",[26,15190,15192,15219],{"className":15191},[29],[26,15193,15195],{"className":15194},[33],[35,15196,15197],{"xmlns":37},[39,15198,15199,15216],{},[42,15200,15201,15203,15205,15208,15210],{},[45,15202,372],{},[89,15204,543],{},[5057,15206,15207],{},"Area",[45,15209,3991],{"mathvariant":2636},[3513,15211,15212,15214],{},[45,15213,10978],{},[554,15215,572],{},[49,15217,15218],{"encoding":51},"E = \\text{Area}\u002FR^2",[26,15220,15222,15240],{"className":15221,"ariaHidden":56},[55],[26,15223,15225,15228,15231,15234,15237],{"className":15224},[60],[26,15226],{"className":15227,"style":65},[64],[26,15229,372],{"className":15230,"style":413},[69,70],[26,15232],{"className":15233,"style":676},[123],[26,15235,543],{"className":15236},[680],[26,15238],{"className":15239,"style":676},[123],[26,15241,15243,15247,15253,15256],{"className":15242},[60],[26,15244],{"className":15245,"style":15246},[64],"height:1.0641em;vertical-align:-0.25em;",[26,15248,15250],{"className":15249},[69,5118],[26,15251,15207],{"className":15252},[69],[26,15254,3991],{"className":15255},[69],[26,15257,15259,15262],{"className":15258},[69],[26,15260,10978],{"className":15261,"style":11134},[69,70],[26,15263,15265],{"className":15264},[417],[26,15266,15268],{"className":15267},[421],[26,15269,15271],{"className":15270},[426],[26,15272,15275],{"className":15273,"style":15274},[430],"height:0.8141em;",[26,15276,15277,15280],{"style":3577},[26,15278],{"className":15279,"style":439},[438],[26,15281,15283],{"className":15282},[443,444,445,446],[26,15284,572],{"className":15285},[69,446],", ensure Area and ",[26,15288,15290,15308],{"className":15289},[29],[26,15291,15293],{"className":15292},[33],[35,15294,15295],{"xmlns":37},[39,15296,15297,15305],{},[42,15298,15299],{},[3513,15300,15301,15303],{},[45,15302,10978],{},[554,15304,572],{},[49,15306,15307],{"encoding":51},"R^2",[26,15309,15311],{"className":15310,"ariaHidden":56},[55],[26,15312,15314,15317],{"className":15313},[60],[26,15315],{"className":15316,"style":15274},[64],[26,15318,15320,15323],{"className":15319},[69],[26,15321,10978],{"className":15322,"style":11134},[69,70],[26,15324,15326],{"className":15325},[417],[26,15327,15329],{"className":15328},[421],[26,15330,15332],{"className":15331},[426],[26,15333,15335],{"className":15334,"style":15274},[430],[26,15336,15337,15340],{"style":3577},[26,15338],{"className":15339,"style":439},[438],[26,15341,15343],{"className":15342},[443,444,445,446],[26,15344,572],{"className":15345},[69,446]," are in the same units (e.g., ",[26,15348,15350,15365],{"className":15349},[29],[26,15351,15353],{"className":15352},[33],[35,15354,15355],{"xmlns":37},[39,15356,15357,15362],{},[42,15358,15359],{},[5057,15360,15361],{},"km",[49,15363,15364],{"encoding":51},"\\text{km}",[26,15366,15368],{"className":15367,"ariaHidden":56},[55],[26,15369,15371,15374],{"className":15370},[60],[26,15372],{"className":15373,"style":7264},[64],[26,15375,15377],{"className":15376},[69,5118],[26,15378,15361],{"className":15379},[69],[26,15381,15383,15401],{"className":15382},[29],[26,15384,15386],{"className":15385},[33],[35,15387,15388],{"xmlns":37},[39,15389,15390,15398],{},[42,15391,15392],{},[3513,15393,15394,15396],{},[5057,15395,15361],{},[554,15397,572],{},[49,15399,15400],{"encoding":51},"\\text{km}^2",[26,15402,15404],{"className":15403,"ariaHidden":56},[55],[26,15405,15407,15410],{"className":15406},[60],[26,15408],{"className":15409,"style":10555},[64],[26,15411,15413,15419],{"className":15412},[69],[26,15414,15416],{"className":15415},[69,5118],[26,15417,15361],{"className":15418},[69],[26,15420,15422],{"className":15421},[417],[26,15423,15425],{"className":15424},[421],[26,15426,15428],{"className":15427},[426],[26,15429,15431],{"className":15430,"style":10555},[430],[26,15432,15433,15436],{"style":10582},[26,15434],{"className":15435,"style":439},[438],[26,15437,15439],{"className":15438},[443,444,445,446],[26,15440,572],{"className":15441},[69,446],[1929,15443,15444,15447],{},[21,15445,15446],{},"Linear Misclosure:"," Applying spherical excess corrections doesn't fix linear errors; it only addresses the geometric contradiction caused by the Earth's shape .",[13,15449,5694],{"id":5693},[5696,15451,15452,15514,15554],{},[5699,15453,15455],{"label":15454},"Do I need to calculate spherical excess for GPS surveys?",[18,15456,15457,15458,15513],{},"GPS coordinates are calculated in 3D Cartesian space (",[26,15459,15461,15483],{"className":15460},[29],[26,15462,15464],{"className":15463},[33],[35,15465,15466],{"xmlns":37},[39,15467,15468,15480],{},[42,15469,15470,15472,15474,15476,15478],{},[45,15471,5906],{},[89,15473,91],{"separator":56},[45,15475,5911],{},[89,15477,91],{"separator":56},[45,15479,5919],{},[49,15481,15482],{"encoding":51},"X, Y, Z",[26,15484,15486],{"className":15485,"ariaHidden":56},[55],[26,15487,15489,15492,15495,15498,15501,15504,15507,15510],{"className":15488},[60],[26,15490],{"className":15491,"style":112},[64],[26,15493,5906],{"className":15494,"style":5935},[69,70],[26,15496,91],{"className":15497},[119],[26,15499],{"className":15500,"style":124},[123],[26,15502,5911],{"className":15503,"style":4156},[69,70],[26,15505,91],{"className":15506},[119],[26,15508],{"className":15509,"style":124},[123],[26,15511,5919],{"className":15512,"style":138},[69,70],") or on the ellipsoid directly, so spherical excess is mathematically \"built-in\" to the processing algorithms. You only need to calculate it manually when using a theodolite for large angular networks .",[5699,15515,15517],{"label":15516},"What is Legendre’s Theorem exactly?",[18,15518,15519,15520,15553],{},"It states that a spherical triangle behaves like a plane triangle if you subtract ",[26,15521,15523,15541],{"className":15522},[29],[26,15524,15526],{"className":15525},[33],[35,15527,15528],{"xmlns":37},[39,15529,15530,15538],{},[42,15531,15532,15534,15536],{},[554,15533,341],{},[45,15535,3991],{"mathvariant":2636},[554,15537,588],{},[49,15539,15540],{"encoding":51},"1\u002F3",[26,15542,15544],{"className":15543,"ariaHidden":56},[55],[26,15545,15547,15550],{"className":15546},[60],[26,15548],{"className":15549,"style":3328},[64],[26,15551,15540],{"className":15552},[69]," of the spherical excess from each of the spherical angles .",[5699,15555,15557],{"label":15556},"Is spherical excess always positive?",[18,15558,15559,15560,5870],{},"Yes. Because the Earth is convex, the observed angles will always sum to more than ",[26,15561,15563,15580],{"className":15562},[29],[26,15564,15566],{"className":15565},[33],[35,15567,15568],{"xmlns":37},[39,15569,15570,15578],{},[42,15571,15572],{},[3513,15573,15574,15576],{},[554,15575,5145],{},[89,15577,3520],{},[49,15579,5150],{"encoding":51},[26,15581,15583],{"className":15582,"ariaHidden":56},[55],[26,15584,15586,15589,15592],{"className":15585},[60],[26,15587],{"className":15588,"style":3551},[64],[26,15590,5163],{"className":15591},[69],[26,15593,15595,15598],{"className":15594},[69],[26,15596,3562],{"className":15597},[69],[26,15599,15601],{"className":15600},[417],[26,15602,15604],{"className":15603},[421],[26,15605,15607],{"className":15606},[426],[26,15608,15610],{"className":15609,"style":3551},[430],[26,15611,15612,15615],{"style":3577},[26,15613],{"className":15614,"style":439},[438],[26,15616,15618],{"className":15617},[443,444,445,446],[26,15619,3520],{"className":15620},[773,446],[13,15622,5757],{"id":5756},[18,15624,15625,15626,15691],{},"Spherical excess is the boundary where plane surveying meets geodesy. By understanding how to calculate ",[26,15627,15629,15650],{"className":15628},[29],[26,15630,15632],{"className":15631},[33],[35,15633,15634],{"xmlns":37},[39,15635,15636,15648],{},[42,15637,15638],{},[3513,15639,15640,15642],{},[45,15641,372],{},[42,15643,15644,15646],{},[89,15645,10678],{"mathvariant":2636},[89,15647,10678],{"mathvariant":2636},[49,15649,10683],{"encoding":51},[26,15651,15653],{"className":15652,"ariaHidden":56},[55],[26,15654,15656,15659],{"className":15655},[60],[26,15657],{"className":15658,"style":10693},[64],[26,15660,15662,15665],{"className":15661},[69],[26,15663,372],{"className":15664,"style":413},[69,70],[26,15666,15668],{"className":15667},[417],[26,15669,15671],{"className":15670},[421],[26,15672,15674],{"className":15673},[426],[26,15675,15677],{"className":15676,"style":10693},[430],[26,15678,15679,15682],{"style":3577},[26,15680],{"className":15681,"style":439},[438],[26,15683,15685],{"className":15684},[443,444,445,446],[26,15686,15688],{"className":15687},[69,446],[26,15689,10726],{"className":15690},[69,446],", the engineering surveyor ensures that large-scale control networks remain mathematically consistent and capable of supporting high-precision construction across vast distances.",[13,15693,5764],{"id":5763},[18,15695,5767,15696,6374],{},[2588,15697,5770],{},{"title":5781,"searchDepth":5782,"depth":5782,"links":15699},[15700,15701,15702,15703,15707,15708,15715,15716,15717,15718,15719],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":194,"depth":5782,"text":195},{"id":344,"depth":5782,"text":345,"children":15704},[15705,15706],{"id":10872,"depth":5791,"text":10873},{"id":11667,"depth":5791,"text":11668},{"id":3282,"depth":5782,"text":3283},{"id":2824,"depth":5782,"text":2825,"children":15709},[15710,15711,15712,15713,15714],{"id":2831,"depth":5791,"text":2832},{"id":14668,"depth":5791,"text":14669},{"id":14742,"depth":5791,"text":14743},{"id":14749,"depth":5791,"text":14750},{"id":14821,"depth":5791,"text":14822},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Control Surveys","A technical guide to calculating spherical excess and adjusting angular observations in primary triangulation networks where the Earth's curvature cannot be ignored.","\u002Fimages\u002Fspherical-excess.jpg",{"authorCredentials":6399,"authorBio":15724,"commentCount":6400,"tags":15725,"relatedTools":15730,"relatedArticles":15733,"sources":15736},"Specialist in geodetic surveying and primary control network adjustment.",[15726,15727,15728,15729,9830],"Geodesy","Spherical Excess","Triangulation","Network Adjustment",[15731,15732],"Theodolite","Least Squares Software",[15734,15735],"wgs84-datum-projections","weighted-observations-mpv",[15737],{"title":5828,"type":5829,"note":15738},"Primary source for spherical excess formulas and Legendre’s theorem.","\u002Fblog\u002Fspherical-excess-large-networks","2024-04-11",16,{"title":10452,"description":15721},{"loc":15739},"blog\u002F24.spherical-excess-large-networks","PYxbA57iyyzpqzu7LWCStZnGumqEOPTf5eJG4fRaJTY",{"id":15747,"title":15748,"author":5850,"body":15749,"category":18743,"description":18744,"extension":5809,"image":18745,"meta":18746,"navigation":5839,"path":18761,"publishedAt":18762,"readTime":18763,"seo":18764,"sitemap":18765,"stem":18766,"__hash__":18767},"blog\u002Fblog\u002F22.wgs84-datum-and-geodetic-projections.md","WGS84 and Projections: The Mathematics of Global to Local Mapping",{"type":10,"value":15750,"toc":18719},[15751,15753,15771,15773,15820,15822,15870,15872,15993,15995,15999,16002,16179,16314,17077,17081,17246,17250,17400,17979,17983,18063,18289,18291,18464,18466,18560,18562,18708,18710,18713,18715],[13,15752,16],{"id":15},[18,15754,15755,15756,15759,15760,15763,15764,15767,15768,151],{},"Every GPS measurement is initially calculated in a global coordinate system called ",[21,15757,15758],{},"WGS84"," (World Geodetic System 1984). However, engineering designs are typically created on flat, rectangular site grids. Mastering the gap between the curved ",[21,15761,15762],{},"ellipsoidal"," world of WGS84 and the flat ",[21,15765,15766],{},"projected"," world of the construction site is fundamental to modern geomatics ",[26,15769,15770],{},"14, 15",[13,15772,155],{"id":154},[18,15774,15775,15776,15813,15814,15817,15818,151],{},"If you treat WGS84 coordinates as if they were on a flat plane, the Earth's curvature will introduce massive distortions over just a few kilometres. Furthermore, WGS84 coordinates refer to a mathematical surface (the ellipsoid) that can be up to ",[26,15777,15779,15795],{"className":15778},[29],[26,15780,15782],{"className":15781},[33],[35,15783,15784],{"xmlns":37},[39,15785,15786,15792],{},[42,15787,15788,15790],{},[554,15789,6159],{},[5057,15791,6986],{},[49,15793,15794],{"encoding":51},"100 \\text{ m}",[26,15796,15798],{"className":15797,"ariaHidden":56},[55],[26,15799,15801,15804,15807],{"className":15800},[60],[26,15802],{"className":15803,"style":4270},[64],[26,15805,6159],{"className":15806},[69],[26,15808,15810],{"className":15809},[69,5118],[26,15811,6986],{"className":15812},[69]," away from the physical \"sea level\" surface (the geoid) ",[26,15815,15816],{},"16, 17",". Understanding this relationship is critical for accurate heighting and distance measurement ",[26,15819,3729],{},[13,15821,173],{"id":172},[18,15823,4917,15824,15826,15827,15830,15831,15860,15861,15863,15864,15867,15868,151],{},[21,15825,15758],{}," system is a geocentric datum, meaning its origin is at the Earth's centre of mass ",[26,15828,15829],{},"15, 18",". It was established using observations from roughly ",[26,15832,15834,15848],{"className":15833},[29],[26,15835,15837],{"className":15836},[33],[35,15838,15839],{"xmlns":37},[39,15840,15841,15846],{},[42,15842,15843],{},[554,15844,15845],{},"1600",[49,15847,15845],{"encoding":51},[26,15849,15851],{"className":15850,"ariaHidden":56},[55],[26,15852,15854,15857],{"className":15853},[60],[26,15855],{"className":15856,"style":4270},[64],[26,15858,15845],{"className":15859},[69]," sites globally and is the standard for all satellite positioning ",[26,15862,5163],{},". For all practical engineering purposes, it is identical to the ",[21,15865,15866],{},"GRS80"," ellipsoid ",[26,15869,7672],{},[13,15871,195],{"id":194},[3763,15873,15874,15883,15980],{},[1929,15875,15876,15879,15880,151],{},[21,15877,15878],{},"Reference Ellipsoid:"," Since the physical Earth is too irregular to define mathematically, we use a smooth, symmetrical ellipsoid that best fits the Earth's shape ",[26,15881,15882],{},"14",[1929,15884,15885,15888,15889,15917,15918,15948,15949,139,15978,151],{},[21,15886,15887],{},"Geodetic Coordinates:"," Position is defined by Latitude (",[26,15890,15892,15905],{"className":15891},[29],[26,15893,15895],{"className":15894},[33],[35,15896,15897],{"xmlns":37},[39,15898,15899,15903],{},[42,15900,15901],{},[45,15902,7357],{},[49,15904,7360],{"encoding":51},[26,15906,15908],{"className":15907,"ariaHidden":56},[55],[26,15909,15911,15914],{"className":15910},[60],[26,15912],{"className":15913,"style":1898},[64],[26,15915,7357],{"className":15916},[69,70],"), Longitude (",[26,15919,15921,15936],{"className":15920},[29],[26,15922,15924],{"className":15923},[33],[35,15925,15926],{"xmlns":37},[39,15927,15928,15933],{},[42,15929,15930],{},[45,15931,15932],{},"λ",[49,15934,15935],{"encoding":51},"\\lambda",[26,15937,15939],{"className":15938,"ariaHidden":56},[55],[26,15940,15942,15945],{"className":15941},[60],[26,15943],{"className":15944,"style":7264},[64],[26,15946,15932],{"className":15947},[69,70],"), and height above the ellipsoid (",[26,15950,15952,15966],{"className":15951},[29],[26,15953,15955],{"className":15954},[33],[35,15956,15957],{"xmlns":37},[39,15958,15959,15964],{},[42,15960,15961],{},[45,15962,15963],{},"h",[49,15965,15963],{"encoding":51},[26,15967,15969],{"className":15968,"ariaHidden":56},[55],[26,15970,15972,15975],{"className":15971},[60],[26,15973],{"className":15974,"style":7264},[64],[26,15976,15963],{"className":15977},[69,70],[26,15979,6250],{},[1929,15981,15982,15985,15986,15989,15990,151],{},[21,15983,15984],{},"Projections:"," To create a map or site plan, we mathematically project these coordinates onto a flat surface (like a cylinder in the ",[21,15987,15988],{},"Transverse Mercator"," projection) ",[26,15991,15992],{},"21",[13,15994,345],{"id":344},[1738,15996,15998],{"id":15997},"_1-wgs84-ellipsoid-parameters","1. WGS84 Ellipsoid Parameters",[18,16000,16001],{},"The size and shape of the WGS84\u002FGRS80 ellipsoid are defined by:",[1926,16003,16004,16105],{},[1929,16005,16006,304,16037,304,16102],{},[21,16007,16008,16009,4551],{},"Semi-major axis (",[26,16010,16012,16025],{"className":16011},[29],[26,16013,16015],{"className":16014},[33],[35,16016,16017],{"xmlns":37},[39,16018,16019,16023],{},[42,16020,16021],{},[45,16022,562],{},[49,16024,562],{"encoding":51},[26,16026,16028],{"className":16027,"ariaHidden":56},[55],[26,16029,16031,16034],{"className":16030},[60],[26,16032],{"className":16033,"style":2617},[64],[26,16035,562],{"className":16036},[69,70],[26,16038,16040,16066],{"className":16039},[29],[26,16041,16043],{"className":16042},[33],[35,16044,16045],{"xmlns":37},[39,16046,16047,16063],{},[42,16048,16049,16051,16053,16056,16058,16061],{},[554,16050,307],{},[89,16052,91],{"separator":56},[554,16054,16055],{},"378",[89,16057,91],{"separator":56},[554,16059,16060],{},"137.0",[5057,16062,6986],{},[49,16064,16065],{"encoding":51},"6,378,137.0 \\text{ m}",[26,16067,16069],{"className":16068,"ariaHidden":56},[55],[26,16070,16072,16075,16078,16081,16084,16087,16090,16093,16096],{"className":16071},[60],[26,16073],{"className":16074,"style":11440},[64],[26,16076,307],{"className":16077},[69],[26,16079,91],{"className":16080},[119],[26,16082],{"className":16083,"style":124},[123],[26,16085,16055],{"className":16086},[69],[26,16088,91],{"className":16089},[119],[26,16091],{"className":16092,"style":124},[123],[26,16094,16060],{"className":16095},[69],[26,16097,16099],{"className":16098},[69,5118],[26,16100,6986],{"className":16101},[69],[26,16103,16104],{},"19, 22",[1929,16106,16107,304,16148,304,16177],{},[21,16108,16109,16110,4551],{},"Inverse Flattening (",[26,16111,16113,16132],{"className":16112},[29],[26,16114,16116],{"className":16115},[33],[35,16117,16118],{"xmlns":37},[39,16119,16120,16129],{},[42,16121,16122,16124,16126],{},[554,16123,341],{},[45,16125,3991],{"mathvariant":2636},[45,16127,16128],{},"f",[49,16130,16131],{"encoding":51},"1\u002Ff",[26,16133,16135],{"className":16134,"ariaHidden":56},[55],[26,16136,16138,16141,16144],{"className":16137},[60],[26,16139],{"className":16140,"style":3328},[64],[26,16142,4111],{"className":16143},[69],[26,16145,16128],{"className":16146,"style":16147},[69,70],"margin-right:0.1076em;",[26,16149,16151,16165],{"className":16150},[29],[26,16152,16154],{"className":16153},[33],[35,16155,16156],{"xmlns":37},[39,16157,16158,16163],{},[42,16159,16160],{},[554,16161,16162],{},"298.257223563",[49,16164,16162],{"encoding":51},[26,16166,16168],{"className":16167,"ariaHidden":56},[55],[26,16169,16171,16174],{"className":16170},[60],[26,16172],{"className":16173,"style":4270},[64],[26,16175,16162],{"className":16176},[69],[26,16178,16104],{},[1738,16180,16182,16183,16248,16249],{"id":16181},"_2-cartesian-xyzx-y-zxyz-to-geodetic-ϕλhphi-lambda-hϕλh","2. Cartesian ",[26,16184,16186,16212],{"className":16185},[29],[26,16187,16189],{"className":16188},[33],[35,16190,16191],{"xmlns":37},[39,16192,16193,16209],{},[42,16194,16195,16197,16199,16201,16203,16205,16207],{},[89,16196,366],{"stretchy":365},[45,16198,5906],{},[89,16200,91],{"separator":56},[45,16202,5911],{},[89,16204,91],{"separator":56},[45,16206,5919],{},[89,16208,386],{"stretchy":365},[49,16210,16211],{"encoding":51},"(X, Y, Z)",[26,16213,16215],{"className":16214,"ariaHidden":56},[55],[26,16216,16218,16221,16224,16227,16230,16233,16236,16239,16242,16245],{"className":16217},[60],[26,16219],{"className":16220,"style":3328},[64],[26,16222,366],{"className":16223},[406],[26,16225,5906],{"className":16226,"style":5935},[69,70],[26,16228,91],{"className":16229},[119],[26,16231],{"className":16232,"style":124},[123],[26,16234,5911],{"className":16235,"style":4156},[69,70],[26,16237,91],{"className":16238},[119],[26,16240],{"className":16241,"style":124},[123],[26,16243,5919],{"className":16244,"style":138},[69,70],[26,16246,386],{"className":16247},[515]," to Geodetic ",[26,16250,16252,16278],{"className":16251},[29],[26,16253,16255],{"className":16254},[33],[35,16256,16257],{"xmlns":37},[39,16258,16259,16275],{},[42,16260,16261,16263,16265,16267,16269,16271,16273],{},[89,16262,366],{"stretchy":365},[45,16264,7357],{},[89,16266,91],{"separator":56},[45,16268,15932],{},[89,16270,91],{"separator":56},[45,16272,15963],{},[89,16274,386],{"stretchy":365},[49,16276,16277],{"encoding":51},"(\\phi, \\lambda, h)",[26,16279,16281],{"className":16280,"ariaHidden":56},[55],[26,16282,16284,16287,16290,16293,16296,16299,16302,16305,16308,16311],{"className":16283},[60],[26,16285],{"className":16286,"style":3328},[64],[26,16288,366],{"className":16289},[406],[26,16291,7357],{"className":16292},[69,70],[26,16294,91],{"className":16295},[119],[26,16297],{"className":16298,"style":124},[123],[26,16300,15932],{"className":16301},[69,70],[26,16303,91],{"className":16304},[119],[26,16306],{"className":16307,"style":124},[123],[26,16309,15963],{"className":16310},[69,70],[26,16312,386],{"className":16313},[515],[18,16315,16316,16317,304,16449,16452,304,16788,16790,304,16981,16983,304,17075,151],{},"To convert satellite Cartesian coordinates to mapping coordinates:\n",[26,16318,16320,16346],{"className":16319},[29],[26,16321,16323],{"className":16322},[33],[35,16324,16325],{"xmlns":37},[39,16326,16327,16343],{},[42,16328,16329,16331,16333,16335,16337],{},[45,16330,7388],{},[89,16332,1962],{},[45,16334,15932],{},[89,16336,543],{},[545,16338,16339,16341],{},[45,16340,5911],{},[45,16342,5906],{},[49,16344,16345],{"encoding":51},"\\tan \\lambda = \\frac{Y}{X}",[26,16347,16349,16373],{"className":16348,"ariaHidden":56},[55],[26,16350,16352,16355,16358,16361,16364,16367,16370],{"className":16351},[60],[26,16353],{"className":16354,"style":7264},[64],[26,16356,7388],{"className":16357},[2079],[26,16359],{"className":16360,"style":124},[123],[26,16362,15932],{"className":16363},[69,70],[26,16365],{"className":16366,"style":676},[123],[26,16368,543],{"className":16369},[680],[26,16371],{"className":16372,"style":676},[123],[26,16374,16376,16380],{"className":16375},[60],[26,16377],{"className":16378,"style":16379},[64],"height:1.2173em;vertical-align:-0.345em;",[26,16381,16383,16386,16446],{"className":16382},[69],[26,16384],{"className":16385},[406,697],[26,16387,16389],{"className":16388},[545],[26,16390,16392,16438],{"className":16391},[421,422],[26,16393,16395,16435],{"className":16394},[426],[26,16396,16399,16413,16421],{"className":16397,"style":16398},[430],"height:0.8723em;",[26,16400,16401,16404],{"style":713},[26,16402],{"className":16403,"style":717},[438],[26,16405,16407],{"className":16406},[443,444,445,446],[26,16408,16410],{"className":16409},[69,446],[26,16411,5906],{"className":16412,"style":5935},[69,70,446],[26,16414,16415,16418],{"style":859},[26,16416],{"className":16417,"style":717},[438],[26,16419],{"className":16420,"style":867},[866],[26,16422,16423,16426],{"style":2129},[26,16424],{"className":16425,"style":717},[438],[26,16427,16429],{"className":16428},[443,444,445,446],[26,16430,16432],{"className":16431},[69,446],[26,16433,5911],{"className":16434,"style":4156},[69,70,446],[26,16436,454],{"className":16437},[453],[26,16439,16441],{"className":16440},[426],[26,16442,16444],{"className":16443,"style":11249},[430],[26,16445],{},[26,16447],{"className":16448},[515,697],[26,16450,16451],{},"23",[26,16453,16455,16529],{"className":16454},[29],[26,16456,16458],{"className":16457},[33],[35,16459,16460],{"xmlns":37},[39,16461,16462,16526],{},[42,16463,16464,16466,16468,16470,16472],{},[45,16465,7388],{},[89,16467,1962],{},[45,16469,7357],{},[89,16471,543],{},[545,16473,16474,16496],{},[42,16475,16476,16478,16480,16487,16490,16492,16494],{},[45,16477,5919],{},[89,16479,565],{},[3513,16481,16482,16485],{},[45,16483,16484],{},"e",[554,16486,572],{},[45,16488,16489],{},"ν",[45,16491,8523],{},[89,16493,1962],{},[45,16495,7357],{},[42,16497,16498,16500,16506,16508,16514],{},[89,16499,366],{"stretchy":365},[3513,16501,16502,16504],{},[45,16503,5906],{},[554,16505,572],{},[89,16507,565],{},[3513,16509,16510,16512],{},[45,16511,5911],{},[554,16513,572],{},[3513,16515,16516,16518],{},[89,16517,386],{"stretchy":365},[42,16519,16520,16522,16524],{},[554,16521,341],{},[45,16523,3991],{"mathvariant":2636},[554,16525,572],{},[49,16527,16528],{"encoding":51},"\\tan \\phi = \\frac{Z + e^2 \\nu \\sin \\phi}{(X^2 + Y^2)^{1\u002F2}}",[26,16530,16532,16556],{"className":16531,"ariaHidden":56},[55],[26,16533,16535,16538,16541,16544,16547,16550,16553],{"className":16534},[60],[26,16536],{"className":16537,"style":1898},[64],[26,16539,7388],{"className":16540},[2079],[26,16542],{"className":16543,"style":124},[123],[26,16545,7357],{"className":16546},[69,70],[26,16548],{"className":16549,"style":676},[123],[26,16551,543],{"className":16552},[680],[26,16554],{"className":16555,"style":676},[123],[26,16557,16559,16563],{"className":16558},[60],[26,16560],{"className":16561,"style":16562},[64],"height:1.6305em;vertical-align:-0.5604em;",[26,16564,16566,16569,16785],{"className":16565},[69],[26,16567],{"className":16568},[406,697],[26,16570,16572],{"className":16571},[545],[26,16573,16575,16776],{"className":16574},[421,422],[26,16576,16578,16773],{"className":16577},[426],[26,16579,16582,16694,16702],{"className":16580,"style":16581},[430],"height:1.07em;",[26,16583,16585,16588],{"style":16584},"top:-2.6146em;",[26,16586],{"className":16587,"style":717},[438],[26,16589,16591],{"className":16590},[443,444,445,446],[26,16592,16594,16597,16626,16629,16658],{"className":16593},[69,446],[26,16595,366],{"className":16596},[406,446],[26,16598,16600,16603],{"className":16599},[69,446],[26,16601,5906],{"className":16602,"style":5935},[69,70,446],[26,16604,16606],{"className":16605},[417],[26,16607,16609],{"className":16608},[421],[26,16610,16612],{"className":16611},[426],[26,16613,16615],{"className":16614,"style":11147},[430],[26,16616,16617,16620],{"style":11150},[26,16618],{"className":16619,"style":749},[438],[26,16621,16623],{"className":16622},[443,753,754,446],[26,16624,572],{"className":16625},[69,446],[26,16627,565],{"className":16628},[773,446],[26,16630,16632,16635],{"className":16631},[69,446],[26,16633,5911],{"className":16634,"style":4156},[69,70,446],[26,16636,16638],{"className":16637},[417],[26,16639,16641],{"className":16640},[421],[26,16642,16644],{"className":16643},[426],[26,16645,16647],{"className":16646,"style":11147},[430],[26,16648,16649,16652],{"style":11150},[26,16650],{"className":16651,"style":749},[438],[26,16653,16655],{"className":16654},[443,753,754,446],[26,16656,572],{"className":16657},[69,446],[26,16659,16661,16664],{"className":16660},[515,446],[26,16662,386],{"className":16663},[515,446],[26,16665,16667],{"className":16666},[417],[26,16668,16670],{"className":16669},[421],[26,16671,16673],{"className":16672},[426],[26,16674,16677],{"className":16675,"style":16676},[430],"height:0.822em;",[26,16678,16680,16684],{"style":16679},"top:-2.822em;margin-right:0.0714em;",[26,16681],{"className":16682,"style":16683},[438],"height:2.5357em;",[26,16685,16687],{"className":16686},[443,753,754,446],[26,16688,16690],{"className":16689},[69,446],[26,16691,16693],{"className":16692},[69,446],"1\u002F2",[26,16695,16696,16699],{"style":859},[26,16697],{"className":16698,"style":717},[438],[26,16700],{"className":16701,"style":867},[866],[26,16703,16704,16707],{"style":11224},[26,16705],{"className":16706,"style":717},[438],[26,16708,16710],{"className":16709},[443,444,445,446],[26,16711,16713,16716,16719,16748,16752,16755,16767,16770],{"className":16712},[69,446],[26,16714,5919],{"className":16715,"style":138},[69,70,446],[26,16717,565],{"className":16718},[773,446],[26,16720,16722,16725],{"className":16721},[69,446],[26,16723,16484],{"className":16724},[69,70,446],[26,16726,16728],{"className":16727},[417],[26,16729,16731],{"className":16730},[421],[26,16732,16734],{"className":16733},[426],[26,16735,16737],{"className":16736,"style":13003},[430],[26,16738,16739,16742],{"style":13006},[26,16740],{"className":16741,"style":749},[438],[26,16743,16745],{"className":16744},[443,753,754,446],[26,16746,572],{"className":16747},[69,446],[26,16749,16489],{"className":16750,"style":16751},[69,70,446],"margin-right:0.0637em;",[26,16753],{"className":16754,"style":2094},[123,446],[26,16756,16758,16761,16764],{"className":16757},[2079,446],[26,16759,11169],{"className":16760},[446],[26,16762,11173],{"className":16763},[446],[26,16765,11177],{"className":16766},[446],[26,16768],{"className":16769,"style":2094},[123,446],[26,16771,7357],{"className":16772},[69,70,446],[26,16774,454],{"className":16775},[453],[26,16777,16779],{"className":16778},[426],[26,16780,16783],{"className":16781,"style":16782},[430],"height:0.5604em;",[26,16784],{},[26,16786],{"className":16787},[515,697],[26,16789,16451],{},[26,16791,16793,16831],{"className":16792},[29],[26,16794,16796],{"className":16795},[33],[35,16797,16798],{"xmlns":37},[39,16799,16800,16828],{},[42,16801,16802,16804,16806,16824,16826],{},[45,16803,15963],{},[89,16805,543],{},[545,16807,16808,16810],{},[45,16809,5906],{},[42,16811,16812,16814,16816,16818,16820,16822],{},[45,16813,7407],{},[89,16815,1962],{},[45,16817,7357],{},[45,16819,7407],{},[89,16821,1962],{},[45,16823,15932],{},[89,16825,1967],{},[45,16827,16489],{},[49,16829,16830],{"encoding":51},"h = \\frac{X}{\\cos \\phi \\cos \\lambda} - \\nu",[26,16832,16834,16852,16972],{"className":16833,"ariaHidden":56},[55],[26,16835,16837,16840,16843,16846,16849],{"className":16836},[60],[26,16838],{"className":16839,"style":7264},[64],[26,16841,15963],{"className":16842},[69,70],[26,16844],{"className":16845,"style":676},[123],[26,16847,543],{"className":16848},[680],[26,16850],{"className":16851,"style":676},[123],[26,16853,16855,16859,16963,16966,16969],{"className":16854},[60],[26,16856],{"className":16857,"style":16858},[64],"height:1.3534em;vertical-align:-0.4811em;",[26,16860,16862,16865,16960],{"className":16861},[69],[26,16863],{"className":16864},[406,697],[26,16866,16868],{"className":16867},[545],[26,16869,16871,16952],{"className":16870},[421,422],[26,16872,16874,16949],{"className":16873},[426],[26,16875,16877,16927,16935],{"className":16876,"style":16398},[430],[26,16878,16879,16882],{"style":713},[26,16880],{"className":16881,"style":717},[438],[26,16883,16885],{"className":16884},[443,444,445,446],[26,16886,16888,16900,16903,16906,16909,16921,16924],{"className":16887},[69,446],[26,16889,16891,16894,16897],{"className":16890},[2079,446],[26,16892,595],{"className":16893},[446],[26,16895,2086],{"className":16896},[446],[26,16898,11169],{"className":16899},[446],[26,16901],{"className":16902,"style":2094},[123,446],[26,16904,7357],{"className":16905},[69,70,446],[26,16907],{"className":16908,"style":2094},[123,446],[26,16910,16912,16915,16918],{"className":16911},[2079,446],[26,16913,595],{"className":16914},[446],[26,16916,2086],{"className":16917},[446],[26,16919,11169],{"className":16920},[446],[26,16922],{"className":16923,"style":2094},[123,446],[26,16925,15932],{"className":16926},[69,70,446],[26,16928,16929,16932],{"style":859},[26,16930],{"className":16931,"style":717},[438],[26,16933],{"className":16934,"style":867},[866],[26,16936,16937,16940],{"style":2129},[26,16938],{"className":16939,"style":717},[438],[26,16941,16943],{"className":16942},[443,444,445,446],[26,16944,16946],{"className":16945},[69,446],[26,16947,5906],{"className":16948,"style":5935},[69,70,446],[26,16950,454],{"className":16951},[453],[26,16953,16955],{"className":16954},[426],[26,16956,16958],{"className":16957,"style":2366},[430],[26,16959],{},[26,16961],{"className":16962},[515,697],[26,16964],{"className":16965,"style":4156},[123],[26,16967,1967],{"className":16968},[773],[26,16970],{"className":16971,"style":4156},[123],[26,16973,16975,16978],{"className":16974},[60],[26,16976],{"className":16977,"style":2617},[64],[26,16979,16489],{"className":16980,"style":16751},[69,70],[26,16982,16451],{},[2588,16984,2590,16985,17014,17015,17074],{},[26,16986,16988,17002],{"className":16987},[29],[26,16989,16991],{"className":16990},[33],[35,16992,16993],{"xmlns":37},[39,16994,16995,16999],{},[42,16996,16997],{},[45,16998,16489],{},[49,17000,17001],{"encoding":51},"\\nu",[26,17003,17005],{"className":17004,"ariaHidden":56},[55],[26,17006,17008,17011],{"className":17007},[60],[26,17009],{"className":17010,"style":2617},[64],[26,17012,16489],{"className":17013,"style":16751},[69,70]," is the radius of curvature in the prime vertical, and ",[26,17016,17018,17036],{"className":17017},[29],[26,17019,17021],{"className":17020},[33],[35,17022,17023],{"xmlns":37},[39,17024,17025,17033],{},[42,17026,17027],{},[3513,17028,17029,17031],{},[45,17030,16484],{},[554,17032,572],{},[49,17034,17035],{"encoding":51},"e^2",[26,17037,17039],{"className":17038,"ariaHidden":56},[55],[26,17040,17042,17045],{"className":17041},[60],[26,17043],{"className":17044,"style":15274},[64],[26,17046,17048,17051],{"className":17047},[69],[26,17049,16484],{"className":17050},[69,70],[26,17052,17054],{"className":17053},[417],[26,17055,17057],{"className":17056},[421],[26,17058,17060],{"className":17059},[426],[26,17061,17063],{"className":17062,"style":15274},[430],[26,17064,17065,17068],{"style":3577},[26,17066],{"className":17067,"style":439},[438],[26,17069,17071],{"className":17070},[443,444,445,446],[26,17072,572],{"className":17073},[69,446]," is the eccentricity squared)",[26,17076,16451],{},[13,17078,17080],{"id":17079},"field-workflow-wgs84-to-site-grid","Field Workflow: WGS84 to Site Grid",[2827,17082,17083,17087,17093,17097,17107,17111,17203,17207],{},[1738,17084,17086],{"id":17085},"capture-data","Capture Data",[18,17088,17089,17090,151],{},"Observe your control points using high-precision GNSS in the WGS84 system ",[26,17091,17092],{},"24",[1738,17094,17096],{"id":17095},"perform-transformation","Perform Transformation",[18,17098,17099,17100,17103,17104,151],{},"Apply a ",[21,17101,17102],{},"Helmert 7-Parameter Transformation"," to shift the origin, rotate the axes, and adjust the scale to match the local datum (e.g., OSGB36) ",[26,17105,17106],{},"25, 26",[1738,17108,17110],{"id":17109},"project-to-plane","Project to Plane",[18,17112,17113,17114,17156,17157,17199,17200,151],{},"Convert the transformed ellipsoidal coordinates (",[26,17115,17117,17135],{"className":17116},[29],[26,17118,17120],{"className":17119},[33],[35,17121,17122],{"xmlns":37},[39,17123,17124,17132],{},[42,17125,17126,17128,17130],{},[45,17127,7357],{},[89,17129,91],{"separator":56},[45,17131,15932],{},[49,17133,17134],{"encoding":51},"\\phi, \\lambda",[26,17136,17138],{"className":17137,"ariaHidden":56},[55],[26,17139,17141,17144,17147,17150,17153],{"className":17140},[60],[26,17142],{"className":17143,"style":1898},[64],[26,17145,7357],{"className":17146},[69,70],[26,17148,91],{"className":17149},[119],[26,17151],{"className":17152,"style":124},[123],[26,17154,15932],{"className":17155},[69,70],") into Northings and Eastings (",[26,17158,17160,17178],{"className":17159},[29],[26,17161,17163],{"className":17162},[33],[35,17164,17165],{"xmlns":37},[39,17166,17167,17175],{},[42,17168,17169,17171,17173],{},[45,17170,372],{},[89,17172,91],{"separator":56},[45,17174,381],{},[49,17176,17177],{"encoding":51},"E, N",[26,17179,17181],{"className":17180,"ariaHidden":56},[55],[26,17182,17184,17187,17190,17193,17196],{"className":17183},[60],[26,17185],{"className":17186,"style":112},[64],[26,17188,372],{"className":17189,"style":413},[69,70],[26,17191,91],{"className":17192},[119],[26,17194],{"className":17195,"style":124},[123],[26,17197,381],{"className":17198,"style":476},[69,70],") using a projection like the Universal Transverse Mercator (UTM) ",[26,17201,17202],{},"21, 27",[1738,17204,17206],{"id":17205},"apply-local-scale-factor","Apply Local Scale Factor",[18,17208,17209,17210,17242,17243,151],{},"Because projections distort distances, multiply all \"grid\" distances by the ",[21,17211,17212,17213,386],{},"Local Scale Factor (",[26,17214,17216,17230],{"className":17215},[29],[26,17217,17219],{"className":17218},[33],[35,17220,17221],{"xmlns":37},[39,17222,17223,17228],{},[42,17224,17225],{},[45,17226,17227],{},"F",[49,17229,17227],{"encoding":51},[26,17231,17233],{"className":17232,"ariaHidden":56},[55],[26,17234,17236,17239],{"className":17235},[60],[26,17237],{"className":17238,"style":65},[64],[26,17240,17227],{"className":17241,"style":71},[69,70]," to get the actual ground distance for setting out ",[26,17244,17245],{},"28, 29",[13,17247,17249],{"id":17248},"step-by-step-example-ellipsoid-elements","Step-by-Step Example: Ellipsoid Elements",[18,17251,17252,17254,17255,17283,17284,17312,17313,17341,17342,4476],{},[21,17253,7512],{}," Given the semi-major axis (",[26,17256,17258,17271],{"className":17257},[29],[26,17259,17261],{"className":17260},[33],[35,17262,17263],{"xmlns":37},[39,17264,17265,17269],{},[42,17266,17267],{},[45,17268,562],{},[49,17270,562],{"encoding":51},[26,17272,17274],{"className":17273,"ariaHidden":56},[55],[26,17275,17277,17280],{"className":17276},[60],[26,17278],{"className":17279,"style":2617},[64],[26,17281,562],{"className":17282},[69,70],") and flattening (",[26,17285,17287,17300],{"className":17286},[29],[26,17288,17290],{"className":17289},[33],[35,17291,17292],{"xmlns":37},[39,17293,17294,17298],{},[42,17295,17296],{},[45,17297,16128],{},[49,17299,16128],{"encoding":51},[26,17301,17303],{"className":17302,"ariaHidden":56},[55],[26,17304,17306,17309],{"className":17305},[60],[26,17307],{"className":17308,"style":1898},[64],[26,17310,16128],{"className":17311,"style":16147},[69,70],"), find the semi-minor axis (",[26,17314,17316,17329],{"className":17315},[29],[26,17317,17319],{"className":17318},[33],[35,17320,17321],{"xmlns":37},[39,17322,17323,17327],{},[42,17324,17325],{},[45,17326,579],{},[49,17328,579],{"encoding":51},[26,17330,17332],{"className":17331,"ariaHidden":56},[55],[26,17333,17335,17338],{"className":17334},[60],[26,17336],{"className":17337,"style":7264},[64],[26,17339,579],{"className":17340},[69,70],") and eccentricity squared (",[26,17343,17345,17362],{"className":17344},[29],[26,17346,17348],{"className":17347},[33],[35,17349,17350],{"xmlns":37},[39,17351,17352,17360],{},[42,17353,17354],{},[3513,17355,17356,17358],{},[45,17357,16484],{},[554,17359,572],{},[49,17361,17035],{"encoding":51},[26,17363,17365],{"className":17364,"ariaHidden":56},[55],[26,17366,17368,17371],{"className":17367},[60],[26,17369],{"className":17370,"style":15274},[64],[26,17372,17374,17377],{"className":17373},[69],[26,17375,16484],{"className":17376},[69,70],[26,17378,17380],{"className":17379},[417],[26,17381,17383],{"className":17382},[421],[26,17384,17386],{"className":17385},[426],[26,17387,17389],{"className":17388,"style":15274},[430],[26,17390,17391,17394],{"style":3577},[26,17392],{"className":17393,"style":439},[438],[26,17395,17397],{"className":17396},[443,444,445,446],[26,17398,572],{"className":17399},[69,446],[3763,17401,17402,17662],{},[1929,17403,17404,17435,304,17660],{},[21,17405,17406,17407,4551],{},"Semi-minor axis (",[26,17408,17410,17423],{"className":17409},[29],[26,17411,17413],{"className":17412},[33],[35,17414,17415],{"xmlns":37},[39,17416,17417,17421],{},[42,17418,17419],{},[45,17420,579],{},[49,17422,579],{"encoding":51},[26,17424,17426],{"className":17425,"ariaHidden":56},[55],[26,17427,17429,17432],{"className":17428},[60],[26,17430],{"className":17431,"style":7264},[64],[26,17433,579],{"className":17434},[69,70],[26,17436,17438,17488],{"className":17437},[29],[26,17439,17441],{"className":17440},[33],[35,17442,17443],{"xmlns":37},[39,17444,17445,17485],{},[42,17446,17447,17449,17451,17463,17465,17467,17469,17471,17473,17475,17477,17479,17481,17483],{},[45,17448,16128],{},[89,17450,543],{},[545,17452,17453,17461],{},[42,17454,17455,17457,17459],{},[45,17456,562],{},[89,17458,1967],{},[45,17460,579],{},[45,17462,562],{},[5057,17464,8682],{},[89,17466,8685],{},[5057,17468,8682],{},[45,17470,579],{},[89,17472,543],{},[45,17474,562],{},[89,17476,366],{"stretchy":365},[554,17478,341],{},[89,17480,1967],{},[45,17482,16128],{},[89,17484,386],{"stretchy":365},[49,17486,17487],{"encoding":51},"f = \\frac{a - b}{a} \\implies b = a(1 - f)",[26,17489,17491,17509,17606,17624,17648],{"className":17490,"ariaHidden":56},[55],[26,17492,17494,17497,17500,17503,17506],{"className":17493},[60],[26,17495],{"className":17496,"style":1898},[64],[26,17498,16128],{"className":17499,"style":16147},[69,70],[26,17501],{"className":17502,"style":676},[123],[26,17504,543],{"className":17505},[680],[26,17507],{"className":17508,"style":676},[123],[26,17510,17512,17516,17591,17594,17597,17600,17603],{"className":17511},[60],[26,17513],{"className":17514,"style":17515},[64],"height:1.2251em;vertical-align:-0.345em;",[26,17517,17519,17522,17588],{"className":17518},[69],[26,17520],{"className":17521},[406,697],[26,17523,17525],{"className":17524},[545],[26,17526,17528,17580],{"className":17527},[421,422],[26,17529,17531,17577],{"className":17530},[426],[26,17532,17535,17549,17557],{"className":17533,"style":17534},[430],"height:0.8801em;",[26,17536,17537,17540],{"style":713},[26,17538],{"className":17539,"style":717},[438],[26,17541,17543],{"className":17542},[443,444,445,446],[26,17544,17546],{"className":17545},[69,446],[26,17547,562],{"className":17548},[69,70,446],[26,17550,17551,17554],{"style":859},[26,17552],{"className":17553,"style":717},[438],[26,17555],{"className":17556,"style":867},[866],[26,17558,17559,17562],{"style":2129},[26,17560],{"className":17561,"style":717},[438],[26,17563,17565],{"className":17564},[443,444,445,446],[26,17566,17568,17571,17574],{"className":17567},[69,446],[26,17569,562],{"className":17570},[69,70,446],[26,17572,1967],{"className":17573},[773,446],[26,17575,579],{"className":17576},[69,70,446],[26,17578,454],{"className":17579},[453],[26,17581,17583],{"className":17582},[426],[26,17584,17586],{"className":17585,"style":11249},[430],[26,17587],{},[26,17589],{"className":17590},[515,697],[26,17592],{"className":17593,"style":676},[123],[26,17595],{"className":17596,"style":676},[123],[26,17598,8685],{"className":17599},[680],[26,17601],{"className":17602,"style":676},[123],[26,17604],{"className":17605,"style":676},[123],[26,17607,17609,17612,17615,17618,17621],{"className":17608},[60],[26,17610],{"className":17611,"style":7264},[64],[26,17613,579],{"className":17614},[69,70],[26,17616],{"className":17617,"style":676},[123],[26,17619,543],{"className":17620},[680],[26,17622],{"className":17623,"style":676},[123],[26,17625,17627,17630,17633,17636,17639,17642,17645],{"className":17626},[60],[26,17628],{"className":17629,"style":3328},[64],[26,17631,562],{"className":17632},[69,70],[26,17634,366],{"className":17635},[406],[26,17637,341],{"className":17638},[69],[26,17640],{"className":17641,"style":4156},[123],[26,17643,1967],{"className":17644},[773],[26,17646],{"className":17647,"style":4156},[123],[26,17649,17651,17654,17657],{"className":17650},[60],[26,17652],{"className":17653,"style":3328},[64],[26,17655,16128],{"className":17656,"style":16147},[69,70],[26,17658,386],{"className":17659},[515],[26,17661,7557],{},[1929,17663,17664,17725,304,17977],{},[21,17665,17666,17667,4551],{},"Eccentricity squared (",[26,17668,17670,17687],{"className":17669},[29],[26,17671,17673],{"className":17672},[33],[35,17674,17675],{"xmlns":37},[39,17676,17677,17685],{},[42,17678,17679],{},[3513,17680,17681,17683],{},[45,17682,16484],{},[554,17684,572],{},[49,17686,17035],{"encoding":51},[26,17688,17690],{"className":17689,"ariaHidden":56},[55],[26,17691,17693,17696],{"className":17692},[60],[26,17694],{"className":17695,"style":15274},[64],[26,17697,17699,17702],{"className":17698},[69],[26,17700,16484],{"className":17701},[69,70],[26,17703,17705],{"className":17704},[417],[26,17706,17708],{"className":17707},[421],[26,17709,17711],{"className":17710},[426],[26,17712,17714],{"className":17713,"style":15274},[430],[26,17715,17716,17719],{"style":3577},[26,17717],{"className":17718,"style":439},[438],[26,17720,17722],{"className":17721},[443,444,445,446],[26,17723,572],{"className":17724},[69,446],[26,17726,17728,17772],{"className":17727},[29],[26,17729,17731],{"className":17730},[33],[35,17732,17733],{"xmlns":37},[39,17734,17735,17769],{},[42,17736,17737,17743,17745],{},[3513,17738,17739,17741],{},[45,17740,16484],{},[554,17742,572],{},[89,17744,543],{},[545,17746,17747,17763],{},[42,17748,17749,17755,17757],{},[3513,17750,17751,17753],{},[45,17752,562],{},[554,17754,572],{},[89,17756,1967],{},[3513,17758,17759,17761],{},[45,17760,579],{},[554,17762,572],{},[3513,17764,17765,17767],{},[45,17766,562],{},[554,17768,572],{},[49,17770,17771],{"encoding":51},"e^2 = \\frac{a^2 - b^2}{a^2}",[26,17773,17775,17819],{"className":17774,"ariaHidden":56},[55],[26,17776,17778,17781,17810,17813,17816],{"className":17777},[60],[26,17779],{"className":17780,"style":15274},[64],[26,17782,17784,17787],{"className":17783},[69],[26,17785,16484],{"className":17786},[69,70],[26,17788,17790],{"className":17789},[417],[26,17791,17793],{"className":17792},[421],[26,17794,17796],{"className":17795},[426],[26,17797,17799],{"className":17798,"style":15274},[430],[26,17800,17801,17804],{"style":3577},[26,17802],{"className":17803,"style":439},[438],[26,17805,17807],{"className":17806},[443,444,445,446],[26,17808,572],{"className":17809},[69,446],[26,17811],{"className":17812,"style":676},[123],[26,17814,543],{"className":17815},[680],[26,17817],{"className":17818,"style":676},[123],[26,17820,17822,17825],{"className":17821},[60],[26,17823],{"className":17824,"style":12882},[64],[26,17826,17828,17831,17974],{"className":17827},[69],[26,17829],{"className":17830},[406,697],[26,17832,17834],{"className":17833},[545],[26,17835,17837,17966],{"className":17836},[421,422],[26,17838,17840,17963],{"className":17839},[426],[26,17841,17843,17883,17891],{"className":17842,"style":12901},[430],[26,17844,17845,17848],{"style":713},[26,17846],{"className":17847,"style":717},[438],[26,17849,17851],{"className":17850},[443,444,445,446],[26,17852,17854],{"className":17853},[69,446],[26,17855,17857,17860],{"className":17856},[69,446],[26,17858,562],{"className":17859},[69,70,446],[26,17861,17863],{"className":17862},[417],[26,17864,17866],{"className":17865},[421],[26,17867,17869],{"className":17868},[426],[26,17870,17872],{"className":17871,"style":11147},[430],[26,17873,17874,17877],{"style":11150},[26,17875],{"className":17876,"style":749},[438],[26,17878,17880],{"className":17879},[443,753,754,446],[26,17881,572],{"className":17882},[69,446],[26,17884,17885,17888],{"style":859},[26,17886],{"className":17887,"style":717},[438],[26,17889],{"className":17890,"style":867},[866],[26,17892,17893,17896],{"style":2129},[26,17894],{"className":17895,"style":717},[438],[26,17897,17899],{"className":17898},[443,444,445,446],[26,17900,17902,17931,17934],{"className":17901},[69,446],[26,17903,17905,17908],{"className":17904},[69,446],[26,17906,562],{"className":17907},[69,70,446],[26,17909,17911],{"className":17910},[417],[26,17912,17914],{"className":17913},[421],[26,17915,17917],{"className":17916},[426],[26,17918,17920],{"className":17919,"style":13003},[430],[26,17921,17922,17925],{"style":13006},[26,17923],{"className":17924,"style":749},[438],[26,17926,17928],{"className":17927},[443,753,754,446],[26,17929,572],{"className":17930},[69,446],[26,17932,1967],{"className":17933},[773,446],[26,17935,17937,17940],{"className":17936},[69,446],[26,17938,579],{"className":17939},[69,70,446],[26,17941,17943],{"className":17942},[417],[26,17944,17946],{"className":17945},[421],[26,17947,17949],{"className":17948},[426],[26,17950,17952],{"className":17951,"style":13003},[430],[26,17953,17954,17957],{"style":13006},[26,17955],{"className":17956,"style":749},[438],[26,17958,17960],{"className":17959},[443,753,754,446],[26,17961,572],{"className":17962},[69,446],[26,17964,454],{"className":17965},[453],[26,17967,17969],{"className":17968},[426],[26,17970,17972],{"className":17971,"style":11249},[430],[26,17973],{},[26,17975],{"className":17976},[515,697],[26,17978,7607],{},[13,17980,17982],{"id":17981},"formula-breakdown","Formula Breakdown",[18,17984,17985,17988,304,18061,11453],{},[21,17986,17987],{},"The Height Problem:",[26,17989,17991,18013],{"className":17990},[29],[26,17992,17994],{"className":17993},[33],[35,17995,17996],{"xmlns":37},[39,17997,17998,18010],{},[42,17999,18000,18002,18004,18006,18008],{},[45,18001,15963],{},[89,18003,543],{},[45,18005,9904],{},[89,18007,565],{},[45,18009,381],{},[49,18011,18012],{"encoding":51},"h = H + N",[26,18014,18016,18034,18052],{"className":18015,"ariaHidden":56},[55],[26,18017,18019,18022,18025,18028,18031],{"className":18018},[60],[26,18020],{"className":18021,"style":7264},[64],[26,18023,15963],{"className":18024},[69,70],[26,18026],{"className":18027,"style":676},[123],[26,18029,543],{"className":18030},[680],[26,18032],{"className":18033,"style":676},[123],[26,18035,18037,18040,18043,18046,18049],{"className":18036},[60],[26,18038],{"className":18039,"style":8591},[64],[26,18041,9904],{"className":18042,"style":9943},[69,70],[26,18044],{"className":18045,"style":4156},[123],[26,18047,565],{"className":18048},[773],[26,18050],{"className":18051,"style":4156},[123],[26,18053,18055,18058],{"className":18054},[60],[26,18056],{"className":18057,"style":65},[64],[26,18059,381],{"className":18060,"style":476},[69,70],[26,18062,3643],{},[1926,18064,18065,18102,18138],{},[1929,18066,18067,18095,18096,18099,18100,151],{},[26,18068,18070,18083],{"className":18069},[29],[26,18071,18073],{"className":18072},[33],[35,18074,18075],{"xmlns":37},[39,18076,18077,18081],{},[42,18078,18079],{},[45,18080,15963],{},[49,18082,15963],{"encoding":51},[26,18084,18086],{"className":18085,"ariaHidden":56},[55],[26,18087,18089,18092],{"className":18088},[60],[26,18090],{"className":18091,"style":7264},[64],[26,18093,15963],{"className":18094},[69,70]," = ",[21,18097,18098],{},"Ellipsoidal Height"," (from GPS) ",[26,18101,6034],{},[1929,18103,18104,18095,18132,18135,18136,151],{},[26,18105,18107,18120],{"className":18106},[29],[26,18108,18110],{"className":18109},[33],[35,18111,18112],{"xmlns":37},[39,18113,18114,18118],{},[42,18115,18116],{},[45,18117,9904],{},[49,18119,9904],{"encoding":51},[26,18121,18123],{"className":18122,"ariaHidden":56},[55],[26,18124,18126,18129],{"className":18125},[60],[26,18127],{"className":18128,"style":65},[64],[26,18130,9904],{"className":18131,"style":9943},[69,70],[21,18133,18134],{},"Orthometric Height"," (Elevation above Mean Sea Level) ",[26,18137,6034],{},[1929,18139,18140,18095,18168,304,18171,7799,18174],{},[26,18141,18143,18156],{"className":18142},[29],[26,18144,18146],{"className":18145},[33],[35,18147,18148],{"xmlns":37},[39,18149,18150,18154],{},[42,18151,18152],{},[45,18153,381],{},[49,18155,381],{"encoding":51},[26,18157,18159],{"className":18158,"ariaHidden":56},[55],[26,18160,18162,18165],{"className":18161},[60],[26,18163],{"className":18164,"style":65},[64],[26,18166,381],{"className":18167,"style":476},[69,70],[21,18169,18170],{},"Geoid-Ellipsoid Separation",[26,18172,18173],{},"12, 30",[5028,18175,18177],{"type":18176},"warning",[18,18178,18179,18180,304,18217,18220,18221,18249,18250,18286,18287,151],{},"In the UK, the WGS84 ellipsoid lies about ",[26,18181,18183,18199],{"className":18182},[29],[26,18184,18186],{"className":18185},[33],[35,18187,18188],{"xmlns":37},[39,18189,18190,18196],{},[42,18191,18192,18194],{},[554,18193,12055],{},[5057,18195,6986],{},[49,18197,18198],{"encoding":51},"50 \\text{ m}",[26,18200,18202],{"className":18201,"ariaHidden":56},[55],[26,18203,18205,18208,18211],{"className":18204},[60],[26,18206],{"className":18207,"style":4270},[64],[26,18209,12055],{"className":18210},[69],[26,18212,18214],{"className":18213},[69,5118],[26,18215,6986],{"className":18216},[69],[21,18218,18219],{},"below"," the geoid. Failing to account for this ",[26,18222,18224,18237],{"className":18223},[29],[26,18225,18227],{"className":18226},[33],[35,18228,18229],{"xmlns":37},[39,18230,18231,18235],{},[42,18232,18233],{},[45,18234,381],{},[49,18236,381],{"encoding":51},[26,18238,18240],{"className":18239,"ariaHidden":56},[55],[26,18241,18243,18246],{"className":18242},[60],[26,18244],{"className":18245,"style":65},[64],[26,18247,381],{"className":18248,"style":476},[69,70]," value will result in elevation errors of ",[26,18251,18253,18268],{"className":18252},[29],[26,18254,18256],{"className":18255},[33],[35,18257,18258],{"xmlns":37},[39,18259,18260,18266],{},[42,18261,18262,18264],{},[554,18263,12055],{},[5057,18265,6986],{},[49,18267,18198],{"encoding":51},[26,18269,18271],{"className":18270,"ariaHidden":56},[55],[26,18272,18274,18277,18280],{"className":18273},[60],[26,18275],{"className":18276,"style":4270},[64],[26,18278,12055],{"className":18279},[69],[26,18281,18283],{"className":18282},[69,5118],[26,18284,6986],{"className":18285},[69],", even if your horizontal position is perfect ",[26,18288,5163],{},[13,18290,5301],{"id":5300},[1926,18292,18293,18356,18427],{},[1929,18294,18295,18298,18299,18352,18353,151],{},[21,18296,18297],{},"Uniform Scale:"," For small engineering sites (",[26,18300,18302,18321],{"className":18301},[29],[26,18303,18305],{"className":18304},[33],[35,18306,18307],{"xmlns":37},[39,18308,18309,18318],{},[42,18310,18311,18314,18316],{},[89,18312,18313],{},"\u003C",[554,18315,3761],{},[5057,18317,10540],{},[49,18319,18320],{"encoding":51},"\u003C 10 \\text{ km}",[26,18322,18324,18337],{"className":18323,"ariaHidden":56},[55],[26,18325,18327,18331,18334],{"className":18326},[60],[26,18328],{"className":18329,"style":18330},[64],"height:0.5782em;vertical-align:-0.0391em;",[26,18332,18313],{"className":18333},[680],[26,18335],{"className":18336,"style":676},[123],[26,18338,18340,18343,18346],{"className":18339},[60],[26,18341],{"className":18342,"style":7264},[64],[26,18344,3761],{"className":18345},[69],[26,18347,18349],{"className":18348},[69,5118],[26,18350,10540],{"className":18351},[69],"), you can often use a single scale factor for the whole site ",[26,18354,18355],{},"31",[1929,18357,18358,18361,18362,18390,18391,18419,18420,18423,18424,151],{},[21,18359,18360],{},"Coordinate Order:"," In mapping, the ",[26,18363,18365,18378],{"className":18364},[29],[26,18366,18368],{"className":18367},[33],[35,18369,18370],{"xmlns":37},[39,18371,18372,18376],{},[42,18373,18374],{},[45,18375,5906],{},[49,18377,5906],{"encoding":51},[26,18379,18381],{"className":18380,"ariaHidden":56},[55],[26,18382,18384,18387],{"className":18383},[60],[26,18385],{"className":18386,"style":65},[64],[26,18388,5906],{"className":18389,"style":5935},[69,70],"-axis is Easting and the ",[26,18392,18394,18407],{"className":18393},[29],[26,18395,18397],{"className":18396},[33],[35,18398,18399],{"xmlns":37},[39,18400,18401,18405],{},[42,18402,18403],{},[45,18404,5911],{},[49,18406,5911],{"encoding":51},[26,18408,18410],{"className":18409,"ariaHidden":56},[55],[26,18411,18413,18416],{"className":18412},[60],[26,18414],{"className":18415,"style":65},[64],[26,18417,5911],{"className":18418,"style":4156},[69,70],"-axis is Northing, which is the ",[21,18421,18422],{},"reverse"," of pure mathematical convention ",[26,18425,18426],{},"32",[1929,18428,18429,18460,18461,151],{},[21,18430,18431,18432,4551],{},"Check the Grid Convergence (",[26,18433,18435,18448],{"className":18434},[29],[26,18436,18438],{"className":18437},[33],[35,18439,18440],{"xmlns":37},[39,18441,18442,18446],{},[42,18443,18444],{},[45,18445,1885],{},[49,18447,2761],{"encoding":51},[26,18449,18451],{"className":18450,"ariaHidden":56},[55],[26,18452,18454,18457],{"className":18453},[60],[26,18455],{"className":18456,"style":2771},[64],[26,18458,1885],{"className":18459,"style":1922},[69,70]," Grid North and True North are only the same on the Central Meridian. Ensure your Total Station orientation accounts for this ",[26,18462,18463],{},"33, 34",[13,18465,5569],{"id":5568},[1926,18467,18468,18551],{},[1929,18469,18470,18473,18474,18547,18548,151],{},[21,18471,18472],{},"Assuming Flatness:"," Neglecting the Earth's radius (",[26,18475,18477,18502],{"className":18476},[29],[26,18478,18480],{"className":18479},[33],[35,18481,18482],{"xmlns":37},[39,18483,18484,18499],{},[42,18485,18486,18488,18490,18492,18494,18497],{},[45,18487,10978],{},[89,18489,10997],{},[554,18491,307],{},[89,18493,91],{"separator":56},[554,18495,18496],{},"380",[5057,18498,10540],{},[49,18500,18501],{"encoding":51},"R \\approx 6,380 \\text{ km}",[26,18503,18505,18523],{"className":18504,"ariaHidden":56},[55],[26,18506,18508,18511,18514,18517,18520],{"className":18507},[60],[26,18509],{"className":18510,"style":65},[64],[26,18512,10978],{"className":18513,"style":11134},[69,70],[26,18515],{"className":18516,"style":676},[123],[26,18518,10997],{"className":18519},[680],[26,18521],{"className":18522,"style":676},[123],[26,18524,18526,18529,18532,18535,18538,18541],{"className":18525},[60],[26,18527],{"className":18528,"style":1898},[64],[26,18530,307],{"className":18531},[69],[26,18533,91],{"className":18534},[119],[26,18536],{"className":18537,"style":124},[123],[26,18539,18496],{"className":18540},[69],[26,18542,18544],{"className":18543},[69,5118],[26,18545,10540],{"className":18546},[69],") in distance reductions ",[26,18549,18550],{},"35",[1929,18552,18553,18556,18557,151],{},[21,18554,18555],{},"Sign Errors in Transformations:"," Helmert transformation parameters often have specific sign conventions for rotations. Always test with a known point ",[26,18558,18559],{},"26",[13,18561,5694],{"id":5693},[5696,18563,18564,18572,18634],{},[5699,18565,18567],{"label":18566},"Is WGS84 the same as the GPS datum?",[18,18568,18569,18570,151],{},"Yes. WGS84 is the coordinate system used by the GPS satellites to describe their orbits and your position ",[26,18571,15829],{},[5699,18573,18575],{"label":18574},"What is a 'False Origin'?",[18,18576,18577,18578,18630,18631,151],{},"It is a constant value added to coordinates (e.g., ",[26,18579,18581,18603],{"className":18580},[29],[26,18582,18584],{"className":18583},[33],[35,18585,18586],{"xmlns":37},[39,18587,18588,18600],{},[42,18589,18590,18593,18595,18598],{},[554,18591,18592],{},"400",[89,18594,91],{"separator":56},[554,18596,18597],{},"000",[5057,18599,6986],{},[49,18601,18602],{"encoding":51},"400,000 \\text{ m}",[26,18604,18606],{"className":18605,"ariaHidden":56},[55],[26,18607,18609,18612,18615,18618,18621,18624],{"className":18608},[60],[26,18610],{"className":18611,"style":11440},[64],[26,18613,18592],{"className":18614},[69],[26,18616,91],{"className":18617},[119],[26,18619],{"className":18620,"style":124},[123],[26,18622,18597],{"className":18623},[69],[26,18625,18627],{"className":18626},[69,5118],[26,18628,6986],{"className":18629},[69]," Easting) to ensure that all coordinates in a specific region remain positive ",[26,18632,18633],{},"33",[5699,18635,18637],{"label":18636},"Can I use the same scale factor for $100 \\text{ km}$?",[18,18638,18639,18640,6802,18668,18705,18706,151],{},"No. On long route projects, you must change the scale factor every ",[26,18641,18643,18656],{"className":18642},[29],[26,18644,18646],{"className":18645},[33],[35,18647,18648],{"xmlns":37},[39,18649,18650,18654],{},[42,18651,18652],{},[554,18653,6471],{},[49,18655,6471],{"encoding":51},[26,18657,18659],{"className":18658,"ariaHidden":56},[55],[26,18660,18662,18665],{"className":18661},[60],[26,18663],{"className":18664,"style":4270},[64],[26,18666,6471],{"className":18667},[69],[26,18669,18671,18687],{"className":18670},[29],[26,18672,18674],{"className":18673},[33],[35,18675,18676],{"xmlns":37},[39,18677,18678,18684],{},[42,18679,18680,18682],{},[554,18681,3761],{},[5057,18683,10540],{},[49,18685,18686],{"encoding":51},"10 \\text{ km}",[26,18688,18690],{"className":18689,"ariaHidden":56},[55],[26,18691,18693,18696,18699],{"className":18692},[60],[26,18694],{"className":18695,"style":7264},[64],[26,18697,3761],{"className":18698},[69],[26,18700,18702],{"className":18701},[69,5118],[26,18703,10540],{"className":18704},[69]," to maintain engineering accuracy ",[26,18707,18355],{},[13,18709,5757],{"id":5756},[18,18711,18712],{},"The WGS84 system provides the global foundation for all modern surveying. By rigorously applying geodetic formulas and understanding the transition from ellipsoid to plane, the surveyor ensures that satellite-derived data remains precise and useful for local engineering.",[13,18714,5764],{"id":5763},[18,18716,5767,18717,6374],{},[2588,18718,5770],{},{"title":5781,"searchDepth":5782,"depth":5782,"links":18720},[18721,18722,18723,18724,18725,18730,18736,18737,18738,18739,18740,18741,18742],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":172,"depth":5782,"text":173},{"id":194,"depth":5782,"text":195},{"id":344,"depth":5782,"text":345,"children":18726},[18727,18728],{"id":15997,"depth":5791,"text":15998},{"id":16181,"depth":5791,"text":18729},"2. Cartesian (X,Y,Z)(X, Y, Z)(X,Y,Z) to Geodetic (ϕ,λ,h)(\\phi, \\lambda, h)(ϕ,λ,h)",{"id":17079,"depth":5782,"text":17080,"children":18731},[18732,18733,18734,18735],{"id":17085,"depth":5791,"text":17086},{"id":17095,"depth":5791,"text":17096},{"id":17109,"depth":5791,"text":17110},{"id":17205,"depth":5791,"text":17206},{"id":17248,"depth":5782,"text":17249},{"id":17981,"depth":5782,"text":17982},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Position","A deep dive into the WGS84 reference system, geodetic coordinates, and the formulas required to project a curved Earth onto a flat engineering site grid.","\u002Fimages\u002Fwgs84-projection.jpg",{"authorCredentials":9826,"commentCount":6400,"tags":18747,"relatedTools":18751,"relatedArticles":18755,"sources":18758},[15758,18748,18749,15726,18750],"Datums","Projections","Mapping Math",[18752,18753,18754],"GNSS","GIS","Coordinate Converter",[18756,18757],"rtk-gps-precision","scale-factors-msl-reduction",[18759],{"title":5828,"type":5829,"note":18760},"Primary source for WGS84 parameters and coordinate conversion formulas.","\u002Fblog\u002Fwgs84-datum-and-geodetic-projections","2024-04-10",22,{"title":15748,"description":18744},{"loc":18761},"blog\u002F22.wgs84-datum-and-geodetic-projections","-eZAK5wrCXVxa3OhFza9MC-QKXXR6CphREKPVz6zy_s",{"id":18769,"title":18770,"author":5850,"body":18771,"category":20291,"description":20292,"extension":5809,"image":20293,"meta":20294,"navigation":5839,"path":20307,"publishedAt":20308,"readTime":5842,"seo":20309,"sitemap":20310,"stem":20311,"__hash__":20312},"blog\u002Fblog\u002F21.rtk-gps-real-time-kinematic.md","Real-Time Kinematic (RTK): Instantaneous Precision in GPS Surveying",{"type":10,"value":18772,"toc":20270},[18773,18775,18786,18788,18800,18802,18816,18818,18835,18837,19565,19714,19716,19763,19767,19825,19909,19911,20112,20114,20182,20184,20200,20202,20259,20261,20264,20266],[13,18774,16],{"id":15},[18,18776,18777,18780,18781,18783,18784,151],{},[21,18778,18779],{},"Real-Time Kinematic (RTK)"," surveying is the pinnacle of satellite-based positioning for the site engineer. Unlike traditional GPS methods that require post-processing of data in an office, RTK provides relative positions instantaneously as the roving receiver occupies a point ",[26,18782,341],{},". By using a dedicated data link between a fixed base station and a mobile rover, the system resolves integer ambiguities \"on-the-fly,\" delivering centimetre-level accuracy in seconds ",[26,18785,142],{},[13,18787,155],{"id":154},[18,18789,18790,18791,18794,18795,18797,18798,151],{},"For the engineering surveyor, time is the most expensive variable. RTK eliminates the \"wait and see\" risk of post-processing. It allows for the immediate ",[21,18792,18793],{},"setting-out"," of design coordinates and the rapid capture of topographic detail ",[26,18796,588],{},". If a point is blocked or a measurement is suspect, the surveyor knows immediately, rather than discovering the error hours later at a desk ",[26,18799,341],{},[13,18801,173],{"id":172},[18,18803,18804,18805,304,18808,18810,18811,18813,18814,151],{},"The major hurdle in high-precision GPS is the time required for a significant change in satellite-receiver geometry to resolve the ",[21,18806,18807],{},"initial integer ambiguities",[26,18809,572],{},". RTK overcomes this by utilizing a base station at a known position to transmit both code and carrier phase data to the rover ",[26,18812,588],{},". This allows the rover's on-board processor to solve for coordinate differences relative to the base in real-time ",[26,18815,341],{},[13,18817,195],{"id":194},[18,18819,18820,18821,18824,18825,18828,18829,18832,18833,151],{},"RTK relies on ",[21,18822,18823],{},"Differential GPS (DGPS)"," principles, where errors common to both receivers (such as satellite clock bias, orbital errors, and atmospheric refraction) are cancelled out over short baselines ",[26,18826,18827],{},"4, 5",". The \"Real-Time\" component is achieved via ",[21,18830,18831],{},"mobile data communication"," (usually radio modems or GSM), which must have sufficient bandwidth to transmit complex carrier phase data every epoch ",[26,18834,341],{},[13,18836,345],{"id":344},[18,18838,18839,18840,18843,18844,304,19563],{},"The core of RTK is the ",[21,18841,18842],{},"Double Difference"," algorithm, which eliminates receiver and satellite clock offsets.\nThe pure phase observation equation is:\n",[26,18845,18847,18984],{"className":18846},[29],[26,18848,18850],{"className":18849},[33],[35,18851,18852],{"xmlns":37},[39,18853,18854,18981],{},[42,18855,18856,18859,18861,18868,18870,18872,18878,18880,18882,18888,18890,18896,18898,18900,18902,18905,18907,18913,18915,18921,18923,18925,18927,18933,18935,18941,18943,18946,18948,18950,18952,18966,18968],{},[45,18857,18858],{"mathvariant":2636},"Φ",[89,18860,366],{"stretchy":365},[368,18862,18863,18865],{},[45,18864,2090],{},[45,18866,18867],{},"r",[89,18869,386],{"stretchy":365},[89,18871,543],{},[545,18873,18874,18876],{},[45,18875,16128],{},[45,18877,595],{},[45,18879,10978],{},[89,18881,366],{"stretchy":365},[368,18883,18884,18886],{},[45,18885,2090],{},[45,18887,11169],{},[89,18889,91],{"separator":56},[368,18891,18892,18894],{},[45,18893,2090],{},[45,18895,18867],{},[89,18897,386],{"stretchy":365},[89,18899,1967],{},[45,18901,16128],{},[89,18903,18904],{"stretchy":365},"[",[45,18906,7782],{},[368,18908,18909,18911],{},[45,18910,2090],{},[45,18912,18867],{},[89,18914,366],{"stretchy":365},[368,18916,18917,18919],{},[45,18918,2090],{},[45,18920,18867],{},[89,18922,386],{"stretchy":365},[89,18924,1967],{},[45,18926,7782],{},[3513,18928,18929,18931],{},[45,18930,2090],{},[45,18932,11169],{},[89,18934,366],{"stretchy":365},[3513,18936,18937,18939],{},[45,18938,2090],{},[45,18940,11169],{},[89,18942,386],{"stretchy":365},[89,18944,18945],{"stretchy":365},"]",[89,18947,565],{},[45,18949,381],{},[89,18951,565],{},[368,18953,18954,18956],{},[45,18955,16484],{},[42,18957,18958,18961,18964],{},[45,18959,18960],{},"I",[45,18962,18963],{},"O",[45,18965,381],{},[89,18967,565],{},[368,18969,18970,18972],{},[45,18971,16484],{},[42,18973,18974,18977,18979],{},[45,18975,18976],{},"T",[45,18978,10978],{},[45,18980,18963],{},[49,18982,18983],{"encoding":51},"\\Phi(t_r) = \\frac{f}{c} R(t_s, t_r) - f[\\delta t_r(t_r) - \\delta t^s(t^s)] + N + e_{ION} + e_{TRO}",[26,18985,18987,19052,19230,19340,19424,19442,19507],{"className":18986,"ariaHidden":56},[55],[26,18988,18990,18993,18996,18999,19040,19043,19046,19049],{"className":18989},[60],[26,18991],{"className":18992,"style":3328},[64],[26,18994,18858],{"className":18995},[69],[26,18997,366],{"className":18998},[406],[26,19000,19002,19005],{"className":19001},[69],[26,19003,2090],{"className":19004},[69,70],[26,19006,19008],{"className":19007},[417],[26,19009,19011,19032],{"className":19010},[421,422],[26,19012,19014,19029],{"className":19013},[426],[26,19015,19017],{"className":19016,"style":431},[430],[26,19018,19020,19023],{"style":19019},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[26,19021],{"className":19022,"style":439},[438],[26,19024,19026],{"className":19025},[443,444,445,446],[26,19027,18867],{"className":19028,"style":7268},[69,70,446],[26,19030,454],{"className":19031},[453],[26,19033,19035],{"className":19034},[426],[26,19036,19038],{"className":19037,"style":2027},[430],[26,19039],{},[26,19041,386],{"className":19042},[515],[26,19044],{"className":19045,"style":676},[123],[26,19047,543],{"className":19048},[680],[26,19050],{"className":19051,"style":676},[123],[26,19053,19055,19058,19126,19129,19132,19172,19175,19178,19218,19221,19224,19227],{"className":19054},[60],[26,19056],{"className":19057,"style":11097},[64],[26,19059,19061,19064,19123],{"className":19060},[69],[26,19062],{"className":19063},[406,697],[26,19065,19067],{"className":19066},[545],[26,19068,19070,19115],{"className":19069},[421,422],[26,19071,19073,19112],{"className":19072},[426],[26,19074,19076,19090,19098],{"className":19075,"style":11116},[430],[26,19077,19078,19081],{"style":713},[26,19079],{"className":19080,"style":717},[438],[26,19082,19084],{"className":19083},[443,444,445,446],[26,19085,19087],{"className":19086},[69,446],[26,19088,595],{"className":19089},[69,70,446],[26,19091,19092,19095],{"style":859},[26,19093],{"className":19094,"style":717},[438],[26,19096],{"className":19097,"style":867},[866],[26,19099,19100,19103],{"style":11224},[26,19101],{"className":19102,"style":717},[438],[26,19104,19106],{"className":19105},[443,444,445,446],[26,19107,19109],{"className":19108},[69,446],[26,19110,16128],{"className":19111,"style":16147},[69,70,446],[26,19113,454],{"className":19114},[453],[26,19116,19118],{"className":19117},[426],[26,19119,19121],{"className":19120,"style":11249},[430],[26,19122],{},[26,19124],{"className":19125},[515,697],[26,19127,10978],{"className":19128,"style":11134},[69,70],[26,19130,366],{"className":19131},[406],[26,19133,19135,19138],{"className":19134},[69],[26,19136,2090],{"className":19137},[69,70],[26,19139,19141],{"className":19140},[417],[26,19142,19144,19164],{"className":19143},[421,422],[26,19145,19147,19161],{"className":19146},[426],[26,19148,19150],{"className":19149,"style":431},[430],[26,19151,19152,19155],{"style":19019},[26,19153],{"className":19154,"style":439},[438],[26,19156,19158],{"className":19157},[443,444,445,446],[26,19159,11169],{"className":19160},[69,70,446],[26,19162,454],{"className":19163},[453],[26,19165,19167],{"className":19166},[426],[26,19168,19170],{"className":19169,"style":2027},[430],[26,19171],{},[26,19173,91],{"className":19174},[119],[26,19176],{"className":19177,"style":124},[123],[26,19179,19181,19184],{"className":19180},[69],[26,19182,2090],{"className":19183},[69,70],[26,19185,19187],{"className":19186},[417],[26,19188,19190,19210],{"className":19189},[421,422],[26,19191,19193,19207],{"className":19192},[426],[26,19194,19196],{"className":19195,"style":431},[430],[26,19197,19198,19201],{"style":19019},[26,19199],{"className":19200,"style":439},[438],[26,19202,19204],{"className":19203},[443,444,445,446],[26,19205,18867],{"className":19206,"style":7268},[69,70,446],[26,19208,454],{"className":19209},[453],[26,19211,19213],{"className":19212},[426],[26,19214,19216],{"className":19215,"style":2027},[430],[26,19217],{},[26,19219,386],{"className":19220},[515],[26,19222],{"className":19223,"style":4156},[123],[26,19225,1967],{"className":19226},[773],[26,19228],{"className":19229,"style":4156},[123],[26,19231,19233,19236,19239,19242,19245,19285,19288,19328,19331,19334,19337],{"className":19232},[60],[26,19234],{"className":19235,"style":3328},[64],[26,19237,16128],{"className":19238,"style":16147},[69,70],[26,19240,18904],{"className":19241},[406],[26,19243,7782],{"className":19244,"style":7798},[69,70],[26,19246,19248,19251],{"className":19247},[69],[26,19249,2090],{"className":19250},[69,70],[26,19252,19254],{"className":19253},[417],[26,19255,19257,19277],{"className":19256},[421,422],[26,19258,19260,19274],{"className":19259},[426],[26,19261,19263],{"className":19262,"style":431},[430],[26,19264,19265,19268],{"style":19019},[26,19266],{"className":19267,"style":439},[438],[26,19269,19271],{"className":19270},[443,444,445,446],[26,19272,18867],{"className":19273,"style":7268},[69,70,446],[26,19275,454],{"className":19276},[453],[26,19278,19280],{"className":19279},[426],[26,19281,19283],{"className":19282,"style":2027},[430],[26,19284],{},[26,19286,366],{"className":19287},[406],[26,19289,19291,19294],{"className":19290},[69],[26,19292,2090],{"className":19293},[69,70],[26,19295,19297],{"className":19296},[417],[26,19298,19300,19320],{"className":19299},[421,422],[26,19301,19303,19317],{"className":19302},[426],[26,19304,19306],{"className":19305,"style":431},[430],[26,19307,19308,19311],{"style":19019},[26,19309],{"className":19310,"style":439},[438],[26,19312,19314],{"className":19313},[443,444,445,446],[26,19315,18867],{"className":19316,"style":7268},[69,70,446],[26,19318,454],{"className":19319},[453],[26,19321,19323],{"className":19322},[426],[26,19324,19326],{"className":19325,"style":2027},[430],[26,19327],{},[26,19329,386],{"className":19330},[515],[26,19332],{"className":19333,"style":4156},[123],[26,19335,1967],{"className":19336},[773],[26,19338],{"className":19339,"style":4156},[123],[26,19341,19343,19346,19349,19379,19382,19411,19415,19418,19421],{"className":19342},[60],[26,19344],{"className":19345,"style":3328},[64],[26,19347,7782],{"className":19348,"style":7798},[69,70],[26,19350,19352,19355],{"className":19351},[69],[26,19353,2090],{"className":19354},[69,70],[26,19356,19358],{"className":19357},[417],[26,19359,19361],{"className":19360},[421],[26,19362,19364],{"className":19363},[426],[26,19365,19368],{"className":19366,"style":19367},[430],"height:0.6644em;",[26,19369,19370,19373],{"style":3577},[26,19371],{"className":19372,"style":439},[438],[26,19374,19376],{"className":19375},[443,444,445,446],[26,19377,11169],{"className":19378},[69,70,446],[26,19380,366],{"className":19381},[406],[26,19383,19385,19388],{"className":19384},[69],[26,19386,2090],{"className":19387},[69,70],[26,19389,19391],{"className":19390},[417],[26,19392,19394],{"className":19393},[421],[26,19395,19397],{"className":19396},[426],[26,19398,19400],{"className":19399,"style":19367},[430],[26,19401,19402,19405],{"style":3577},[26,19403],{"className":19404,"style":439},[438],[26,19406,19408],{"className":19407},[443,444,445,446],[26,19409,11169],{"className":19410},[69,70,446],[26,19412,19414],{"className":19413},[515],")]",[26,19416],{"className":19417,"style":4156},[123],[26,19419,565],{"className":19420},[773],[26,19422],{"className":19423,"style":4156},[123],[26,19425,19427,19430,19433,19436,19439],{"className":19426},[60],[26,19428],{"className":19429,"style":8591},[64],[26,19431,381],{"className":19432,"style":476},[69,70],[26,19434],{"className":19435,"style":4156},[123],[26,19437,565],{"className":19438},[773],[26,19440],{"className":19441,"style":4156},[123],[26,19443,19445,19449,19498,19501,19504],{"className":19444},[60],[26,19446],{"className":19447,"style":19448},[64],"height:0.7333em;vertical-align:-0.15em;",[26,19450,19452,19455],{"className":19451},[69],[26,19453,16484],{"className":19454},[69,70],[26,19456,19458],{"className":19457},[417],[26,19459,19461,19490],{"className":19460},[421,422],[26,19462,19464,19487],{"className":19463},[426],[26,19465,19467],{"className":19466,"style":10121},[430],[26,19468,19469,19472],{"style":19019},[26,19470],{"className":19471,"style":439},[438],[26,19473,19475],{"className":19474},[443,444,445,446],[26,19476,19478,19481,19484],{"className":19477},[69,446],[26,19479,18960],{"className":19480,"style":5935},[69,70,446],[26,19482,18963],{"className":19483,"style":7268},[69,70,446],[26,19485,381],{"className":19486,"style":476},[69,70,446],[26,19488,454],{"className":19489},[453],[26,19491,19493],{"className":19492},[426],[26,19494,19496],{"className":19495,"style":2027},[430],[26,19497],{},[26,19499],{"className":19500,"style":4156},[123],[26,19502,565],{"className":19503},[773],[26,19505],{"className":19506,"style":4156},[123],[26,19508,19510,19514],{"className":19509},[60],[26,19511],{"className":19512,"style":19513},[64],"height:0.5806em;vertical-align:-0.15em;",[26,19515,19517,19520],{"className":19516},[69],[26,19518,16484],{"className":19519},[69,70],[26,19521,19523],{"className":19522},[417],[26,19524,19526,19555],{"className":19525},[421,422],[26,19527,19529,19552],{"className":19528},[426],[26,19530,19532],{"className":19531,"style":10121},[430],[26,19533,19534,19537],{"style":19019},[26,19535],{"className":19536,"style":439},[438],[26,19538,19540],{"className":19539},[443,444,445,446],[26,19541,19543,19546,19549],{"className":19542},[69,446],[26,19544,18976],{"className":19545,"style":71},[69,70,446],[26,19547,10978],{"className":19548,"style":11134},[69,70,446],[26,19550,18963],{"className":19551,"style":7268},[69,70,446],[26,19553,454],{"className":19554},[453],[26,19556,19558],{"className":19557},[426],[26,19559,19561],{"className":19560,"style":2027},[430],[26,19562],{},[26,19564,307],{},[18,19566,19567,19568,275,19596,19624,19625,275,19653,19681,19682,19710,19711,151],{},"By differencing observations between two receivers (",[26,19569,19571,19584],{"className":19570},[29],[26,19572,19574],{"className":19573},[33],[35,19575,19576],{"xmlns":37},[39,19577,19578,19582],{},[42,19579,19580],{},[45,19581,87],{},[49,19583,87],{"encoding":51},[26,19585,19587],{"className":19586,"ariaHidden":56},[55],[26,19588,19590,19593],{"className":19589},[60],[26,19591],{"className":19592,"style":65},[64],[26,19594,87],{"className":19595},[69,70],[26,19597,19599,19612],{"className":19598},[29],[26,19600,19602],{"className":19601},[33],[35,19603,19604],{"xmlns":37},[39,19605,19606,19610],{},[42,19607,19608],{},[45,19609,94],{},[49,19611,94],{"encoding":51},[26,19613,19615],{"className":19614,"ariaHidden":56},[55],[26,19616,19618,19621],{"className":19617},[60],[26,19619],{"className":19620,"style":65},[64],[26,19622,94],{"className":19623,"style":128},[69,70],") and two satellites (",[26,19626,19628,19641],{"className":19627},[29],[26,19629,19631],{"className":19630},[33],[35,19632,19633],{"xmlns":37},[39,19634,19635,19639],{},[42,19636,19637],{},[554,19638,341],{},[49,19640,341],{"encoding":51},[26,19642,19644],{"className":19643,"ariaHidden":56},[55],[26,19645,19647,19650],{"className":19646},[60],[26,19648],{"className":19649,"style":4270},[64],[26,19651,341],{"className":19652},[69],[26,19654,19656,19669],{"className":19655},[29],[26,19657,19659],{"className":19658},[33],[35,19660,19661],{"xmlns":37},[39,19662,19663,19667],{},[42,19664,19665],{},[554,19666,572],{},[49,19668,572],{"encoding":51},[26,19670,19672],{"className":19671,"ariaHidden":56},[55],[26,19673,19675,19678],{"className":19674},[60],[26,19676],{"className":19677,"style":4270},[64],[26,19679,572],{"className":19680},[69],"), the resultant equation removes the clock errors and leaves the baseline vector as the primary unknown, provided the integer ambiguity (",[26,19683,19685,19698],{"className":19684},[29],[26,19686,19688],{"className":19687},[33],[35,19689,19690],{"xmlns":37},[39,19691,19692,19696],{},[42,19693,19694],{},[45,19695,381],{},[49,19697,381],{"encoding":51},[26,19699,19701],{"className":19700,"ariaHidden":56},[55],[26,19702,19704,19707],{"className":19703},[60],[26,19705],{"className":19706,"style":65},[64],[26,19708,381],{"className":19709,"style":476},[69,70],") can be resolved ",[26,19712,19713],{},"4, 7",[13,19715,2825],{"id":2824},[2827,19717,19718,19722,19727,19731,19736,19740,19745,19749,19754,19758],{},[1738,19719,19721],{"id":19720},"base-station-setup","Base Station Setup",[18,19723,19724,19725,151],{},"Set up a high-quality dual-frequency receiver over a point with known coordinates. Connect the transmitter (radio modem) and ensure a clear line of sight for the antenna ",[26,19726,588],{},[1738,19728,19730],{"id":19729},"rover-initialization","Rover Initialization",[18,19732,19733,19734,151],{},"Power on the roving receiver. It must \"lock on\" to a minimum of five satellites to begin ambiguity resolution. The data link must be stable to receive the base station's corrections ",[26,19735,588],{},[1738,19737,19739],{"id":19738},"ambiguity-resolution-initialization","Ambiguity Resolution (Initialization)",[18,19741,19742,19743,151],{},"Wait for the system to achieve a \"Fixed\" solution. Modern \"On-the-Fly\" (OTF) systems can resolve these ambiguities while the rover is moving ",[26,19744,572],{},[1738,19746,19748],{"id":19747},"capture-or-set-out","Capture or Set-Out",[18,19750,19751,19752,151],{},"Once \"Fixed,\" the rover can capture topographic points or use graphical on-board software to guide the surveyor to design coordinates ",[26,19753,588],{},[1738,19755,19757],{"id":19756},"verification","Verification",[18,19759,19760,19761,151],{},"Periodically check the RTK solution by occupying a different known control point to ensure the ambiguity resolution remains correct ",[26,19762,588],{},[13,19764,19766],{"id":19765},"step-by-step-example-setting-out","Step-by-Step Example: Setting Out",[18,19768,19769,19771,19772,19824],{},[21,19770,3288],{}," Set out a building corner with design coordinates ",[26,19773,19775,19797],{"className":19774},[29],[26,19776,19778],{"className":19777},[33],[35,19779,19780],{"xmlns":37},[39,19781,19782,19794],{},[42,19783,19784,19786,19788,19790,19792],{},[89,19785,366],{"stretchy":365},[45,19787,372],{},[89,19789,91],{"separator":56},[45,19791,381],{},[89,19793,386],{"stretchy":365},[49,19795,19796],{"encoding":51},"(E, N)",[26,19798,19800],{"className":19799,"ariaHidden":56},[55],[26,19801,19803,19806,19809,19812,19815,19818,19821],{"className":19802},[60],[26,19804],{"className":19805,"style":3328},[64],[26,19807,366],{"className":19808},[406],[26,19810,372],{"className":19811,"style":413},[69,70],[26,19813,91],{"className":19814},[119],[26,19816],{"className":19817,"style":124},[123],[26,19819,381],{"className":19820,"style":476},[69,70],[26,19822,386],{"className":19823},[515]," using RTK.",[3763,19826,19827,19887,19895,19901],{},[1929,19828,19829,19832,19833,19884,19885,151],{},[21,19830,19831],{},"Input Coordinates:"," Key the design ",[26,19834,19836,19857],{"className":19835},[29],[26,19837,19839],{"className":19838},[33],[35,19840,19841],{"xmlns":37},[39,19842,19843,19855],{},[42,19844,19845,19847,19849,19851,19853],{},[89,19846,366],{"stretchy":365},[45,19848,372],{},[89,19850,91],{"separator":56},[45,19852,381],{},[89,19854,386],{"stretchy":365},[49,19856,19796],{"encoding":51},[26,19858,19860],{"className":19859,"ariaHidden":56},[55],[26,19861,19863,19866,19869,19872,19875,19878,19881],{"className":19862},[60],[26,19864],{"className":19865,"style":3328},[64],[26,19867,366],{"className":19868},[406],[26,19870,372],{"className":19871,"style":413},[69,70],[26,19873,91],{"className":19874},[119],[26,19876],{"className":19877,"style":124},[123],[26,19879,381],{"className":19880,"style":476},[69,70],[26,19882,386],{"className":19883},[515]," into the palm-sized processor ",[26,19886,588],{},[1929,19888,19889,19892,19893,151],{},[21,19890,19891],{},"Navigate:"," The graphical output shows the surveyor's current position as a cross and the target as a dot ",[26,19894,588],{},[1929,19896,19897,19900],{},[21,19898,19899],{},"Refine:"," Move the pole-antenna in the direction indicated by the software's arrows.",[1929,19902,19903,19906,19907,151],{},[21,19904,19905],{},"Finalize:"," When the cross and dot coincide, the pole is at the exact setting-out position. Record the \"as-built\" position for a quality check ",[26,19908,588],{},[13,19910,17982],{"id":17981},[18,19912,19913,19916,19917,20022,20023,20060,20061,304,20103,20105,20106,20109,20110,151],{},[21,19914,19915],{},"Accuracy of RTK:","\nThe system typically achieves a precision of:\n",[26,19918,19920,19953],{"className":19919},[29],[26,19921,19923],{"className":19922},[33],[35,19924,19925],{"xmlns":37},[39,19926,19927,19950],{},[42,19928,19929,19931,19933,19935,19937,19939,19941,19943,19945,19948],{},[89,19930,10350],{},[89,19932,366],{"stretchy":365},[554,19934,3761],{},[5057,19936,6122],{},[89,19938,565],{},[554,19940,341],{},[89,19942,1967],{},[554,19944,572],{},[5057,19946,19947],{}," ppm",[89,19949,386],{"stretchy":365},[49,19951,19952],{"encoding":51},"\\pm (10 \\text{ mm} + 1-2 \\text{ ppm})",[26,19954,19956,19986,20004],{"className":19955,"ariaHidden":56},[55],[26,19957,19959,19962,19965,19968,19971,19977,19980,19983],{"className":19958},[60],[26,19960],{"className":19961,"style":3328},[64],[26,19963,10350],{"className":19964},[69],[26,19966,366],{"className":19967},[406],[26,19969,3761],{"className":19970},[69],[26,19972,19974],{"className":19973},[69,5118],[26,19975,6122],{"className":19976},[69],[26,19978],{"className":19979,"style":4156},[123],[26,19981,565],{"className":19982},[773],[26,19984],{"className":19985,"style":4156},[123],[26,19987,19989,19992,19995,19998,20001],{"className":19988},[60],[26,19990],{"className":19991,"style":4702},[64],[26,19993,341],{"className":19994},[69],[26,19996],{"className":19997,"style":4156},[123],[26,19999,1967],{"className":20000},[773],[26,20002],{"className":20003,"style":4156},[123],[26,20005,20007,20010,20013,20019],{"className":20006},[60],[26,20008],{"className":20009,"style":3328},[64],[26,20011,572],{"className":20012},[69],[26,20014,20016],{"className":20015},[69,5118],[26,20017,19947],{"className":20018},[69],[26,20020,386],{"className":20021},[515],"\nThis means for a ",[26,20024,20026,20042],{"className":20025},[29],[26,20027,20029],{"className":20028},[33],[35,20030,20031],{"xmlns":37},[39,20032,20033,20039],{},[42,20034,20035,20037],{},[554,20036,341],{},[5057,20038,10540],{},[49,20040,20041],{"encoding":51},"1 \\text{ km}",[26,20043,20045],{"className":20044,"ariaHidden":56},[55],[26,20046,20048,20051,20054],{"className":20047},[60],[26,20049],{"className":20050,"style":7264},[64],[26,20052,341],{"className":20053},[69],[26,20055,20057],{"className":20056},[69,5118],[26,20058,10540],{"className":20059},[69]," baseline, the expected error is approximately ",[26,20062,20064,20082],{"className":20063},[29],[26,20065,20067],{"className":20066},[33],[35,20068,20069],{"xmlns":37},[39,20070,20071,20079],{},[42,20072,20073,20075,20077],{},[89,20074,10350],{},[554,20076,3643],{},[5057,20078,6122],{},[49,20080,20081],{"encoding":51},"\\pm 12 \\text{ mm}",[26,20083,20085],{"className":20084,"ariaHidden":56},[55],[26,20086,20088,20091,20094,20097],{"className":20087},[60],[26,20089],{"className":20090,"style":4702},[64],[26,20092,10350],{"className":20093},[69],[26,20095,3643],{"className":20096},[69],[26,20098,20100],{"className":20099},[69,5118],[26,20101,6122],{"className":20102},[69],[26,20104,14162],{},". Note that this is relative to the ",[21,20107,20108],{},"Base Station",", so any error in the base station's coordinates will shift the entire site survey ",[26,20111,5565],{},[13,20113,5301],{"id":5300},[1926,20115,20116,20124,20170],{},[1929,20117,20118,20121,20122,151],{},[21,20119,20120],{},"Loss of Lock:"," If you go under a tree or near a tall building, you may lose the \"Fixed\" status. You must remain static for a short time or return to an open area to regain the integer resolution ",[26,20123,588],{},[1929,20125,20126,20129,20130,20167,20168,151],{},[21,20127,20128],{},"Radio Range:"," The communication link is the \"weakest link.\" Use high-gain antennas if working over ",[26,20131,20133,20149],{"className":20132},[29],[26,20134,20136],{"className":20135},[33],[35,20137,20138],{"xmlns":37},[39,20139,20140,20146],{},[42,20141,20142,20144],{},[554,20143,6471],{},[5057,20145,10540],{},[49,20147,20148],{"encoding":51},"5 \\text{ km}",[26,20150,20152],{"className":20151,"ariaHidden":56},[55],[26,20153,20155,20158,20161],{"className":20154},[60],[26,20156],{"className":20157,"style":7264},[64],[26,20159,6471],{"className":20160},[69],[26,20162,20164],{"className":20163},[69,5118],[26,20165,10540],{"className":20166},[69]," or in undulating terrain ",[26,20169,341],{},[1929,20171,20172,20175,20176,20179,20180,151],{},[21,20173,20174],{},"GDOP Awareness:"," Avoid working when the ",[21,20177,20178],{},"Geometric Dilution of Precision (GDOP)"," is high (e.g., satellites in a straight line), as this degrades the 3D intersection accuracy ",[26,20181,3761],{},[13,20183,5569],{"id":5568},[1926,20185,20186,20192],{},[1929,20187,20188,20191],{},[21,20189,20190],{},"Ignoring Base Stability:"," If the base tripod is bumped or settles into soft ground, all subsequent RTK points will be wrong. Always sandbag the base tripod.",[1929,20193,20194,20197,20198,151],{},[21,20195,20196],{},"Mixing Datums:"," Ensure the base station is using the same local transformation parameters as the design plan (e.g., WGS84 to National Grid) ",[26,20199,3555],{},[13,20201,5694],{"id":5693},[5696,20203,20204,20212,20253],{},[5699,20205,20207],{"label":20206},"What is the difference between 'Fixed' and 'Float'?",[18,20208,20209,20210,151],{},"A \"Fixed\" solution means the integer ambiguities have been resolved to whole numbers, providing centimetre accuracy. A \"Float\" solution means the ambiguities are still being estimated as decimals, resulting in decimetre or metre-level error ",[26,20211,6471],{},[5699,20213,20215],{"label":20214},"Can I use RTK for vertical control?",[18,20216,20217,20218,20220,20221,20249,20250,151],{},"Yes, but GPS heights are ",[21,20219,15762],{},". You must apply a geoid-ellipsoid separation (",[26,20222,20224,20237],{"className":20223},[29],[26,20225,20227],{"className":20226},[33],[35,20228,20229],{"xmlns":37},[39,20230,20231,20235],{},[42,20232,20233],{},[45,20234,381],{},[49,20236,381],{"encoding":51},[26,20238,20240],{"className":20239,"ariaHidden":56},[55],[26,20241,20243,20246],{"className":20242},[60],[26,20244],{"className":20245,"style":65},[64],[26,20247,381],{"className":20248,"style":476},[69,70],") model to get orthometric heights (elevations) suitable for drainage ",[26,20251,20252],{},"12, 13",[5699,20254,20256],{"label":20255},"Does RTK work in tunnels?",[18,20257,20258],{},"No. GPS signals require a clear line of sight to the satellites. For underground work, use Weisbach Triangles or Gyro-theodolites .",[13,20260,5757],{"id":5756},[18,20262,20263],{},"RTK GPS has revolutionized site surveying by providing instantaneous, high-precision results. By mastering the data link and understanding the constraints of ambiguity resolution, the engineering surveyor can perform complex setting-out tasks with unprecedented speed and confidence.",[13,20265,5764],{"id":5763},[18,20267,5767,20268,6374],{},[2588,20269,5770],{},{"title":5781,"searchDepth":5782,"depth":5782,"links":20271},[20272,20273,20274,20275,20276,20277,20284,20285,20286,20287,20288,20289,20290],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":172,"depth":5782,"text":173},{"id":194,"depth":5782,"text":195},{"id":344,"depth":5782,"text":345},{"id":2824,"depth":5782,"text":2825,"children":20278},[20279,20280,20281,20282,20283],{"id":19720,"depth":5791,"text":19721},{"id":19729,"depth":5791,"text":19730},{"id":19738,"depth":5791,"text":19739},{"id":19747,"depth":5791,"text":19748},{"id":19756,"depth":5791,"text":19757},{"id":19765,"depth":5782,"text":19766},{"id":17981,"depth":5782,"text":17982},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Satellite Positioning","A technical guide to RTK GPS procedures, mobile data communication requirements, and real-time ambiguity resolution for engineering site work.","\u002Fimages\u002Frtk-gps.jpg",{"authorCredentials":6399,"authorBio":20295,"commentCount":6400,"tags":20296,"relatedTools":20298,"relatedArticles":20302,"sources":20304},"Specialist in satellite positioning and real-time dimensional control.",[10051,10433,20291,7103,20297],"Real-Time",[20299,20300,20301],"GNSS Receiver","Radio Modem","Data Controller",[15734,20303],"coordinate-transformations-helmert",[20305],{"title":5828,"type":5829,"note":20306},"Primary source for RTK theory and site application.","\u002Fblog\u002Frtk-gps-real-time-kinematic","2024-04-09",{"title":18770,"description":20292},{"loc":20307},"blog\u002F21.rtk-gps-real-time-kinematic","nhE0qeuACn3gtZvCGpmtnP7wbMLEkoZSq79iQeAB_pc",{"id":20314,"title":20315,"author":5850,"body":20316,"category":23603,"description":23604,"extension":5809,"image":23605,"meta":23606,"navigation":5839,"path":23619,"publishedAt":23620,"readTime":5842,"seo":23621,"sitemap":23622,"stem":23623,"__hash__":23624},"blog\u002Fblog\u002F20.stadia-tacheometry-optical-measurement.md","Stadia Tacheometry: Mastering Optical Distance Measurement",{"type":10,"value":20317,"toc":23581},[20318,20320,20330,20332,20337,20339,20378,20380,20384,20551,20764,20768,20802,21065,21067,21333,21335,21406,21807,22961,22963,23196,23198,23396,23398,23499,23501,23575,23577],[13,20319,16],{"id":15},[18,20321,20322,20323,20326,20327,151],{},"Before the widespread adoption of EDM, ",[21,20324,20325],{},"Stadia Tacheometry"," was the primary method for rapid topographic detailing. By observing the interval between two \"stadia hairs\" on a theodolite's reticule, a surveyor can mathematically derive both the horizontal distance to a staff and the difference in elevation ",[26,20328,20329],{},"18, 19",[13,20331,155],{"id":154},[18,20333,20334,20335,151],{},"While Total Stations are now standard, tacheometry remains a fundamental skill for understanding how optics relate to geometry. It is still used for \"rough-in\" surveys, verifying electronic distances, and in specialized optical instruments where electronics might fail ",[26,20336,6250],{},[13,20338,195],{"id":194},[18,20340,20341,20342,20345,20346,20375,20376,151],{},"Tacheometry is based on the properties of ",[21,20343,20344],{},"Similar Triangles",". In a fixed-hair telescope, the angle subtended by the stadia hairs is constant. As the staff moves further away, the \"Stadia Intercept\" (",[26,20347,20349,20363],{"className":20348},[29],[26,20350,20352],{"className":20351},[33],[35,20353,20354],{"xmlns":37},[39,20355,20356,20361],{},[42,20357,20358],{},[45,20359,20360],{},"S",[49,20362,20360],{"encoding":51},[26,20364,20366],{"className":20365,"ariaHidden":56},[55],[26,20367,20369,20372],{"className":20368},[60],[26,20370],{"className":20371,"style":65},[64],[26,20373,20360],{"className":20374,"style":413},[69,70],")—the distance between the top and bottom hair readings—increases proportionally ",[26,20377,5163],{},[13,20379,345],{"id":344},[1738,20381,20383],{"id":20382},"_1-the-basic-horizontal-formula","1. The Basic Horizontal Formula",[18,20385,20386,20387,20389,20390,11453],{},"For a horizontal line of sight ",[26,20388,20329],{},":\n",[26,20391,20393,20426],{"className":20392},[29],[26,20394,20396],{"className":20395},[33],[35,20397,20398],{"xmlns":37},[39,20399,20400,20423],{},[42,20401,20402,20405,20407,20413,20415,20417],{},[45,20403,20404],{},"D",[89,20406,543],{},[368,20408,20409,20411],{},[45,20410,323],{},[554,20412,341],{},[45,20414,20360],{},[89,20416,565],{},[368,20418,20419,20421],{},[45,20420,323],{},[554,20422,572],{},[49,20424,20425],{"encoding":51},"D = K_1 S + K_2",[26,20427,20429,20447,20505],{"className":20428,"ariaHidden":56},[55],[26,20430,20432,20435,20438,20441,20444],{"className":20431},[60],[26,20433],{"className":20434,"style":65},[64],[26,20436,20404],{"className":20437,"style":7268},[69,70],[26,20439],{"className":20440,"style":676},[123],[26,20442,543],{"className":20443},[680],[26,20445],{"className":20446,"style":676},[123],[26,20448,20450,20453,20493,20496,20499,20502],{"className":20449},[60],[26,20451],{"className":20452,"style":1986},[64],[26,20454,20456,20459],{"className":20455},[69],[26,20457,323],{"className":20458,"style":138},[69,70],[26,20460,20462],{"className":20461},[417],[26,20463,20465,20485],{"className":20464},[421,422],[26,20466,20468,20482],{"className":20467},[426],[26,20469,20471],{"className":20470,"style":2005},[430],[26,20472,20473,20476],{"style":2008},[26,20474],{"className":20475,"style":439},[438],[26,20477,20479],{"className":20478},[443,444,445,446],[26,20480,341],{"className":20481},[69,446],[26,20483,454],{"className":20484},[453],[26,20486,20488],{"className":20487},[426],[26,20489,20491],{"className":20490,"style":2027},[430],[26,20492],{},[26,20494,20360],{"className":20495,"style":413},[69,70],[26,20497],{"className":20498,"style":4156},[123],[26,20500,565],{"className":20501},[773],[26,20503],{"className":20504,"style":4156},[123],[26,20506,20508,20511],{"className":20507},[60],[26,20509],{"className":20510,"style":1986},[64],[26,20512,20514,20517],{"className":20513},[69],[26,20515,323],{"className":20516,"style":138},[69,70],[26,20518,20520],{"className":20519},[417],[26,20521,20523,20543],{"className":20522},[421,422],[26,20524,20526,20540],{"className":20525},[426],[26,20527,20529],{"className":20528,"style":2005},[430],[26,20530,20531,20534],{"style":2008},[26,20532],{"className":20533,"style":439},[438],[26,20535,20537],{"className":20536},[443,444,445,446],[26,20538,572],{"className":20539},[69,446],[26,20541,454],{"className":20542},[453],[26,20544,20546],{"className":20545},[426],[26,20547,20549],{"className":20548,"style":2027},[430],[26,20550],{},[1926,20552,20553,20584,20615,20689],{},[1929,20554,20555,20583],{},[26,20556,20558,20571],{"className":20557},[29],[26,20559,20561],{"className":20560},[33],[35,20562,20563],{"xmlns":37},[39,20564,20565,20569],{},[42,20566,20567],{},[45,20568,20404],{},[49,20570,20404],{"encoding":51},[26,20572,20574],{"className":20573,"ariaHidden":56},[55],[26,20575,20577,20580],{"className":20576},[60],[26,20578],{"className":20579,"style":65},[64],[26,20581,20404],{"className":20582,"style":7268},[69,70]," = Horizontal distance.",[1929,20585,20586,20614],{},[26,20587,20589,20602],{"className":20588},[29],[26,20590,20592],{"className":20591},[33],[35,20593,20594],{"xmlns":37},[39,20595,20596,20600],{},[42,20597,20598],{},[45,20599,20360],{},[49,20601,20360],{"encoding":51},[26,20603,20605],{"className":20604,"ariaHidden":56},[55],[26,20606,20608,20611],{"className":20607},[60],[26,20609],{"className":20610,"style":65},[64],[26,20612,20360],{"className":20613,"style":413},[69,70]," = Stadia intercept (Top reading - Bottom reading).",[1929,20616,20617,20686,20687,4476],{},[26,20618,20620,20637],{"className":20619},[29],[26,20621,20623],{"className":20622},[33],[35,20624,20625],{"xmlns":37},[39,20626,20627,20635],{},[42,20628,20629],{},[368,20630,20631,20633],{},[45,20632,323],{},[554,20634,341],{},[49,20636,3917],{"encoding":51},[26,20638,20640],{"className":20639,"ariaHidden":56},[55],[26,20641,20643,20646],{"className":20642},[60],[26,20644],{"className":20645,"style":1986},[64],[26,20647,20649,20652],{"className":20648},[69],[26,20650,323],{"className":20651,"style":138},[69,70],[26,20653,20655],{"className":20654},[417],[26,20656,20658,20678],{"className":20657},[421,422],[26,20659,20661,20675],{"className":20660},[426],[26,20662,20664],{"className":20663,"style":2005},[430],[26,20665,20666,20669],{"style":2008},[26,20667],{"className":20668,"style":439},[438],[26,20670,20672],{"className":20671},[443,444,445,446],[26,20673,341],{"className":20674},[69,446],[26,20676,454],{"className":20677},[453],[26,20679,20681],{"className":20680},[426],[26,20682,20684],{"className":20683,"style":2027},[430],[26,20685],{}," = Multiplying constant (usually ",[21,20688,6159],{},[1929,20690,20691,20760,20761,20763],{},[26,20692,20694,20711],{"className":20693},[29],[26,20695,20697],{"className":20696},[33],[35,20698,20699],{"xmlns":37},[39,20700,20701,20709],{},[42,20702,20703],{},[368,20704,20705,20707],{},[45,20706,323],{},[554,20708,572],{},[49,20710,4298],{"encoding":51},[26,20712,20714],{"className":20713,"ariaHidden":56},[55],[26,20715,20717,20720],{"className":20716},[60],[26,20718],{"className":20719,"style":1986},[64],[26,20721,20723,20726],{"className":20722},[69],[26,20724,323],{"className":20725,"style":138},[69,70],[26,20727,20729],{"className":20728},[417],[26,20730,20732,20752],{"className":20731},[421,422],[26,20733,20735,20749],{"className":20734},[426],[26,20736,20738],{"className":20737,"style":2005},[430],[26,20739,20740,20743],{"style":2008},[26,20741],{"className":20742,"style":439},[438],[26,20744,20746],{"className":20745},[443,444,445,446],[26,20747,572],{"className":20748},[69,446],[26,20750,454],{"className":20751},[453],[26,20753,20755],{"className":20754},[426],[26,20756,20758],{"className":20757,"style":2027},[430],[26,20759],{}," = Additive constant (usually ",[21,20762,3562],{}," in modern anallactic telescopes).",[1738,20765,20767],{"id":20766},"_2-the-inclined-sight-formula","2. The Inclined Sight Formula",[18,20769,20770,20771,20799,20800,520],{},"In engineering, sights are rarely horizontal. When the telescope is tilted at an angle ",[26,20772,20774,20787],{"className":20773},[29],[26,20775,20777],{"className":20776},[33],[35,20778,20779],{"xmlns":37},[39,20780,20781,20785],{},[42,20782,20783],{},[45,20784,7251],{},[49,20786,7254],{"encoding":51},[26,20788,20790],{"className":20789,"ariaHidden":56},[55],[26,20791,20793,20796],{"className":20792},[60],[26,20794],{"className":20795,"style":7264},[64],[26,20797,7251],{"className":20798,"style":7268},[69,70],", the formulas must account for the slant ",[26,20801,15992],{},[1926,20803,20804,20943],{},[1929,20805,20806,20837],{},[21,20807,20808,20809,4551],{},"Horizontal Distance (",[26,20810,20812,20825],{"className":20811},[29],[26,20813,20815],{"className":20814},[33],[35,20816,20817],{"xmlns":37},[39,20818,20819,20823],{},[42,20820,20821],{},[45,20822,20404],{},[49,20824,20404],{"encoding":51},[26,20826,20828],{"className":20827,"ariaHidden":56},[55],[26,20829,20831,20834],{"className":20830},[60],[26,20832],{"className":20833,"style":65},[64],[26,20835,20404],{"className":20836,"style":7268},[69,70],[26,20838,20840,20872],{"className":20839},[29],[26,20841,20843],{"className":20842},[33],[35,20844,20845],{"xmlns":37},[39,20846,20847,20869],{},[42,20848,20849,20851,20853,20855,20857,20867],{},[45,20850,20404],{},[89,20852,543],{},[554,20854,6159],{},[45,20856,20360],{},[3513,20858,20859,20865],{},[42,20860,20861,20863],{},[45,20862,7407],{},[89,20864,1962],{},[554,20866,572],{},[45,20868,7251],{},[49,20870,20871],{"encoding":51},"D = 100 S \\cos^2 \\theta",[26,20873,20875,20893],{"className":20874,"ariaHidden":56},[55],[26,20876,20878,20881,20884,20887,20890],{"className":20877},[60],[26,20879],{"className":20880,"style":65},[64],[26,20882,20404],{"className":20883,"style":7268},[69,70],[26,20885],{"className":20886,"style":676},[123],[26,20888,543],{"className":20889},[680],[26,20891],{"className":20892,"style":676},[123],[26,20894,20896,20899,20902,20905,20908,20937,20940],{"className":20895},[60],[26,20897],{"className":20898,"style":15274},[64],[26,20900,6159],{"className":20901},[69],[26,20903,20360],{"className":20904,"style":413},[69,70],[26,20906],{"className":20907,"style":124},[123],[26,20909,20911,20914],{"className":20910},[2079],[26,20912,7407],{"className":20913},[2079],[26,20915,20917],{"className":20916},[417],[26,20918,20920],{"className":20919},[421],[26,20921,20923],{"className":20922},[426],[26,20924,20926],{"className":20925,"style":15274},[430],[26,20927,20928,20931],{"style":3577},[26,20929],{"className":20930,"style":439},[438],[26,20932,20934],{"className":20933},[443,444,445,446],[26,20935,572],{"className":20936},[69,446],[26,20938],{"className":20939,"style":124},[123],[26,20941,7251],{"className":20942,"style":7268},[69,70],[1929,20944,20945,20981],{},[21,20946,20947,20948,4551],{},"Vertical Height (",[26,20949,20951,20966],{"className":20950},[29],[26,20952,20954],{"className":20953},[33],[35,20955,20956],{"xmlns":37},[39,20957,20958,20964],{},[42,20959,20960,20962],{},[45,20961,10006],{"mathvariant":2636},[45,20963,9904],{},[49,20965,10011],{"encoding":51},[26,20967,20969],{"className":20968,"ariaHidden":56},[55],[26,20970,20972,20975,20978],{"className":20971},[60],[26,20973],{"className":20974,"style":65},[64],[26,20976,10006],{"className":20977},[69],[26,20979,9904],{"className":20980,"style":9943},[69,70],[26,20982,20984,21014],{"className":20983},[29],[26,20985,20987],{"className":20986},[33],[35,20988,20989],{"xmlns":37},[39,20990,20991,21011],{},[42,20992,20993,20995,20997,20999,21001,21003,21005,21007,21009],{},[45,20994,10006],{"mathvariant":2636},[45,20996,9904],{},[89,20998,543],{},[554,21000,12055],{},[45,21002,20360],{},[45,21004,8523],{},[89,21006,1962],{},[554,21008,572],{},[45,21010,7251],{},[49,21012,21013],{"encoding":51},"\\Delta H = 50 S \\sin 2\\theta",[26,21015,21017,21038],{"className":21016,"ariaHidden":56},[55],[26,21018,21020,21023,21026,21029,21032,21035],{"className":21019},[60],[26,21021],{"className":21022,"style":65},[64],[26,21024,10006],{"className":21025},[69],[26,21027,9904],{"className":21028,"style":9943},[69,70],[26,21030],{"className":21031,"style":676},[123],[26,21033,543],{"className":21034},[680],[26,21036],{"className":21037,"style":676},[123],[26,21039,21041,21044,21047,21050,21053,21056,21059,21062],{"className":21040},[60],[26,21042],{"className":21043,"style":7264},[64],[26,21045,12055],{"className":21046},[69],[26,21048,20360],{"className":21049,"style":413},[69,70],[26,21051],{"className":21052,"style":124},[123],[26,21054,8523],{"className":21055},[2079],[26,21057],{"className":21058,"style":124},[123],[26,21060,572],{"className":21061},[69],[26,21063,7251],{"className":21064,"style":7268},[69,70],[13,21066,2825],{"id":2824},[2827,21068,21069,21073,21150,21154,21157,21161,21171,21175,21208,21212],{},[1738,21070,21072],{"id":21071},"instrument-setup","Instrument Setup",[18,21074,21075,21076,21147,21148,151],{},"Set up the theodolite over a known point. Measure the height of the instrument (",[26,21077,21079,21097],{"className":21078},[29],[26,21080,21082],{"className":21081},[33],[35,21083,21084],{"xmlns":37},[39,21085,21086,21094],{},[42,21087,21088],{},[368,21089,21090,21092],{},[45,21091,15963],{},[45,21093,11173],{},[49,21095,21096],{"encoding":51},"h_i",[26,21098,21100],{"className":21099,"ariaHidden":56},[55],[26,21101,21103,21106],{"className":21102},[60],[26,21104],{"className":21105,"style":7300},[64],[26,21107,21109,21112],{"className":21108},[69],[26,21110,15963],{"className":21111},[69,70],[26,21113,21115],{"className":21114},[417],[26,21116,21118,21139],{"className":21117},[421,422],[26,21119,21121,21136],{"className":21120},[426],[26,21122,21125],{"className":21123,"style":21124},[430],"height:0.3117em;",[26,21126,21127,21130],{"style":19019},[26,21128],{"className":21129,"style":439},[438],[26,21131,21133],{"className":21132},[443,444,445,446],[26,21134,11173],{"className":21135},[69,70,446],[26,21137,454],{"className":21138},[453],[26,21140,21142],{"className":21141},[426],[26,21143,21145],{"className":21144,"style":2027},[430],[26,21146],{},") above the peg ",[26,21149,15992],{},[1738,21151,21153],{"id":21152},"sight-the-staff","Sight the Staff",[18,21155,21156],{},"Direct the telescope to a staff held vertically at the target point.",[1738,21158,21160],{"id":21159},"record-three-hair-readings","Record Three Hair Readings",[18,21162,21163,21164,21167,21168,151],{},"Note the Top, Middle, and Bottom stadia hair readings.\n",[2588,21165,21166],{},"Check:"," The middle reading should be the exact mean of the top and bottom readings ",[26,21169,21170],{},"18, 22",[1738,21172,21174],{"id":21173},"measure-vertical-angle","Measure Vertical Angle",[18,21176,21177,21178,139,21206,151],{},"Note the vertical circle reading to determine the angle of inclination (",[26,21179,21181,21194],{"className":21180},[29],[26,21182,21184],{"className":21183},[33],[35,21185,21186],{"xmlns":37},[39,21187,21188,21192],{},[42,21189,21190],{},[45,21191,7251],{},[49,21193,7254],{"encoding":51},[26,21195,21197],{"className":21196,"ariaHidden":56},[55],[26,21198,21200,21203],{"className":21199},[60],[26,21201],{"className":21202,"style":7264},[64],[26,21204,7251],{"className":21205,"style":7268},[69,70],[26,21207,15992],{},[1738,21209,21211],{"id":21210},"reduction","Reduction",[18,21213,9361,21214,275,21285,21329,21330,151],{},[26,21215,21217,21241],{"className":21216},[29],[26,21218,21220],{"className":21219},[33],[35,21221,21222],{"xmlns":37},[39,21223,21224,21238],{},[42,21225,21226,21236],{},[3513,21227,21228,21234],{},[42,21229,21230,21232],{},[45,21231,7407],{},[89,21233,1962],{},[554,21235,572],{},[45,21237,7251],{},[49,21239,21240],{"encoding":51},"\\cos^2 \\theta",[26,21242,21244],{"className":21243,"ariaHidden":56},[55],[26,21245,21247,21250,21279,21282],{"className":21246},[60],[26,21248],{"className":21249,"style":15274},[64],[26,21251,21253,21256],{"className":21252},[2079],[26,21254,7407],{"className":21255},[2079],[26,21257,21259],{"className":21258},[417],[26,21260,21262],{"className":21261},[421],[26,21263,21265],{"className":21264},[426],[26,21266,21268],{"className":21267,"style":15274},[430],[26,21269,21270,21273],{"style":3577},[26,21271],{"className":21272,"style":439},[438],[26,21274,21276],{"className":21275},[443,444,445,446],[26,21277,572],{"className":21278},[69,446],[26,21280],{"className":21281,"style":124},[123],[26,21283,7251],{"className":21284,"style":7268},[69,70],[26,21286,21288,21308],{"className":21287},[29],[26,21289,21291],{"className":21290},[33],[35,21292,21293],{"xmlns":37},[39,21294,21295,21305],{},[42,21296,21297,21299,21301,21303],{},[45,21298,8523],{},[89,21300,1962],{},[554,21302,572],{},[45,21304,7251],{},[49,21306,21307],{"encoding":51},"\\sin 2\\theta",[26,21309,21311],{"className":21310,"ariaHidden":56},[55],[26,21312,21314,21317,21320,21323,21326],{"className":21313},[60],[26,21315],{"className":21316,"style":7264},[64],[26,21318,8523],{"className":21319},[2079],[26,21321],{"className":21322,"style":124},[123],[26,21324,572],{"className":21325},[69],[26,21327,7251],{"className":21328,"style":7268},[69,70]," formulas to calculate distance and level ",[26,21331,21332],{},"21, 22",[13,21334,3283],{"id":3282},[18,21336,21337,21339,21340,4476],{},[21,21338,7512],{}," Find the distance and level of point B from station A (",[26,21341,21343,21367],{"className":21342},[29],[26,21344,21346],{"className":21345},[33],[35,21347,21348],{"xmlns":37},[39,21349,21350,21364],{},[42,21351,21352,21354,21357,21359,21362],{},[45,21353,10978],{},[45,21355,21356],{},"L",[89,21358,543],{},[554,21360,21361],{},"100.00",[5057,21363,6986],{},[49,21365,21366],{"encoding":51},"RL = 100.00\\text{ m}",[26,21368,21370,21391],{"className":21369,"ariaHidden":56},[55],[26,21371,21373,21376,21379,21382,21385,21388],{"className":21372},[60],[26,21374],{"className":21375,"style":65},[64],[26,21377,10978],{"className":21378,"style":11134},[69,70],[26,21380,21356],{"className":21381},[69,70],[26,21383],{"className":21384,"style":676},[123],[26,21386,543],{"className":21387},[680],[26,21389],{"className":21390,"style":676},[123],[26,21392,21394,21397,21400],{"className":21393},[60],[26,21395],{"className":21396,"style":4270},[64],[26,21398,21361],{"className":21399},[69],[26,21401,21403],{"className":21402},[69,5118],[26,21404,6986],{"className":21405},[69],[1926,21407,21408,21511,21686],{},[1929,21409,21410,151],{},[26,21411,21413,21438],{"className":21412},[29],[26,21414,21416],{"className":21415},[33],[35,21417,21418],{"xmlns":37},[39,21419,21420,21435],{},[42,21421,21422,21428,21430,21433],{},[368,21423,21424,21426],{},[45,21425,15963],{},[45,21427,11173],{},[89,21429,543],{},[554,21431,21432],{},"1.50",[5057,21434,6986],{},[49,21436,21437],{"encoding":51},"h_i = 1.50\\text{ m}",[26,21439,21441,21496],{"className":21440,"ariaHidden":56},[55],[26,21442,21444,21447,21487,21490,21493],{"className":21443},[60],[26,21445],{"className":21446,"style":7300},[64],[26,21448,21450,21453],{"className":21449},[69],[26,21451,15963],{"className":21452},[69,70],[26,21454,21456],{"className":21455},[417],[26,21457,21459,21479],{"className":21458},[421,422],[26,21460,21462,21476],{"className":21461},[426],[26,21463,21465],{"className":21464,"style":21124},[430],[26,21466,21467,21470],{"style":19019},[26,21468],{"className":21469,"style":439},[438],[26,21471,21473],{"className":21472},[443,444,445,446],[26,21474,11173],{"className":21475},[69,70,446],[26,21477,454],{"className":21478},[453],[26,21480,21482],{"className":21481},[426],[26,21483,21485],{"className":21484,"style":2027},[430],[26,21486],{},[26,21488],{"className":21489,"style":676},[123],[26,21491,543],{"className":21492},[680],[26,21494],{"className":21495,"style":676},[123],[26,21497,21499,21502,21505],{"className":21498},[60],[26,21500],{"className":21501,"style":4270},[64],[26,21503,21432],{"className":21504},[69],[26,21506,21508],{"className":21507},[69,5118],[26,21509,6986],{"className":21510},[69],[1929,21512,21513,21514,151],{},"Vertical Angle ",[26,21515,21517,21557],{"className":21516},[29],[26,21518,21520],{"className":21519},[33],[35,21521,21522],{"xmlns":37},[39,21523,21524,21554],{},[42,21525,21526,21528,21530,21532,21538,21544],{},[45,21527,7251],{},[89,21529,543],{},[89,21531,565],{},[3513,21533,21534,21536],{},[554,21535,6471],{},[89,21537,3520],{},[3513,21539,21540,21542],{},[554,21541,13258],{},[89,21543,10678],{"mathvariant":2636,"lspace":12064,"rspace":12064},[3513,21545,21546,21548],{},[554,21547,13258],{},[42,21549,21550,21552],{},[89,21551,10678],{"mathvariant":2636},[89,21553,10678],{"mathvariant":2636},[49,21555,21556],{"encoding":51},"\\theta = +5^\\circ 00' 00''",[26,21558,21560,21578],{"className":21559,"ariaHidden":56},[55],[26,21561,21563,21566,21569,21572,21575],{"className":21562},[60],[26,21564],{"className":21565,"style":7264},[64],[26,21567,7251],{"className":21568,"style":7268},[69,70],[26,21570],{"className":21571,"style":676},[123],[26,21573,543],{"className":21574},[680],[26,21576],{"className":21577,"style":676},[123],[26,21579,21581,21584,21587,21616,21619,21651,21654],{"className":21580},[60],[26,21582],{"className":21583,"style":13320},[64],[26,21585,565],{"className":21586},[69],[26,21588,21590,21593],{"className":21589},[69],[26,21591,6471],{"className":21592},[69],[26,21594,21596],{"className":21595},[417],[26,21597,21599],{"className":21598},[421],[26,21600,21602],{"className":21601},[426],[26,21603,21605],{"className":21604,"style":3551},[430],[26,21606,21607,21610],{"style":3577},[26,21608],{"className":21609,"style":439},[438],[26,21611,21613],{"className":21612},[443,444,445,446],[26,21614,3520],{"className":21615},[773,446],[26,21617,3562],{"className":21618},[69],[26,21620,21622,21625],{"className":21621},[69],[26,21623,3562],{"className":21624},[69],[26,21626,21628],{"className":21627},[417],[26,21629,21631],{"className":21630},[421],[26,21632,21634],{"className":21633},[426],[26,21635,21637],{"className":21636,"style":10693},[430],[26,21638,21639,21642],{"style":3577},[26,21640],{"className":21641,"style":439},[438],[26,21643,21645],{"className":21644},[443,444,445,446],[26,21646,21648],{"className":21647},[69,446],[26,21649,10678],{"className":21650},[69,446],[26,21652,3562],{"className":21653},[69],[26,21655,21657,21660],{"className":21656},[69],[26,21658,3562],{"className":21659},[69],[26,21661,21663],{"className":21662},[417],[26,21664,21666],{"className":21665},[421],[26,21667,21669],{"className":21668},[426],[26,21670,21672],{"className":21671,"style":10693},[430],[26,21673,21674,21677],{"style":3577},[26,21675],{"className":21676,"style":439},[438],[26,21678,21680],{"className":21679},[443,444,445,446],[26,21681,21683],{"className":21682},[69,446],[26,21684,10726],{"className":21685},[69,446],[1929,21687,21688,21689,21727,21728,21766,21767,304,21805,151],{},"Stadia Readings: Top = ",[26,21690,21692,21709],{"className":21691},[29],[26,21693,21695],{"className":21694},[33],[35,21696,21697],{"xmlns":37},[39,21698,21699,21706],{},[42,21700,21701,21704],{},[554,21702,21703],{},"2.412",[5057,21705,6986],{},[49,21707,21708],{"encoding":51},"2.412\\text{ m}",[26,21710,21712],{"className":21711,"ariaHidden":56},[55],[26,21713,21715,21718,21721],{"className":21714},[60],[26,21716],{"className":21717,"style":4270},[64],[26,21719,21703],{"className":21720},[69],[26,21722,21724],{"className":21723},[69,5118],[26,21725,6986],{"className":21726},[69],", Mid = ",[26,21729,21731,21748],{"className":21730},[29],[26,21732,21734],{"className":21733},[33],[35,21735,21736],{"xmlns":37},[39,21737,21738,21745],{},[42,21739,21740,21743],{},[554,21741,21742],{},"1.926",[5057,21744,6986],{},[49,21746,21747],{"encoding":51},"1.926\\text{ m}",[26,21749,21751],{"className":21750,"ariaHidden":56},[55],[26,21752,21754,21757,21760],{"className":21753},[60],[26,21755],{"className":21756,"style":4270},[64],[26,21758,21742],{"className":21759},[69],[26,21761,21763],{"className":21762},[69,5118],[26,21764,6986],{"className":21765},[69],", Bottom = ",[26,21768,21770,21787],{"className":21769},[29],[26,21771,21773],{"className":21772},[33],[35,21774,21775],{"xmlns":37},[39,21776,21777,21784],{},[42,21778,21779,21782],{},[554,21780,21781],{},"1.440",[5057,21783,6986],{},[49,21785,21786],{"encoding":51},"1.440\\text{ m}",[26,21788,21790],{"className":21789,"ariaHidden":56},[55],[26,21791,21793,21796,21799],{"className":21792},[60],[26,21794],{"className":21795,"style":4270},[64],[26,21797,21781],{"className":21798},[69],[26,21800,21802],{"className":21801},[69,5118],[26,21803,6986],{"className":21804},[69],[26,21806,8924],{},[3763,21808,21809,21946,22230,22501],{},[1929,21810,21811,21842],{},[21,21812,21813,21814,4551],{},"Calculate Intercept (",[26,21815,21817,21830],{"className":21816},[29],[26,21818,21820],{"className":21819},[33],[35,21821,21822],{"xmlns":37},[39,21823,21824,21828],{},[42,21825,21826],{},[45,21827,20360],{},[49,21829,20360],{"encoding":51},[26,21831,21833],{"className":21832,"ariaHidden":56},[55],[26,21834,21836,21839],{"className":21835},[60],[26,21837],{"className":21838,"style":65},[64],[26,21840,20360],{"className":21841,"style":413},[69,70],[26,21843,21845,21874],{"className":21844},[29],[26,21846,21848],{"className":21847},[33],[35,21849,21850],{"xmlns":37},[39,21851,21852,21871],{},[42,21853,21854,21856,21858,21860,21862,21864,21866,21869],{},[45,21855,20360],{},[89,21857,543],{},[554,21859,21703],{},[89,21861,1967],{},[554,21863,21781],{},[89,21865,543],{},[554,21867,21868],{},"0.972",[5057,21870,6986],{},[49,21872,21873],{"encoding":51},"S = 2.412 - 1.440 = 0.972\\text{ m}",[26,21875,21877,21895,21913,21931],{"className":21876,"ariaHidden":56},[55],[26,21878,21880,21883,21886,21889,21892],{"className":21879},[60],[26,21881],{"className":21882,"style":65},[64],[26,21884,20360],{"className":21885,"style":413},[69,70],[26,21887],{"className":21888,"style":676},[123],[26,21890,543],{"className":21891},[680],[26,21893],{"className":21894,"style":676},[123],[26,21896,21898,21901,21904,21907,21910],{"className":21897},[60],[26,21899],{"className":21900,"style":4702},[64],[26,21902,21703],{"className":21903},[69],[26,21905],{"className":21906,"style":4156},[123],[26,21908,1967],{"className":21909},[773],[26,21911],{"className":21912,"style":4156},[123],[26,21914,21916,21919,21922,21925,21928],{"className":21915},[60],[26,21917],{"className":21918,"style":4270},[64],[26,21920,21781],{"className":21921},[69],[26,21923],{"className":21924,"style":676},[123],[26,21926,543],{"className":21927},[680],[26,21929],{"className":21930,"style":676},[123],[26,21932,21934,21937,21940],{"className":21933},[60],[26,21935],{"className":21936,"style":4270},[64],[26,21938,21868],{"className":21939},[69],[26,21941,21943],{"className":21942},[69,5118],[26,21944,6986],{"className":21945},[69],[1929,21947,21948,21979],{},[21,21949,21950,21951,4551],{},"Calculate Horizontal Distance (",[26,21952,21954,21967],{"className":21953},[29],[26,21955,21957],{"className":21956},[33],[35,21958,21959],{"xmlns":37},[39,21960,21961,21965],{},[42,21962,21963],{},[45,21964,20404],{},[49,21966,20404],{"encoding":51},[26,21968,21970],{"className":21969,"ariaHidden":56},[55],[26,21971,21973,21976],{"className":21972},[60],[26,21974],{"className":21975,"style":65},[64],[26,21977,20404],{"className":21978,"style":7268},[69,70],[26,21980,21982,22043],{"className":21981},[29],[26,21983,21985],{"className":21984},[33],[35,21986,21987],{"xmlns":37},[39,21988,21989,22040],{},[42,21990,21991,21993,21995,21997,21999,22001,22003,22013,22015,22021,22023,22025,22028,22030,22033,22035,22038],{},[45,21992,20404],{},[89,21994,543],{},[554,21996,6159],{},[89,21998,4578],{},[554,22000,21868],{},[89,22002,4578],{},[3513,22004,22005,22011],{},[42,22006,22007,22009],{},[45,22008,7407],{},[89,22010,1962],{},[554,22012,572],{},[89,22014,366],{"stretchy":365},[3513,22016,22017,22019],{},[554,22018,6471],{},[89,22020,3520],{},[89,22022,386],{"stretchy":365},[89,22024,543],{},[554,22026,22027],{},"97.2",[89,22029,4578],{},[554,22031,22032],{},"0.9924",[89,22034,543],{},[554,22036,22037],{},"96.46",[5057,22039,6986],{},[49,22041,22042],{"encoding":51},"D = 100 \\times 0.972 \\times \\cos^2(5^\\circ) = 97.2 \\times 0.9924 = 96.46\\text{ m}",[26,22044,22046,22064,22082,22100,22179,22197,22215],{"className":22045,"ariaHidden":56},[55],[26,22047,22049,22052,22055,22058,22061],{"className":22048},[60],[26,22050],{"className":22051,"style":65},[64],[26,22053,20404],{"className":22054,"style":7268},[69,70],[26,22056],{"className":22057,"style":676},[123],[26,22059,543],{"className":22060},[680],[26,22062],{"className":22063,"style":676},[123],[26,22065,22067,22070,22073,22076,22079],{"className":22066},[60],[26,22068],{"className":22069,"style":4702},[64],[26,22071,6159],{"className":22072},[69],[26,22074],{"className":22075,"style":4156},[123],[26,22077,4578],{"className":22078},[773],[26,22080],{"className":22081,"style":4156},[123],[26,22083,22085,22088,22091,22094,22097],{"className":22084},[60],[26,22086],{"className":22087,"style":4702},[64],[26,22089,21868],{"className":22090},[69],[26,22092],{"className":22093,"style":4156},[123],[26,22095,4578],{"className":22096},[773],[26,22098],{"className":22099,"style":4156},[123],[26,22101,22103,22106,22135,22138,22167,22170,22173,22176],{"className":22102},[60],[26,22104],{"className":22105,"style":15246},[64],[26,22107,22109,22112],{"className":22108},[2079],[26,22110,7407],{"className":22111},[2079],[26,22113,22115],{"className":22114},[417],[26,22116,22118],{"className":22117},[421],[26,22119,22121],{"className":22120},[426],[26,22122,22124],{"className":22123,"style":15274},[430],[26,22125,22126,22129],{"style":3577},[26,22127],{"className":22128,"style":439},[438],[26,22130,22132],{"className":22131},[443,444,445,446],[26,22133,572],{"className":22134},[69,446],[26,22136,366],{"className":22137},[406],[26,22139,22141,22144],{"className":22140},[69],[26,22142,6471],{"className":22143},[69],[26,22145,22147],{"className":22146},[417],[26,22148,22150],{"className":22149},[421],[26,22151,22153],{"className":22152},[426],[26,22154,22156],{"className":22155,"style":3551},[430],[26,22157,22158,22161],{"style":3577},[26,22159],{"className":22160,"style":439},[438],[26,22162,22164],{"className":22163},[443,444,445,446],[26,22165,3520],{"className":22166},[773,446],[26,22168,386],{"className":22169},[515],[26,22171],{"className":22172,"style":676},[123],[26,22174,543],{"className":22175},[680],[26,22177],{"className":22178,"style":676},[123],[26,22180,22182,22185,22188,22191,22194],{"className":22181},[60],[26,22183],{"className":22184,"style":4702},[64],[26,22186,22027],{"className":22187},[69],[26,22189],{"className":22190,"style":4156},[123],[26,22192,4578],{"className":22193},[773],[26,22195],{"className":22196,"style":4156},[123],[26,22198,22200,22203,22206,22209,22212],{"className":22199},[60],[26,22201],{"className":22202,"style":4270},[64],[26,22204,22032],{"className":22205},[69],[26,22207],{"className":22208,"style":676},[123],[26,22210,543],{"className":22211},[680],[26,22213],{"className":22214,"style":676},[123],[26,22216,22218,22221,22224],{"className":22217},[60],[26,22219],{"className":22220,"style":4270},[64],[26,22222,22037],{"className":22223},[69],[26,22225,22227],{"className":22226},[69,5118],[26,22228,6986],{"className":22229},[69],[1929,22231,22232,22268],{},[21,22233,22234,22235,4551],{},"Calculate Vertical Component (",[26,22236,22238,22253],{"className":22237},[29],[26,22239,22241],{"className":22240},[33],[35,22242,22243],{"xmlns":37},[39,22244,22245,22251],{},[42,22246,22247,22249],{},[45,22248,10006],{"mathvariant":2636},[45,22250,9904],{},[49,22252,10011],{"encoding":51},[26,22254,22256],{"className":22255,"ariaHidden":56},[55],[26,22257,22259,22262,22265],{"className":22258},[60],[26,22260],{"className":22261,"style":65},[64],[26,22263,10006],{"className":22264},[69],[26,22266,9904],{"className":22267,"style":9943},[69,70],[26,22269,22271,22332],{"className":22270},[29],[26,22272,22274],{"className":22273},[33],[35,22275,22276],{"xmlns":37},[39,22277,22278,22329],{},[42,22279,22280,22282,22284,22286,22288,22290,22292,22294,22296,22298,22300,22306,22308,22310,22312,22314,22320,22322,22324,22327],{},[45,22281,10006],{"mathvariant":2636},[45,22283,9904],{},[89,22285,543],{},[554,22287,6159],{},[89,22289,4578],{},[554,22291,21868],{},[89,22293,4578],{},[45,22295,7407],{},[89,22297,1962],{},[89,22299,366],{"stretchy":365},[3513,22301,22302,22304],{},[554,22303,6471],{},[89,22305,3520],{},[89,22307,386],{"stretchy":365},[45,22309,8523],{},[89,22311,1962],{},[89,22313,366],{"stretchy":365},[3513,22315,22316,22318],{},[554,22317,6471],{},[89,22319,3520],{},[89,22321,386],{"stretchy":365},[89,22323,543],{},[554,22325,22326],{},"8.47",[5057,22328,6986],{},[49,22330,22331],{"encoding":51},"\\Delta H = 100 \\times 0.972 \\times \\cos(5^\\circ) \\sin(5^\\circ) = 8.47\\text{ m}",[26,22333,22335,22356,22374,22392,22486],{"className":22334,"ariaHidden":56},[55],[26,22336,22338,22341,22344,22347,22350,22353],{"className":22337},[60],[26,22339],{"className":22340,"style":65},[64],[26,22342,10006],{"className":22343},[69],[26,22345,9904],{"className":22346,"style":9943},[69,70],[26,22348],{"className":22349,"style":676},[123],[26,22351,543],{"className":22352},[680],[26,22354],{"className":22355,"style":676},[123],[26,22357,22359,22362,22365,22368,22371],{"className":22358},[60],[26,22360],{"className":22361,"style":4702},[64],[26,22363,6159],{"className":22364},[69],[26,22366],{"className":22367,"style":4156},[123],[26,22369,4578],{"className":22370},[773],[26,22372],{"className":22373,"style":4156},[123],[26,22375,22377,22380,22383,22386,22389],{"className":22376},[60],[26,22378],{"className":22379,"style":4702},[64],[26,22381,21868],{"className":22382},[69],[26,22384],{"className":22385,"style":4156},[123],[26,22387,4578],{"className":22388},[773],[26,22390],{"className":22391,"style":4156},[123],[26,22393,22395,22398,22401,22404,22433,22436,22439,22442,22445,22474,22477,22480,22483],{"className":22394},[60],[26,22396],{"className":22397,"style":3328},[64],[26,22399,7407],{"className":22400},[2079],[26,22402,366],{"className":22403},[406],[26,22405,22407,22410],{"className":22406},[69],[26,22408,6471],{"className":22409},[69],[26,22411,22413],{"className":22412},[417],[26,22414,22416],{"className":22415},[421],[26,22417,22419],{"className":22418},[426],[26,22420,22422],{"className":22421,"style":3551},[430],[26,22423,22424,22427],{"style":3577},[26,22425],{"className":22426,"style":439},[438],[26,22428,22430],{"className":22429},[443,444,445,446],[26,22431,3520],{"className":22432},[773,446],[26,22434,386],{"className":22435},[515],[26,22437],{"className":22438,"style":124},[123],[26,22440,8523],{"className":22441},[2079],[26,22443,366],{"className":22444},[406],[26,22446,22448,22451],{"className":22447},[69],[26,22449,6471],{"className":22450},[69],[26,22452,22454],{"className":22453},[417],[26,22455,22457],{"className":22456},[421],[26,22458,22460],{"className":22459},[426],[26,22461,22463],{"className":22462,"style":3551},[430],[26,22464,22465,22468],{"style":3577},[26,22466],{"className":22467,"style":439},[438],[26,22469,22471],{"className":22470},[443,444,445,446],[26,22472,3520],{"className":22473},[773,446],[26,22475,386],{"className":22476},[515],[26,22478],{"className":22479,"style":676},[123],[26,22481,543],{"className":22482},[680],[26,22484],{"className":22485,"style":676},[123],[26,22487,22489,22492,22495],{"className":22488},[60],[26,22490],{"className":22491,"style":4270},[64],[26,22493,22326],{"className":22494},[69],[26,22496,22498],{"className":22497},[69,5118],[26,22499,6986],{"className":22500},[69],[1929,22502,22503,22506,22765,304,22959,151],{},[21,22504,22505],{},"Final RL of B:",[26,22507,22509,22558],{"className":22508},[29],[26,22510,22512],{"className":22511},[33],[35,22513,22514],{"xmlns":37},[39,22515,22516,22555],{},[42,22517,22518,22520,22526,22528,22530,22536,22538,22544,22546,22548,22550,22552],{},[45,22519,10978],{},[368,22521,22522,22524],{},[45,22523,21356],{},[45,22525,94],{},[89,22527,543],{},[45,22529,10978],{},[368,22531,22532,22534],{},[45,22533,21356],{},[45,22535,87],{},[89,22537,565],{},[368,22539,22540,22542],{},[45,22541,15963],{},[45,22543,11173],{},[89,22545,565],{},[45,22547,10006],{"mathvariant":2636},[45,22549,9904],{},[89,22551,1967],{},[5057,22553,22554],{},"Mid Reading",[49,22556,22557],{"encoding":51},"RL_B = RL_A + h_i + \\Delta H - \\text{Mid Reading}",[26,22559,22561,22619,22677,22732,22753],{"className":22560,"ariaHidden":56},[55],[26,22562,22564,22567,22570,22610,22613,22616],{"className":22563},[60],[26,22565],{"className":22566,"style":1986},[64],[26,22568,10978],{"className":22569,"style":11134},[69,70],[26,22571,22573,22576],{"className":22572},[69],[26,22574,21356],{"className":22575},[69,70],[26,22577,22579],{"className":22578},[417],[26,22580,22582,22602],{"className":22581},[421,422],[26,22583,22585,22599],{"className":22584},[426],[26,22586,22588],{"className":22587,"style":10121},[430],[26,22589,22590,22593],{"style":19019},[26,22591],{"className":22592,"style":439},[438],[26,22594,22596],{"className":22595},[443,444,445,446],[26,22597,94],{"className":22598,"style":128},[69,70,446],[26,22600,454],{"className":22601},[453],[26,22603,22605],{"className":22604},[426],[26,22606,22608],{"className":22607,"style":2027},[430],[26,22609],{},[26,22611],{"className":22612,"style":676},[123],[26,22614,543],{"className":22615},[680],[26,22617],{"className":22618,"style":676},[123],[26,22620,22622,22625,22628,22668,22671,22674],{"className":22621},[60],[26,22623],{"className":22624,"style":1986},[64],[26,22626,10978],{"className":22627,"style":11134},[69,70],[26,22629,22631,22634],{"className":22630},[69],[26,22632,21356],{"className":22633},[69,70],[26,22635,22637],{"className":22636},[417],[26,22638,22640,22660],{"className":22639},[421,422],[26,22641,22643,22657],{"className":22642},[426],[26,22644,22646],{"className":22645,"style":10121},[430],[26,22647,22648,22651],{"style":19019},[26,22649],{"className":22650,"style":439},[438],[26,22652,22654],{"className":22653},[443,444,445,446],[26,22655,87],{"className":22656},[69,70,446],[26,22658,454],{"className":22659},[453],[26,22661,22663],{"className":22662},[426],[26,22664,22666],{"className":22665,"style":2027},[430],[26,22667],{},[26,22669],{"className":22670,"style":4156},[123],[26,22672,565],{"className":22673},[773],[26,22675],{"className":22676,"style":4156},[123],[26,22678,22680,22683,22723,22726,22729],{"className":22679},[60],[26,22681],{"className":22682,"style":7300},[64],[26,22684,22686,22689],{"className":22685},[69],[26,22687,15963],{"className":22688},[69,70],[26,22690,22692],{"className":22691},[417],[26,22693,22695,22715],{"className":22694},[421,422],[26,22696,22698,22712],{"className":22697},[426],[26,22699,22701],{"className":22700,"style":21124},[430],[26,22702,22703,22706],{"style":19019},[26,22704],{"className":22705,"style":439},[438],[26,22707,22709],{"className":22708},[443,444,445,446],[26,22710,11173],{"className":22711},[69,70,446],[26,22713,454],{"className":22714},[453],[26,22716,22718],{"className":22717},[426],[26,22719,22721],{"className":22720,"style":2027},[430],[26,22722],{},[26,22724],{"className":22725,"style":4156},[123],[26,22727,565],{"className":22728},[773],[26,22730],{"className":22731,"style":4156},[123],[26,22733,22735,22738,22741,22744,22747,22750],{"className":22734},[60],[26,22736],{"className":22737,"style":8591},[64],[26,22739,10006],{"className":22740},[69],[26,22742,9904],{"className":22743,"style":9943},[69,70],[26,22745],{"className":22746,"style":4156},[123],[26,22748,1967],{"className":22749},[773],[26,22751],{"className":22752,"style":4156},[123],[26,22754,22756,22759],{"className":22755},[60],[26,22757],{"className":22758,"style":1898},[64],[26,22760,22762],{"className":22761},[69,5118],[26,22763,22554],{"className":22764},[69],[26,22766,22768,22811],{"className":22767},[29],[26,22769,22771],{"className":22770},[33],[35,22772,22773],{"xmlns":37},[39,22774,22775,22808],{},[42,22776,22777,22779,22785,22787,22789,22791,22793,22795,22797,22799,22801,22803,22806],{},[45,22778,10978],{},[368,22780,22781,22783],{},[45,22782,21356],{},[45,22784,94],{},[89,22786,543],{},[554,22788,21361],{},[89,22790,565],{},[554,22792,21432],{},[89,22794,565],{},[554,22796,22326],{},[89,22798,1967],{},[554,22800,21742],{},[89,22802,543],{},[554,22804,22805],{},"108.044",[5057,22807,6986],{},[49,22809,22810],{"encoding":51},"RL_B = 100.00 + 1.50 + 8.47 - 1.926 = 108.044\\text{ m}",[26,22812,22814,22872,22890,22908,22926,22944],{"className":22813,"ariaHidden":56},[55],[26,22815,22817,22820,22823,22863,22866,22869],{"className":22816},[60],[26,22818],{"className":22819,"style":1986},[64],[26,22821,10978],{"className":22822,"style":11134},[69,70],[26,22824,22826,22829],{"className":22825},[69],[26,22827,21356],{"className":22828},[69,70],[26,22830,22832],{"className":22831},[417],[26,22833,22835,22855],{"className":22834},[421,422],[26,22836,22838,22852],{"className":22837},[426],[26,22839,22841],{"className":22840,"style":10121},[430],[26,22842,22843,22846],{"style":19019},[26,22844],{"className":22845,"style":439},[438],[26,22847,22849],{"className":22848},[443,444,445,446],[26,22850,94],{"className":22851,"style":128},[69,70,446],[26,22853,454],{"className":22854},[453],[26,22856,22858],{"className":22857},[426],[26,22859,22861],{"className":22860,"style":2027},[430],[26,22862],{},[26,22864],{"className":22865,"style":676},[123],[26,22867,543],{"className":22868},[680],[26,22870],{"className":22871,"style":676},[123],[26,22873,22875,22878,22881,22884,22887],{"className":22874},[60],[26,22876],{"className":22877,"style":4702},[64],[26,22879,21361],{"className":22880},[69],[26,22882],{"className":22883,"style":4156},[123],[26,22885,565],{"className":22886},[773],[26,22888],{"className":22889,"style":4156},[123],[26,22891,22893,22896,22899,22902,22905],{"className":22892},[60],[26,22894],{"className":22895,"style":4702},[64],[26,22897,21432],{"className":22898},[69],[26,22900],{"className":22901,"style":4156},[123],[26,22903,565],{"className":22904},[773],[26,22906],{"className":22907,"style":4156},[123],[26,22909,22911,22914,22917,22920,22923],{"className":22910},[60],[26,22912],{"className":22913,"style":4702},[64],[26,22915,22326],{"className":22916},[69],[26,22918],{"className":22919,"style":4156},[123],[26,22921,1967],{"className":22922},[773],[26,22924],{"className":22925,"style":4156},[123],[26,22927,22929,22932,22935,22938,22941],{"className":22928},[60],[26,22930],{"className":22931,"style":4270},[64],[26,22933,21742],{"className":22934},[69],[26,22936],{"className":22937,"style":676},[123],[26,22939,543],{"className":22940},[680],[26,22942],{"className":22943,"style":676},[123],[26,22945,22947,22950,22953],{"className":22946},[60],[26,22948],{"className":22949,"style":4270},[64],[26,22951,22805],{"className":22952},[69],[26,22954,22956],{"className":22955},[69,5118],[26,22957,6986],{"className":22958},[69],[26,22960,21332],{},[13,22962,5301],{"id":5300},[1926,22964,22965,23083],{},[1929,22966,22967,22970,22971,23040,23041,23079,23080,151],{},[21,22968,22969],{},"Constant Checks:"," In modern telescopes, the additive constant (",[26,22972,22974,22991],{"className":22973},[29],[26,22975,22977],{"className":22976},[33],[35,22978,22979],{"xmlns":37},[39,22980,22981,22989],{},[42,22982,22983],{},[368,22984,22985,22987],{},[45,22986,323],{},[554,22988,572],{},[49,22990,4298],{"encoding":51},[26,22992,22994],{"className":22993,"ariaHidden":56},[55],[26,22995,22997,23000],{"className":22996},[60],[26,22998],{"className":22999,"style":1986},[64],[26,23001,23003,23006],{"className":23002},[69],[26,23004,323],{"className":23005,"style":138},[69,70],[26,23007,23009],{"className":23008},[417],[26,23010,23012,23032],{"className":23011},[421,422],[26,23013,23015,23029],{"className":23014},[426],[26,23016,23018],{"className":23017,"style":2005},[430],[26,23019,23020,23023],{"style":2008},[26,23021],{"className":23022,"style":439},[438],[26,23024,23026],{"className":23025},[443,444,445,446],[26,23027,572],{"className":23028},[69,446],[26,23030,454],{"className":23031},[453],[26,23033,23035],{"className":23034},[426],[26,23036,23038],{"className":23037,"style":2027},[430],[26,23039],{},") is zero because of an internal \"anallactic lens.\" Always check your instrument specifications to confirm if you need to add a value (typically ",[26,23042,23044,23061],{"className":23043},[29],[26,23045,23047],{"className":23046},[33],[35,23048,23049],{"xmlns":37},[39,23050,23051,23058],{},[42,23052,23053,23056],{},[554,23054,23055],{},"0.3",[5057,23057,6986],{},[49,23059,23060],{"encoding":51},"0.3\\text{ m}",[26,23062,23064],{"className":23063,"ariaHidden":56},[55],[26,23065,23067,23070,23073],{"className":23066},[60],[26,23068],{"className":23069,"style":4270},[64],[26,23071,23055],{"className":23072},[69],[26,23074,23076],{"className":23075},[69,5118],[26,23077,6986],{"className":23078},[69]," for older external-focusing telescopes) ",[26,23081,23082],{},"19, 21",[1929,23084,23085,23088,23089,23126,23127,23155,23156,304,23194,151],{},[21,23086,23087],{},"The 1% Rule:"," A handy field check: at ",[26,23090,23092,23108],{"className":23091},[29],[26,23093,23095],{"className":23094},[33],[35,23096,23097],{"xmlns":37},[39,23098,23099,23105],{},[42,23100,23101,23103],{},[554,23102,6159],{},[5057,23104,6986],{},[49,23106,23107],{"encoding":51},"100\\text{ m}",[26,23109,23111],{"className":23110,"ariaHidden":56},[55],[26,23112,23114,23117,23120],{"className":23113},[60],[26,23115],{"className":23116,"style":4270},[64],[26,23118,6159],{"className":23119},[69],[26,23121,23123],{"className":23122},[69,5118],[26,23124,6986],{"className":23125},[69],", the stadia intercept (",[26,23128,23130,23143],{"className":23129},[29],[26,23131,23133],{"className":23132},[33],[35,23134,23135],{"xmlns":37},[39,23136,23137,23141],{},[42,23138,23139],{},[45,23140,20360],{},[49,23142,20360],{"encoding":51},[26,23144,23146],{"className":23145,"ariaHidden":56},[55],[26,23147,23149,23152],{"className":23148},[60],[26,23150],{"className":23151,"style":65},[64],[26,23153,20360],{"className":23154,"style":413},[69,70],") should be exactly ",[26,23157,23159,23176],{"className":23158},[29],[26,23160,23162],{"className":23161},[33],[35,23163,23164],{"xmlns":37},[39,23165,23166,23173],{},[42,23167,23168,23171],{},[554,23169,23170],{},"1.000",[5057,23172,6986],{},[49,23174,23175],{"encoding":51},"1.000\\text{ m}",[26,23177,23179],{"className":23178,"ariaHidden":56},[55],[26,23180,23182,23185,23188],{"className":23181},[60],[26,23183],{"className":23184,"style":4270},[64],[26,23186,23170],{"className":23187},[69],[26,23189,23191],{"className":23190},[69,5118],[26,23192,6986],{"className":23193},[69],[26,23195,5163],{},[13,23197,5569],{"id":5568},[1926,23199,23200,23237],{},[1929,23201,23202,23205,23206,23234,23235,151],{},[21,23203,23204],{},"Staff Not Vertical:"," If the staff leans, the intercept (",[26,23207,23209,23222],{"className":23208},[29],[26,23210,23212],{"className":23211},[33],[35,23213,23214],{"xmlns":37},[39,23215,23216,23220],{},[42,23217,23218],{},[45,23219,20360],{},[49,23221,20360],{"encoding":51},[26,23223,23225],{"className":23224,"ariaHidden":56},[55],[26,23226,23228,23231],{"className":23227},[60],[26,23229],{"className":23230,"style":65},[64],[26,23232,20360],{"className":23233,"style":413},[69,70],") will be too large, resulting in a distance that is too long. Use a staff level bubble ",[26,23236,16451],{},[1929,23238,23239,23242,23243,23271,23272,23331,23332,23393,23394,151],{},[21,23240,23241],{},"Confusing Zenith with Vertical Angle:"," Surveying formulas use the angle from the horizontal (",[26,23244,23246,23259],{"className":23245},[29],[26,23247,23249],{"className":23248},[33],[35,23250,23251],{"xmlns":37},[39,23252,23253,23257],{},[42,23254,23255],{},[45,23256,7251],{},[49,23258,7254],{"encoding":51},[26,23260,23262],{"className":23261,"ariaHidden":56},[55],[26,23263,23265,23268],{"className":23264},[60],[26,23266],{"className":23267,"style":7264},[64],[26,23269,7251],{"className":23270,"style":7268},[69,70],"). If your instrument reads Zenith (",[26,23273,23275,23293],{"className":23274},[29],[26,23276,23278],{"className":23277},[33],[35,23279,23280],{"xmlns":37},[39,23281,23282,23290],{},[42,23283,23284],{},[3513,23285,23286,23288],{},[554,23287,3562],{},[89,23289,3520],{},[49,23291,23292],{"encoding":51},"0^\\circ",[26,23294,23296],{"className":23295,"ariaHidden":56},[55],[26,23297,23299,23302],{"className":23298},[60],[26,23300],{"className":23301,"style":3551},[64],[26,23303,23305,23308],{"className":23304},[69],[26,23306,3562],{"className":23307},[69],[26,23309,23311],{"className":23310},[417],[26,23312,23314],{"className":23313},[421],[26,23315,23317],{"className":23316},[426],[26,23318,23320],{"className":23319,"style":3551},[430],[26,23321,23322,23325],{"style":3577},[26,23323],{"className":23324,"style":439},[438],[26,23326,23328],{"className":23327},[443,444,445,446],[26,23329,3520],{"className":23330},[773,446]," at the top), you must subtract it from ",[26,23333,23335,23352],{"className":23334},[29],[26,23336,23338],{"className":23337},[33],[35,23339,23340],{"xmlns":37},[39,23341,23342,23350],{},[42,23343,23344],{},[3513,23345,23346,23348],{},[554,23347,7178],{},[89,23349,3520],{},[49,23351,7183],{"encoding":51},[26,23353,23355],{"className":23354,"ariaHidden":56},[55],[26,23356,23358,23361,23364],{"className":23357},[60],[26,23359],{"className":23360,"style":3551},[64],[26,23362,5565],{"className":23363},[69],[26,23365,23367,23370],{"className":23366},[69],[26,23368,3562],{"className":23369},[69],[26,23371,23373],{"className":23372},[417],[26,23374,23376],{"className":23375},[421],[26,23377,23379],{"className":23378},[426],[26,23380,23382],{"className":23381,"style":3551},[430],[26,23383,23384,23387],{"style":3577},[26,23385],{"className":23386,"style":439},[438],[26,23388,23390],{"className":23389},[443,444,445,446],[26,23391,3520],{"className":23392},[773,446]," before using the tacheometry formulas ",[26,23395,17092],{},[13,23397,5694],{"id":5693},[5696,23399,23400,23483,23491],{},[5699,23401,23403],{"label":23402},"What is the maximum range of tacheometry?",[18,23404,23405,23406,5193,23442,23480,23481,151],{},"For accurate results, sights should be limited to ",[26,23407,23409,23424],{"className":23408},[29],[26,23410,23412],{"className":23411},[33],[35,23413,23414],{"xmlns":37},[39,23415,23416,23422],{},[42,23417,23418,23420],{},[554,23419,6159],{},[5057,23421,6986],{},[49,23423,23107],{"encoding":51},[26,23425,23427],{"className":23426,"ariaHidden":56},[55],[26,23428,23430,23433,23436],{"className":23429},[60],[26,23431],{"className":23432,"style":4270},[64],[26,23434,6159],{"className":23435},[69],[26,23437,23439],{"className":23438},[69,5118],[26,23440,6986],{"className":23441},[69],[26,23443,23445,23462],{"className":23444},[29],[26,23446,23448],{"className":23447},[33],[35,23449,23450],{"xmlns":37},[39,23451,23452,23459],{},[42,23453,23454,23457],{},[554,23455,23456],{},"150",[5057,23458,6986],{},[49,23460,23461],{"encoding":51},"150\\text{ m}",[26,23463,23465],{"className":23464,"ariaHidden":56},[55],[26,23466,23468,23471,23474],{"className":23467},[60],[26,23469],{"className":23470,"style":4270},[64],[26,23472,23456],{"className":23473},[69],[26,23475,23477],{"className":23476},[69,5118],[26,23478,6986],{"className":23479},[69],". Beyond this, the staff graduations become difficult to read precisely ",[26,23482,6250],{},[5699,23484,23486],{"label":23485},"Why are there three hair readings?",[18,23487,23488,23489,151],{},"The top and bottom hairs give the distance. The middle hair gives the height. Taking all three serves as a vital arithmetical check in the field book ",[26,23490,5163],{},[5699,23492,23494],{"label":23493},"What is 'Stadia'?",[18,23495,23496,23497,151],{},"The term \"Stadia\" refers to the fixed horizontal cross-hairs etched onto the reticule of the telescope ",[26,23498,5163],{},[13,23500,5757],{"id":5756},[18,23502,23503,23504,23574],{},"Stadia tacheometry is a brilliant application of optical geometry. By mastering the ",[26,23505,23507,23530],{"className":23506},[29],[26,23508,23510],{"className":23509},[33],[35,23511,23512],{"xmlns":37},[39,23513,23514,23528],{},[42,23515,23516,23526],{},[3513,23517,23518,23524],{},[42,23519,23520,23522],{},[45,23521,7407],{},[89,23523,1962],{},[554,23525,572],{},[45,23527,7251],{},[49,23529,21240],{"encoding":51},[26,23531,23533],{"className":23532,"ariaHidden":56},[55],[26,23534,23536,23539,23568,23571],{"className":23535},[60],[26,23537],{"className":23538,"style":15274},[64],[26,23540,23542,23545],{"className":23541},[2079],[26,23543,7407],{"className":23544},[2079],[26,23546,23548],{"className":23547},[417],[26,23549,23551],{"className":23550},[421],[26,23552,23554],{"className":23553},[426],[26,23555,23557],{"className":23556,"style":15274},[430],[26,23558,23559,23562],{"style":3577},[26,23560],{"className":23561,"style":439},[438],[26,23563,23565],{"className":23564},[443,444,445,446],[26,23566,572],{"className":23567},[69,446],[26,23569],{"className":23570,"style":124},[123],[26,23572,7251],{"className":23573,"style":7268},[69,70]," reduction, you gain a deep understanding of the relationship between angular measurement and spatial distance—the core of all surveying science.",[13,23576,5764],{"id":5763},[18,23578,5767,23579,6374],{},[2588,23580,5770],{},{"title":5781,"searchDepth":5782,"depth":5782,"links":23582},[23583,23584,23585,23586,23590,23597,23598,23599,23600,23601,23602],{"id":15,"depth":5782,"text":16},{"id":154,"depth":5782,"text":155},{"id":194,"depth":5782,"text":195},{"id":344,"depth":5782,"text":345,"children":23587},[23588,23589],{"id":20382,"depth":5791,"text":20383},{"id":20766,"depth":5791,"text":20767},{"id":2824,"depth":5782,"text":2825,"children":23591},[23592,23593,23594,23595,23596],{"id":21071,"depth":5791,"text":21072},{"id":21152,"depth":5791,"text":21153},{"id":21159,"depth":5791,"text":21160},{"id":21173,"depth":5791,"text":21174},{"id":21210,"depth":5791,"text":21211},{"id":3282,"depth":5782,"text":3283},{"id":5300,"depth":5782,"text":5301},{"id":5568,"depth":5782,"text":5569},{"id":5693,"depth":5782,"text":5694},{"id":5756,"depth":5782,"text":5757},{"id":5763,"depth":5782,"text":5764},"Vertical Control","A technical exploration of tacheometry theory and formulas for determining distances and elevations using a theodolite and stadia hair readings.","\u002Fimages\u002Ftacheometry.jpg",{"authorCredentials":7107,"commentCount":6400,"tags":23607,"relatedTools":23611,"relatedArticles":23613,"sources":23616},[23608,23609,23610,15731,9830],"Tacheometry","Stadia","Optical Distance",[15731,23612],"Levelling Staff",[23614,23615],"combination-of-errors-propagation","precise-levelling-techniques",[23617],{"title":5828,"type":5829,"note":23618},"Primary source for the mathematical derivation of stadia formulas.","\u002Fblog\u002Fstadia-tacheometry-optical-measurement","2024-04-08",{"title":20315,"description":23604},{"loc":23619},"blog\u002F20.stadia-tacheometry-optical-measurement","EgjddQZKVvHFUA0Q-Dm7bKCNOU3bhHQudD9bCusvEjc",28,1783851942723]