[{"data":1,"prerenderedAt":4417},["ShallowReactive",2],{"blog-\u002Fblog\u002Fresection-three-point-problem":3,"blog-surround-\u002Fblog\u002Fresection-three-point-problem":4406},{"id":4,"title":5,"author":6,"body":7,"category":4376,"description":4377,"extension":4378,"image":4379,"meta":4380,"navigation":4398,"path":4399,"publishedAt":4400,"readTime":4401,"seo":4402,"sitemap":4403,"stem":4404,"__hash__":4405},"blog\u002Fblog\u002F10.resection-three-point-problem.md","Resection: Fixing Position via the Three-Point Problem","Surveyiiing Expert",{"type":8,"value":9,"toc":4352},"minimark",[10,15,151,155,162,166,229,233,238,497,1641,1645,2097,2101,2577,2582,3416,3420,3876,3880,4110,4114,4236,4240,4331,4335,4338,4342],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,22,23,70,71,137,138,142,143,146,147,150],"p",{},[19,20,21],"strong",{},"Resection"," is a method of fixing the coordinates of an unknown station (",[24,25,28,50],"span",{"className":26},[27],"katex",[24,29,32],{"className":30},[31],"katex-mathml",[33,34,36],"math",{"xmlns":35},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[37,38,39,46],"semantics",{},[40,41,42],"mrow",{},[43,44,45],"mi",{},"P",[47,48,45],"annotation",{"encoding":49},"application\u002Fx-tex",[24,51,55],{"className":52,"ariaHidden":54},[53],"katex-html","true",[24,56,59,64],{"className":57},[58],"base",[24,60],{"className":61,"style":63},[62],"strut","height:0.6833em;",[24,65,45],{"className":66,"style":69},[67,68],"mord","mathnormal","margin-right:0.1389em;",") by observing horizontal angles to at least three known control points (",[24,72,74,101],{"className":73},[27],[24,75,77],{"className":76},[31],[33,78,79],{"xmlns":35},[37,80,81,98],{},[40,82,83,86,90,93,95],{},[43,84,85],{},"A",[87,88,89],"mo",{"separator":54},",",[43,91,92],{},"B",[87,94,89],{"separator":54},[43,96,97],{},"C",[47,99,100],{"encoding":49},"A, B, C",[24,102,104],{"className":103,"ariaHidden":54},[53],[24,105,107,111,114,118,123,127,130,133],{"className":106},[58],[24,108],{"className":109,"style":110},[62],"height:0.8778em;vertical-align:-0.1944em;",[24,112,85],{"className":113},[67,68],[24,115,89],{"className":116},[117],"mpunct",[24,119],{"className":120,"style":122},[121],"mspace","margin-right:0.1667em;",[24,124,92],{"className":125,"style":126},[67,68],"margin-right:0.0502em;",[24,128,89],{"className":129},[117],[24,131],{"className":132,"style":122},[121],[24,134,97],{"className":135,"style":136},[67,68],"margin-right:0.0715em;","). Unlike intersection, where you sight ",[139,140,141],"em",{},"toward"," the new point from known stations, resection allows you to occupy the unknown point and observe ",[139,144,145],{},"outward"," ",[24,148,149],{},"27, 57",".",[11,152,154],{"id":153},"why-this-matters","Why This Matters",[16,156,157,158,161],{},"Resection is the ultimate \"set out\" tool. If you need to set out a building but your control stations are blocked by piles of dirt or machinery, you can set up anywhere with a clear view of three existing points and fix your position in minutes ",[24,159,160],{},"57",". It eliminates the need to run a traverse to get \"on-station.\"",[11,163,165],{"id":164},"theory","Theory",[16,167,168,169,197,198,146,226,150],{},"Resection works by solving the geometry of two overlapping circles. Each measured angle between two points defines a circular arc upon which point ",[24,170,172,185],{"className":171},[27],[24,173,175],{"className":174},[31],[33,176,177],{"xmlns":35},[37,178,179,183],{},[40,180,181],{},[43,182,45],{},[47,184,45],{"encoding":49},[24,186,188],{"className":187,"ariaHidden":54},[53],[24,189,191,194],{"className":190},[58],[24,192],{"className":193,"style":63},[62],[24,195,45],{"className":196,"style":69},[67,68]," must lie. 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N",[24,265,267],{"className":266,"ariaHidden":54},[53],[24,268,270,273,277,280,283],{"className":269},[58],[24,271],{"className":272,"style":110},[62],[24,274,255],{"className":275,"style":276},[67,68],"margin-right:0.0576em;",[24,278,89],{"className":279},[117],[24,281],{"className":282,"style":122},[121],[24,284,260],{"className":285,"style":286},[67,68],"margin-right:0.109em;",") for ",[24,289,291,312],{"className":290},[27],[24,292,294],{"className":293},[31],[33,295,296],{"xmlns":35},[37,297,298,310],{},[40,299,300,302,304,306,308],{},[43,301,85],{},[87,303,89],{"separator":54},[43,305,92],{},[87,307,89],{"separator":54},[43,309,97],{},[47,311,100],{"encoding":49},[24,313,315],{"className":314,"ariaHidden":54},[53],[24,316,318,321,324,327,330,333,336,339],{"className":317},[58],[24,319],{"className":320,"style":110},[62],[24,322,85],{"className":323},[67,68],[24,325,89],{"className":326},[117],[24,328],{"className":329,"style":122},[121],[24,331,92],{"className":332,"style":126},[67,68],[24,334,89],{"className":335},[117],[24,337],{"className":338,"style":122},[121],[24,340,97],{"className":341,"style":136},[67,68]," and observed angles 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B",[24,398,400],{"className":399,"ariaHidden":54},[53],[24,401,403,406,409,412,415],{"className":402},[58],[24,404],{"className":405,"style":110},[62],[24,407,85],{"className":408},[67,68],[24,410,89],{"className":411},[117],[24,413],{"className":414,"style":122},[121],[24,416,92],{"className":417,"style":126},[67,68],") and ",[24,420,422,437],{"className":421},[27],[24,423,425],{"className":424},[31],[33,426,427],{"xmlns":35},[37,428,429,434],{},[40,430,431],{},[43,432,433],{},"β",[47,435,436],{"encoding":49},"\\beta",[24,438,440],{"className":439,"ariaHidden":54},[53],[24,441,443,447],{"className":442},[58],[24,444],{"className":445,"style":446},[62],"height:0.8889em;vertical-align:-0.1944em;",[24,448,433],{"className":449,"style":450},[67,68],"margin-right:0.0528em;",[24,452,454,472],{"className":453},[27],[24,455,457],{"className":456},[31],[33,458,459],{"xmlns":35},[37,460,461,469],{},[40,462,463,465,467],{},[43,464,92],{},[87,466,89],{"separator":54},[43,468,97],{},[47,470,471],{"encoding":49},"B, C",[24,473,475],{"className":474,"ariaHidden":54},[53],[24,476,478,481,484,487,490],{"className":477},[58],[24,479],{"className":480,"style":110},[62],[24,482,92],{"className":483,"style":126},[67,68],[24,485,89],{"className":486},[117],[24,488],{"className":489,"style":122},[121],[24,491,97],{"className":492,"style":136},[67,68],") ",[24,495,228],{},":",[498,499,500,646,1012,1467],"ol",{},[501,502,503,506,507,541,542,576,577,541,610,146,643,150],"li",{},[19,504,505],{},"Calculate Grid Bearings and Distances:"," Find bearings ",[24,508,510,526],{"className":509},[27],[24,511,513],{"className":512},[31],[33,514,515],{"xmlns":35},[37,516,517,523],{},[40,518,519,521],{},[43,520,85],{},[43,522,92],{},[47,524,525],{"encoding":49},"AB",[24,527,529],{"className":528,"ariaHidden":54},[53],[24,530,532,535,538],{"className":531},[58],[24,533],{"className":534,"style":63},[62],[24,536,85],{"className":537},[67,68],[24,539,92],{"className":540,"style":126},[67,68]," and ",[24,543,545,561],{"className":544},[27],[24,546,548],{"className":547},[31],[33,549,550],{"xmlns":35},[37,551,552,558],{},[40,553,554,556],{},[43,555,92],{},[43,557,97],{},[47,559,560],{"encoding":49},"BC",[24,562,564],{"className":563,"ariaHidden":54},[53],[24,565,567,570,573],{"className":566},[58],[24,568],{"className":569,"style":63},[62],[24,571,92],{"className":572,"style":126},[67,68],[24,574,97],{"className":575,"style":136},[67,68]," and distances ",[24,578,580,595],{"className":579},[27],[24,581,583],{"className":582},[31],[33,584,585],{"xmlns":35},[37,586,587,593],{},[40,588,589,591],{},[43,590,85],{},[43,592,92],{},[47,594,525],{"encoding":49},[24,596,598],{"className":597,"ariaHidden":54},[53],[24,599,601,604,607],{"className":600},[58],[24,602],{"className":603,"style":63},[62],[24,605,85],{"className":606},[67,68],[24,608,92],{"className":609,"style":126},[67,68],[24,611,613,628],{"className":612},[27],[24,614,616],{"className":615},[31],[33,617,618],{"xmlns":35},[37,619,620,626],{},[40,621,622,624],{},[43,623,92],{},[43,625,97],{},[47,627,560],{"encoding":49},[24,629,631],{"className":630,"ariaHidden":54},[53],[24,632,634,637,640],{"className":633},[58],[24,635],{"className":636,"style":63},[62],[24,638,92],{"className":639,"style":126},[67,68],[24,641,97],{"className":642,"style":136},[67,68],[24,644,645],{},"58, 59",[501,647,648,680,146,914,917,918],{},[19,649,650,651,496],{},"Calculate Constant ",[24,652,654,668],{"className":653},[27],[24,655,657],{"className":656},[31],[33,658,659],{"xmlns":35},[37,660,661,666],{},[40,662,663],{},[43,664,665],{},"S",[47,667,665],{"encoding":49},[24,669,671],{"className":670,"ariaHidden":54},[53],[24,672,674,677],{"className":673},[58],[24,675],{"className":676,"style":63},[62],[24,678,665],{"className":679,"style":276},[67,68],[24,681,683,737],{"className":682},[27],[24,684,686],{"className":685},[31],[33,687,688],{"xmlns":35},[37,689,690,734],{},[40,691,692,694,697,707,710,714,716,719,721,723,731],{},[43,693,665],{},[87,695,696],{},"=",[698,699,700,704],"msup",{},[701,702,703],"mn",{},"360",[87,705,706],{},"∘",[87,708,709],{},"−",[87,711,713],{"stretchy":712},"false","(",[43,715,357],{},[87,717,718],{},"+",[43,720,433],{},[87,722,718],{},[724,725,726,728],"mover",{"accent":54},[43,727,92],{},[87,729,730],{},"^",[87,732,733],{"stretchy":712},")",[47,735,736],{"encoding":49},"S = 360^\\circ - (\\alpha + \\beta + \\hat{B})",[24,738,740,760,824,847,865],{"className":739,"ariaHidden":54},[53],[24,741,743,746,749,753,757],{"className":742},[58],[24,744],{"className":745,"style":63},[62],[24,747,665],{"className":748,"style":276},[67,68],[24,750],{"className":751,"style":752},[121],"margin-right:0.2778em;",[24,754,696],{"className":755},[756],"mrel",[24,758],{"className":759,"style":752},[121],[24,761,763,767,771,814,818,821],{"className":762},[58],[24,764],{"className":765,"style":766},[62],"height:0.7574em;vertical-align:-0.0833em;",[24,768,770],{"className":769},[67],"36",[24,772,774,778],{"className":773},[67],[24,775,777],{"className":776},[67],"0",[24,779,782],{"className":780},[781],"msupsub",[24,783,786],{"className":784},[785],"vlist-t",[24,787,790],{"className":788},[789],"vlist-r",[24,791,795],{"className":792,"style":794},[793],"vlist","height:0.6741em;",[24,796,798,803],{"style":797},"top:-3.063em;margin-right:0.05em;",[24,799],{"className":800,"style":802},[801],"pstrut","height:2.7em;",[24,804,810],{"className":805},[806,807,808,809],"sizing","reset-size6","size3","mtight",[24,811,706],{"className":812},[813,809],"mbin",[24,815],{"className":816,"style":817},[121],"margin-right:0.2222em;",[24,819,709],{"className":820},[813],[24,822],{"className":823,"style":817},[121],[24,825,827,831,835,838,841,844],{"className":826},[58],[24,828],{"className":829,"style":830},[62],"height:1em;vertical-align:-0.25em;",[24,832,713],{"className":833},[834],"mopen",[24,836,357],{"className":837,"style":374},[67,68],[24,839],{"className":840,"style":817},[121],[24,842,718],{"className":843},[813],[24,845],{"className":846,"style":817},[121],[24,848,850,853,856,859,862],{"className":849},[58],[24,851],{"className":852,"style":446},[62],[24,854,433],{"className":855,"style":450},[67,68],[24,857],{"className":858,"style":817},[121],[24,860,718],{"className":861},[813],[24,863],{"className":864,"style":817},[121],[24,866,868,872,910],{"className":867},[58],[24,869],{"className":870,"style":871},[62],"height:1.1968em;vertical-align:-0.25em;",[24,873,876],{"className":874},[67,875],"accent",[24,877,879],{"className":878},[785],[24,880,882],{"className":881},[789],[24,883,886,896],{"className":884,"style":885},[793],"height:0.9468em;",[24,887,889,893],{"style":888},"top:-3em;",[24,890],{"className":891,"style":892},[801],"height:3em;",[24,894,92],{"className":895,"style":126},[67,68],[24,897,899,902],{"style":898},"top:-3.2523em;",[24,900],{"className":901,"style":892},[801],[24,903,907],{"className":904,"style":906},[905],"accent-body","left:-0.1667em;",[24,908,730],{"className":909},[67],[24,911,733],{"className":912},[913],"mclose",[24,915,916],{},"59",".\n",[139,919,920,921,982,983,1011],{},"(Where ",[24,922,924,942],{"className":923},[27],[24,925,927],{"className":926},[31],[33,928,929],{"xmlns":35},[37,930,931,939],{},[40,932,933],{},[724,934,935,937],{"accent":54},[43,936,92],{},[87,938,730],{},[47,940,941],{"encoding":49},"\\hat{B}",[24,943,945],{"className":944,"ariaHidden":54},[53],[24,946,948,951],{"className":947},[58],[24,949],{"className":950,"style":885},[62],[24,952,954],{"className":953},[67,875],[24,955,957],{"className":956},[785],[24,958,960],{"className":959},[789],[24,961,963,971],{"className":962,"style":885},[793],[24,964,965,968],{"style":888},[24,966],{"className":967,"style":892},[801],[24,969,92],{"className":970,"style":126},[67,68],[24,972,973,976],{"style":898},[24,974],{"className":975,"style":892},[801],[24,977,979],{"className":978,"style":906},[905],[24,980,730],{"className":981},[67]," is the internal angle at 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",[24,1051,1053,1071],{"className":1052},[27],[24,1054,1056],{"className":1055},[31],[33,1057,1058],{"xmlns":35},[37,1059,1060,1068],{},[40,1061,1062,1064,1066],{},[43,1063,92],{},[43,1065,85],{},[43,1067,45],{},[47,1069,1070],{"encoding":49},"BAP",[24,1072,1074],{"className":1073,"ariaHidden":54},[53],[24,1075,1077,1080,1083,1086],{"className":1076},[58],[24,1078],{"className":1079,"style":63},[62],[24,1081,92],{"className":1082,"style":126},[67,68],[24,1084,85],{"className":1085},[67,68],[24,1087,45],{"className":1088,"style":69},[67,68],"):",[24,1091,1093,1168],{"className":1092},[27],[24,1094,1096],{"className":1095},[31],[33,1097,1098],{"xmlns":35},[37,1099,1100,1165],{},[40,1101,1102,1105,1108,1110,1112],{},[43,1103,1104],{},"cot",[87,1106,1107],{},"⁡",[43,1109,1031],{},[87,1111,696],{},[1113,1114,1115,1157],"mfrac",{},[40,1116,1117,1119,1146,1148,1151,1153,1155],{},[87,1118,713],{"fence":54},[1113,1120,1121,1134],{},[40,1122,1123,1125,1127,1130,1132],{},[43,1124,85],{},[43,1126,92],{},[43,1128,1129],{},"sin",[87,1131,1107],{},[43,1133,433],{},[40,1135,1136,1138,1140,1142,1144],{},[43,1137,92],{},[43,1139,97],{},[43,1141,1129],{},[87,1143,1107],{},[43,1145,357],{},[87,1147,718],{},[43,1149,1150],{},"cos",[87,1152,1107],{},[43,1154,665],{},[87,1156,733],{"fence":54},[40,1158,1159,1161,1163],{},[43,1160,1129],{},[87,1162,1107],{},[43,1164,665],{},[47,1166,1167],{"encoding":49},"\\cot \\theta = \\frac{\\left( \\frac{AB \\sin \\beta}{BC \\sin \\alpha} + \\cos S \\right)}{\\sin S}",[24,1169,1171,1196],{"className":1170,"ariaHidden":54},[53],[24,1172,1174,1177,1181,1184,1187,1190,1193],{"className":1173},[58],[24,1175],{"className":1176,"style":1044},[62],[24,1178,1104],{"className":1179},[1180],"mop",[24,1182],{"className":1183,"style":122},[121],[24,1185,1031],{"className":1186,"style":1048},[67,68],[24,1188],{"className":1189,"style":752},[121],[24,1191,696],{"className":1192},[756],[24,1194],{"className":1195,"style":752},[121],[24,1197,1199,1203],{"className":1198},[58],[24,1200],{"className":1201,"style":1202},[62],"height:1.655em;vertical-align:-0.345em;",[24,1204,1206,1210,1462],{"className":1205},[67],[24,1207],{"className":1208},[834,1209],"nulldelimiter",[24,1211,1213],{"className":1212},[1113],[24,1214,1217,1453],{"className":1215},[785,1216],"vlist-t2",[24,1218,1220,1450],{"className":1219},[789],[24,1221,1224,1258,1269],{"className":1222,"style":1223},[793],"height:1.31em;",[24,1225,1227,1230],{"style":1226},"top:-2.655em;",[24,1228],{"className":1229,"style":892},[801],[24,1231,1233],{"className":1232},[806,807,808,809],[24,1234,1236,1251,1255],{"className":1235},[67,809],[24,1237,1239,1243,1247],{"className":1238},[1180,809],[24,1240,1242],{"className":1241},[809],"s",[24,1244,1246],{"className":1245},[809],"i",[24,1248,1250],{"className":1249},[809],"n",[24,1252],{"className":1253,"style":1254},[121,809],"margin-right:0.1952em;",[24,1256,665],{"className":1257,"style":276},[67,68,809],[24,1259,1261,1264],{"style":1260},"top:-3.23em;",[24,1262],{"className":1263,"style":892},[801],[24,1265],{"className":1266,"style":1268},[1267],"frac-line","border-bottom-width:0.04em;",[24,1270,1272,1275],{"style":1271},"top:-3.6125em;",[24,1273],{"className":1274,"style":892},[801],[24,1276,1278],{"className":1277},[806,807,808,809],[24,1279,1281],{"className":1280},[67,809],[24,1282,1285,1295,1421,1424,1438,1441,1444],{"className":1283},[1284,809],"minner",[24,1286,1292],{"className":1287,"style":1291},[834,806,1288,1289,809,1290],"reset-size3","size6","delimcenter","top:0.075em;",[24,1293,713],{"className":1294},[809],[24,1296,1298,1301,1418],{"className":1297},[67,809],[24,1299],{"className":1300},[834,1209,806,1288,1289],[24,1302,1304],{"className":1303},[1113],[24,1305,1307,1409],{"className":1306},[785,1216],[24,1308,1310,1404],{"className":1309},[789],[24,1311,1314,1355,1365],{"className":1312,"style":1313},[793],"height:0.9584em;",[24,1315,1317,1320],{"style":1316},"top:-2.656em;",[24,1318],{"className":1319,"style":892},[801],[24,1321,1324],{"className":1322},[806,1288,1323,809],"size1",[24,1325,1327,1330,1333,1337,1349,1352],{"className":1326},[67,809],[24,1328,92],{"className":1329,"style":126},[67,68,809],[24,1331,97],{"className":1332,"style":136},[67,68,809],[24,1334],{"className":1335,"style":1336},[121,809],"margin-right:0.2453em;",[24,1338,1340,1343,1346],{"className":1339},[1180,809],[24,1341,1242],{"className":1342},[809],[24,1344,1246],{"className":1345},[809],[24,1347,1250],{"className":1348},[809],[24,1350],{"className":1351,"style":1336},[121,809],[24,1353,357],{"className":1354,"style":374},[67,68,809],[24,1356,1358,1361],{"style":1357},"top:-3.2255em;",[24,1359],{"className":1360,"style":892},[801],[24,1362],{"className":1363,"style":1364},[1267,809],"border-bottom-width:0.049em;",[24,1366,1368,1371],{"style":1367},"top:-3.4624em;",[24,1369],{"className":1370,"style":892},[801],[24,1372,1374],{"className":1373},[806,1288,1323,809],[24,1375,1377,1380,1383,1386,1398,1401],{"className":1376},[67,809],[24,1378,85],{"className":1379},[67,68,809],[24,1381,92],{"className":1382,"style":126},[67,68,809],[24,1384],{"className":1385,"style":1336},[121,809],[24,1387,1389,1392,1395],{"className":1388},[1180,809],[24,1390,1242],{"className":1391},[809],[24,1393,1246],{"className":1394},[809],[24,1396,1250],{"className":1397},[809],[24,1399],{"className":1400,"style":1336},[121,809],[24,1402,433],{"className":1403,"style":450},[67,68,809],[24,1405,1408],{"className":1406},[1407],"vlist-s","​",[24,1410,1412],{"className":1411},[789],[24,1413,1416],{"className":1414,"style":1415},[793],"height:0.344em;",[24,1417],{},[24,1419],{"className":1420},[913,1209,806,1288,1289],[24,1422,718],{"className":1423},[813,809],[24,1425,1427,1431,1435],{"className":1426},[1180,809],[24,1428,1430],{"className":1429},[809],"c",[24,1432,1434],{"className":1433},[809],"o",[24,1436,1242],{"className":1437},[809],[24,1439],{"className":1440,"style":1254},[121,809],[24,1442,665],{"className":1443,"style":276},[67,68,809],[24,1445,1447],{"className":1446,"style":1291},[913,806,1288,1289,809,1290],[24,1448,733],{"className":1449},[809],[24,1451,1408],{"className":1452},[1407],[24,1454,1456],{"className":1455},[789],[24,1457,1460],{"className":1458,"style":1459},[793],"height:0.345em;",[24,1461],{},[24,1463],{"className":1464},[913,1209],[24,1466,916],{},[501,1468,1469,1472,1473,1501,1502,1541,1542,1576,1577,1610,1611,146,1639,150],{},[19,1470,1471],{},"Final Coordinates:"," Once ",[24,1474,1476,1489],{"className":1475},[27],[24,1477,1479],{"className":1478},[31],[33,1480,1481],{"xmlns":35},[37,1482,1483,1487],{},[40,1484,1485],{},[43,1486,1031],{},[47,1488,1034],{"encoding":49},[24,1490,1492],{"className":1491,"ariaHidden":54},[53],[24,1493,1495,1498],{"className":1494},[58],[24,1496],{"className":1497,"style":1044},[62],[24,1499,1031],{"className":1500,"style":1048},[67,68]," is known, the triangle ",[24,1503,1505,1523],{"className":1504},[27],[24,1506,1508],{"className":1507},[31],[33,1509,1510],{"xmlns":35},[37,1511,1512,1520],{},[40,1513,1514,1516,1518],{},[43,1515,85],{},[43,1517,92],{},[43,1519,45],{},[47,1521,1522],{"encoding":49},"ABP",[24,1524,1526],{"className":1525,"ariaHidden":54},[53],[24,1527,1529,1532,1535,1538],{"className":1528},[58],[24,1530],{"className":1531,"style":63},[62],[24,1533,85],{"className":1534},[67,68],[24,1536,92],{"className":1537,"style":126},[67,68],[24,1539,45],{"className":1540,"style":69},[67,68]," is solved for length ",[24,1543,1545,1561],{"className":1544},[27],[24,1546,1548],{"className":1547},[31],[33,1549,1550],{"xmlns":35},[37,1551,1552,1558],{},[40,1553,1554,1556],{},[43,1555,85],{},[43,1557,45],{},[47,1559,1560],{"encoding":49},"AP",[24,1562,1564],{"className":1563,"ariaHidden":54},[53],[24,1565,1567,1570,1573],{"className":1566},[58],[24,1568],{"className":1569,"style":63},[62],[24,1571,85],{"className":1572},[67,68],[24,1574,45],{"className":1575,"style":69},[67,68]," and bearing ",[24,1578,1580,1595],{"className":1579},[27],[24,1581,1583],{"className":1582},[31],[33,1584,1585],{"xmlns":35},[37,1586,1587,1593],{},[40,1588,1589,1591],{},[43,1590,85],{},[43,1592,45],{},[47,1594,1560],{"encoding":49},[24,1596,1598],{"className":1597,"ariaHidden":54},[53],[24,1599,1601,1604,1607],{"className":1600},[58],[24,1602],{"className":1603,"style":63},[62],[24,1605,85],{"className":1606},[67,68],[24,1608,45],{"className":1609,"style":69},[67,68]," to get the coordinates of ",[24,1612,1614,1627],{"className":1613},[27],[24,1615,1617],{"className":1616},[31],[33,1618,1619],{"xmlns":35},[37,1620,1621,1625],{},[40,1622,1623],{},[43,1624,45],{},[47,1626,45],{"encoding":49},[24,1628,1630],{"className":1629,"ariaHidden":54},[53],[24,1631,1633,1636],{"className":1632},[58],[24,1634],{"className":1635,"style":63},[62],[24,1637,45],{"className":1638,"style":69},[67,68],[24,1640,916],{},[11,1642,1644],{"id":1643},"field-workflow","Field Workflow",[1646,1647,1648,1652,1733,1737,1943,1947,2057,2061],"steps",{},[234,1649,1651],{"id":1650},"select-three-known-points","Select Three Known Points",[16,1653,1654,1655,541,1702,1730,1731,150],{},"Points ",[24,1656,1658,1678],{"className":1657},[27],[24,1659,1661],{"className":1660},[31],[33,1662,1663],{"xmlns":35},[37,1664,1665,1675],{},[40,1666,1667,1669,1671,1673],{},[43,1668,85],{},[87,1670,89],{"separator":54},[43,1672,92],{},[87,1674,89],{"separator":54},[47,1676,1677],{"encoding":49},"A, B,",[24,1679,1681],{"className":1680,"ariaHidden":54},[53],[24,1682,1684,1687,1690,1693,1696,1699],{"className":1683},[58],[24,1685],{"className":1686,"style":110},[62],[24,1688,85],{"className":1689},[67,68],[24,1691,89],{"className":1692},[117],[24,1694],{"className":1695,"style":122},[121],[24,1697,92],{"className":1698,"style":126},[67,68],[24,1700,89],{"className":1701},[117],[24,1703,1705,1718],{"className":1704},[27],[24,1706,1708],{"className":1707},[31],[33,1709,1710],{"xmlns":35},[37,1711,1712,1716],{},[40,1713,1714],{},[43,1715,97],{},[47,1717,97],{"encoding":49},[24,1719,1721],{"className":1720,"ariaHidden":54},[53],[24,1722,1724,1727],{"className":1723},[58],[24,1725],{"className":1726,"style":63},[62],[24,1728,97],{"className":1729,"style":136},[67,68]," should be established control stations with high-order coordinates ",[24,1732,149],{},[234,1734,1736],{"id":1735},"angle-observations","Angle Observations",[16,1738,1739,1740,1768,1769,541,1797,1825,1826,1854,1855,541,1883,1825,1911,1939,1940,150],{},"Set up the theodolite at the unknown point ",[24,1741,1743,1756],{"className":1742},[27],[24,1744,1746],{"className":1745},[31],[33,1747,1748],{"xmlns":35},[37,1749,1750,1754],{},[40,1751,1752],{},[43,1753,45],{},[47,1755,45],{"encoding":49},[24,1757,1759],{"className":1758,"ariaHidden":54},[53],[24,1760,1762,1765],{"className":1761},[58],[24,1763],{"className":1764,"style":63},[62],[24,1766,45],{"className":1767,"style":69},[67,68],". Measure the horizontal angles between ",[24,1770,1772,1785],{"className":1771},[27],[24,1773,1775],{"className":1774},[31],[33,1776,1777],{"xmlns":35},[37,1778,1779,1783],{},[40,1780,1781],{},[43,1782,85],{},[47,1784,85],{"encoding":49},[24,1786,1788],{"className":1787,"ariaHidden":54},[53],[24,1789,1791,1794],{"className":1790},[58],[24,1792],{"className":1793,"style":63},[62],[24,1795,85],{"className":1796},[67,68],[24,1798,1800,1813],{"className":1799},[27],[24,1801,1803],{"className":1802},[31],[33,1804,1805],{"xmlns":35},[37,1806,1807,1811],{},[40,1808,1809],{},[43,1810,92],{},[47,1812,92],{"encoding":49},[24,1814,1816],{"className":1815,"ariaHidden":54},[53],[24,1817,1819,1822],{"className":1818},[58],[24,1820],{"className":1821,"style":63},[62],[24,1823,92],{"className":1824,"style":126},[67,68]," (",[24,1827,1829,1842],{"className":1828},[27],[24,1830,1832],{"className":1831},[31],[33,1833,1834],{"xmlns":35},[37,1835,1836,1840],{},[40,1837,1838],{},[43,1839,357],{},[47,1841,360],{"encoding":49},[24,1843,1845],{"className":1844,"ariaHidden":54},[53],[24,1846,1848,1851],{"className":1847},[58],[24,1849],{"className":1850,"style":370},[62],[24,1852,357],{"className":1853,"style":374},[67,68],") and between ",[24,1856,1858,1871],{"className":1857},[27],[24,1859,1861],{"className":1860},[31],[33,1862,1863],{"xmlns":35},[37,1864,1865,1869],{},[40,1866,1867],{},[43,1868,92],{},[47,1870,92],{"encoding":49},[24,1872,1874],{"className":1873,"ariaHidden":54},[53],[24,1875,1877,1880],{"className":1876},[58],[24,1878],{"className":1879,"style":63},[62],[24,1881,92],{"className":1882,"style":126},[67,68],[24,1884,1886,1899],{"className":1885},[27],[24,1887,1889],{"className":1888},[31],[33,1890,1891],{"xmlns":35},[37,1892,1893,1897],{},[40,1894,1895],{},[43,1896,97],{},[47,1898,97],{"encoding":49},[24,1900,1902],{"className":1901,"ariaHidden":54},[53],[24,1903,1905,1908],{"className":1904},[58],[24,1906],{"className":1907,"style":63},[62],[24,1909,97],{"className":1910,"style":136},[67,68],[24,1912,1914,1927],{"className":1913},[27],[24,1915,1917],{"className":1916},[31],[33,1918,1919],{"xmlns":35},[37,1920,1921,1925],{},[40,1922,1923],{},[43,1924,433],{},[47,1926,436],{"encoding":49},[24,1928,1930],{"className":1929,"ariaHidden":54},[53],[24,1931,1933,1936],{"className":1932},[58],[24,1934],{"className":1935,"style":446},[62],[24,1937,433],{"className":1938,"style":450},[67,68],"). Use multiple sets on both faces for precision ",[24,1941,1942],{},"60, 61",[234,1944,1946],{"id":1945},"geometry-check","Geometry Check",[16,1948,1949,1950,1978,1979,541,2025,2053,2054,150],{},"Ensure point ",[24,1951,1953,1966],{"className":1952},[27],[24,1954,1956],{"className":1955},[31],[33,1957,1958],{"xmlns":35},[37,1959,1960,1964],{},[40,1961,1962],{},[43,1963,45],{},[47,1965,45],{"encoding":49},[24,1967,1969],{"className":1968,"ariaHidden":54},[53],[24,1970,1972,1975],{"className":1971},[58],[24,1973],{"className":1974,"style":63},[62],[24,1976,45],{"className":1977,"style":69},[67,68]," does not lie on the \"Circle of Danger\" (the circle passing through ",[24,1980,1982,2001],{"className":1981},[27],[24,1983,1985],{"className":1984},[31],[33,1986,1987],{"xmlns":35},[37,1988,1989,1999],{},[40,1990,1991,1993,1995,1997],{},[43,1992,85],{},[87,1994,89],{"separator":54},[43,1996,92],{},[87,1998,89],{"separator":54},[47,2000,1677],{"encoding":49},[24,2002,2004],{"className":2003,"ariaHidden":54},[53],[24,2005,2007,2010,2013,2016,2019,2022],{"className":2006},[58],[24,2008],{"className":2009,"style":110},[62],[24,2011,85],{"className":2012},[67,68],[24,2014,89],{"className":2015},[117],[24,2017],{"className":2018,"style":122},[121],[24,2020,92],{"className":2021,"style":126},[67,68],[24,2023,89],{"className":2024},[117],[24,2026,2028,2041],{"className":2027},[27],[24,2029,2031],{"className":2030},[31],[33,2032,2033],{"xmlns":35},[37,2034,2035,2039],{},[40,2036,2037],{},[43,2038,97],{},[47,2040,97],{"encoding":49},[24,2042,2044],{"className":2043,"ariaHidden":54},[53],[24,2045,2047,2050],{"className":2046},[58],[24,2048],{"className":2049,"style":63},[62],[24,2051,97],{"className":2052,"style":136},[67,68],"). If it does, the math fails, and an infinite number of positions are possible ",[24,2055,2056],{},"58, 62",[234,2058,2060],{"id":2059},"addition-of-a-fourth-point","Addition of a Fourth Point",[16,2062,2063,2064,2093,2094,150],{},"Always sight a fourth point (",[24,2065,2067,2081],{"className":2066},[27],[24,2068,2070],{"className":2069},[31],[33,2071,2072],{"xmlns":35},[37,2073,2074,2079],{},[40,2075,2076],{},[43,2077,2078],{},"D",[47,2080,2078],{"encoding":49},[24,2082,2084],{"className":2083,"ariaHidden":54},[53],[24,2085,2087,2090],{"className":2086},[58],[24,2088],{"className":2089,"style":63},[62],[24,2091,2078],{"className":2092,"style":1048},[67,68],") if possible. This provides an independent check and prevents \"Circle of Danger\" errors ",[24,2095,2096],{},"57, 58",[11,2098,2100],{"id":2099},"step-by-step-example","Step-by-Step Example",[16,2102,2103,2106,2107,541,2166,2225,2226,2363,2364,2546,2547,146,2575,150],{},[19,2104,2105],{},"Problem:"," Points ",[24,2108,2110,2136],{"className":2109},[27],[24,2111,2113],{"className":2112},[31],[33,2114,2115],{"xmlns":35},[37,2116,2117,2133],{},[40,2118,2119,2121,2123,2126,2128,2131],{},[43,2120,85],{},[87,2122,713],{"stretchy":712},[701,2124,2125],{},"1234.96",[87,2127,89],{"separator":54},[701,2129,2130],{},"17594.48",[87,2132,733],{"stretchy":712},[47,2134,2135],{"encoding":49},"A(1234.96, 17594.48)",[24,2137,2139],{"className":2138,"ariaHidden":54},[53],[24,2140,2142,2145,2148,2151,2154,2157,2160,2163],{"className":2141},[58],[24,2143],{"className":2144,"style":830},[62],[24,2146,85],{"className":2147},[67,68],[24,2149,713],{"className":2150},[834],[24,2152,2125],{"className":2153},[67],[24,2155,89],{"className":2156},[117],[24,2158],{"className":2159,"style":122},[121],[24,2161,2130],{"className":2162},[67],[24,2164,733],{"className":2165},[913],[24,2167,2169,2195],{"className":2168},[27],[24,2170,2172],{"className":2171},[31],[33,2173,2174],{"xmlns":35},[37,2175,2176,2192],{},[40,2177,2178,2180,2182,2185,2187,2190],{},[43,2179,92],{},[87,2181,713],{"stretchy":712},[701,2183,2184],{},"7994.42",[87,2186,89],{"separator":54},[701,2188,2189],{},"24343.45",[87,2191,733],{"stretchy":712},[47,2193,2194],{"encoding":49},"B(7994.42, 24343.45)",[24,2196,2198],{"className":2197,"ariaHidden":54},[53],[24,2199,2201,2204,2207,2210,2213,2216,2219,2222],{"className":2200},[58],[24,2202],{"className":2203,"style":830},[62],[24,2205,92],{"className":2206,"style":126},[67,68],[24,2208,713],{"className":2209},[834],[24,2211,2184],{"className":2212},[67],[24,2214,89],{"className":2215},[117],[24,2217],{"className":2218,"style":122},[121],[24,2220,2189],{"className":2221},[67],[24,2223,733],{"className":2224},[913]," are sighted. ",[24,2227,2229,2269],{"className":2228},[27],[24,2230,2232],{"className":2231},[31],[33,2233,2234],{"xmlns":35},[37,2235,2236,2266],{},[40,2237,2238,2240,2257,2259,2262],{},[43,2239,85],{},[2241,2242,2243,2245],"msub",{},[43,2244,92],{},[40,2246,2247,2250,2252,2254],{},[43,2248,2249],{},"d",[43,2251,1246],{},[43,2253,1242],{},[43,2255,2256],{},"t",[87,2258,696],{},[701,2260,2261],{},"9551.91",[2263,2264,2265],"mtext",{}," m",[47,2267,2268],{"encoding":49},"AB_{dist} = 9551.91\\text{ m}",[24,2270,2272,2346],{"className":2271,"ariaHidden":54},[53],[24,2273,2275,2279,2282,2337,2340,2343],{"className":2274},[58],[24,2276],{"className":2277,"style":2278},[62],"height:0.8333em;vertical-align:-0.15em;",[24,2280,85],{"className":2281},[67,68],[24,2283,2285,2288],{"className":2284},[67],[24,2286,92],{"className":2287,"style":126},[67,68],[24,2289,2291],{"className":2290},[781],[24,2292,2294,2328],{"className":2293},[785,1216],[24,2295,2297,2325],{"className":2296},[789],[24,2298,2301],{"className":2299,"style":2300},[793],"height:0.3361em;",[24,2302,2304,2307],{"style":2303},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[24,2305],{"className":2306,"style":802},[801],[24,2308,2310],{"className":2309},[806,807,808,809],[24,2311,2313,2316,2319,2322],{"className":2312},[67,809],[24,2314,2249],{"className":2315},[67,68,809],[24,2317,1246],{"className":2318},[67,68,809],[24,2320,1242],{"className":2321},[67,68,809],[24,2323,2256],{"className":2324},[67,68,809],[24,2326,1408],{"className":2327},[1407],[24,2329,2331],{"className":2330},[789],[24,2332,2335],{"className":2333,"style":2334},[793],"height:0.15em;",[24,2336],{},[24,2338],{"className":2339,"style":752},[121],[24,2341,696],{"className":2342},[756],[24,2344],{"className":2345,"style":752},[121],[24,2347,2349,2353,2356],{"className":2348},[58],[24,2350],{"className":2351,"style":2352},[62],"height:0.6444em;",[24,2354,2261],{"className":2355},[67],[24,2357,2360],{"className":2358},[67,2359],"text",[24,2361,2265],{"className":2362},[67],". Observed angle ",[24,2365,2367,2411],{"className":2366},[27],[24,2368,2370],{"className":2369},[31],[33,2371,2372],{"xmlns":35},[37,2373,2374,2408],{},[40,2375,2376,2378,2380,2387,2397],{},[43,2377,357],{},[87,2379,696],{},[698,2381,2382,2385],{},[701,2383,2384],{},"61",[87,2386,706],{},[698,2388,2389,2392],{},[701,2390,2391],{},"41",[87,2393,2396],{"mathvariant":2394,"lspace":2395,"rspace":2395},"normal","0em","′",[698,2398,2399,2402],{},[701,2400,2401],{},"46.6",[40,2403,2404,2406],{},[87,2405,2396],{"mathvariant":2394},[87,2407,2396],{"mathvariant":2394},[47,2409,2410],{"encoding":49},"\\alpha = 61^\\circ 41' 46.6''",[24,2412,2414,2432],{"className":2413,"ariaHidden":54},[53],[24,2415,2417,2420,2423,2426,2429],{"className":2416},[58],[24,2418],{"className":2419,"style":370},[62],[24,2421,357],{"className":2422,"style":374},[67,68],[24,2424],{"className":2425,"style":752},[121],[24,2427,696],{"className":2428},[756],[24,2430],{"className":2431,"style":752},[121],[24,2433,2435,2439,2443,2473,2477,2509,2513],{"className":2434},[58],[24,2436],{"className":2437,"style":2438},[62],"height:0.7519em;",[24,2440,2442],{"className":2441},[67],"6",[24,2444,2446,2450],{"className":2445},[67],[24,2447,2449],{"className":2448},[67],"1",[24,2451,2453],{"className":2452},[781],[24,2454,2456],{"className":2455},[785],[24,2457,2459],{"className":2458},[789],[24,2460,2462],{"className":2461,"style":794},[793],[24,2463,2464,2467],{"style":797},[24,2465],{"className":2466,"style":802},[801],[24,2468,2470],{"className":2469},[806,807,808,809],[24,2471,706],{"className":2472},[813,809],[24,2474,2476],{"className":2475},[67],"4",[24,2478,2480,2483],{"className":2479},[67],[24,2481,2449],{"className":2482},[67],[24,2484,2486],{"className":2485},[781],[24,2487,2489],{"className":2488},[785],[24,2490,2492],{"className":2491},[789],[24,2493,2495],{"className":2494,"style":2438},[793],[24,2496,2497,2500],{"style":797},[24,2498],{"className":2499,"style":802},[801],[24,2501,2503],{"className":2502},[806,807,808,809],[24,2504,2506],{"className":2505},[67,809],[24,2507,2396],{"className":2508},[67,809],[24,2510,2512],{"className":2511},[67],"46.",[24,2514,2516,2519],{"className":2515},[67],[24,2517,2442],{"className":2518},[67],[24,2520,2522],{"className":2521},[781],[24,2523,2525],{"className":2524},[785],[24,2526,2528],{"className":2527},[789],[24,2529,2531],{"className":2530,"style":2438},[793],[24,2532,2533,2536],{"style":797},[24,2534],{"className":2535,"style":802},[801],[24,2537,2539],{"className":2538},[806,807,808,809],[24,2540,2542],{"className":2541},[67,809],[24,2543,2545],{"className":2544},[67,809],"′′",". Solve for ",[24,2548,2550,2563],{"className":2549},[27],[24,2551,2553],{"className":2552},[31],[33,2554,2555],{"xmlns":35},[37,2556,2557,2561],{},[40,2558,2559],{},[43,2560,45],{},[47,2562,45],{"encoding":49},[24,2564,2566],{"className":2565,"ariaHidden":54},[53],[24,2567,2569,2572],{"className":2568},[58],[24,2570],{"className":2571,"style":63},[62],[24,2573,45],{"className":2574,"style":69},[67,68],[24,2576,228],{},[16,2578,2579],{},[139,2580,2581],{},"(Note: Full resection requires point C; this example simplifies to the triangle solution step).",[498,2583,2584,2806,3000,3255],{},[501,2585,2586,2622,2623,146,2804,150],{},[19,2587,2588,2589,496],{},"Find the Bearing ",[24,2590,2592,2607],{"className":2591},[27],[24,2593,2595],{"className":2594},[31],[33,2596,2597],{"xmlns":35},[37,2598,2599,2605],{},[40,2600,2601,2603],{},[43,2602,85],{},[43,2604,92],{},[47,2606,525],{"encoding":49},[24,2608,2610],{"className":2609,"ariaHidden":54},[53],[24,2611,2613,2616,2619],{"className":2612},[58],[24,2614],{"className":2615,"style":63},[62],[24,2617,85],{"className":2618},[67,68],[24,2620,92],{"className":2621,"style":126},[67,68]," From coordinates, ",[24,2624,2626,2669],{"className":2625},[27],[24,2627,2629],{"className":2628},[31],[33,2630,2631],{"xmlns":35},[37,2632,2633,2666],{},[40,2634,2635,2637,2639,2641,2648,2655],{},[43,2636,85],{},[43,2638,92],{},[87,2640,696],{},[698,2642,2643,2646],{},[701,2644,2645],{},"45",[87,2647,706],{},[698,2649,2650,2653],{},[701,2651,2652],{},"02",[87,2654,2396],{"mathvariant":2394,"lspace":2395,"rspace":2395},[698,2656,2657,2660],{},[701,2658,2659],{},"40.2",[40,2661,2662,2664],{},[87,2663,2396],{"mathvariant":2394},[87,2665,2396],{"mathvariant":2394},[47,2667,2668],{"encoding":49},"AB = 45^\\circ 02' 40.2''",[24,2670,2672,2693],{"className":2671,"ariaHidden":54},[53],[24,2673,2675,2678,2681,2684,2687,2690],{"className":2674},[58],[24,2676],{"className":2677,"style":63},[62],[24,2679,85],{"className":2680},[67,68],[24,2682,92],{"className":2683,"style":126},[67,68],[24,2685],{"className":2686,"style":752},[121],[24,2688,696],{"className":2689},[756],[24,2691],{"className":2692,"style":752},[121],[24,2694,2696,2699,2702,2732,2735,2768,2772],{"className":2695},[58],[24,2697],{"className":2698,"style":2438},[62],[24,2700,2476],{"className":2701},[67],[24,2703,2705,2709],{"className":2704},[67],[24,2706,2708],{"className":2707},[67],"5",[24,2710,2712],{"className":2711},[781],[24,2713,2715],{"className":2714},[785],[24,2716,2718],{"className":2717},[789],[24,2719,2721],{"className":2720,"style":794},[793],[24,2722,2723,2726],{"style":797},[24,2724],{"className":2725,"style":802},[801],[24,2727,2729],{"className":2728},[806,807,808,809],[24,2730,706],{"className":2731},[813,809],[24,2733,777],{"className":2734},[67],[24,2736,2738,2742],{"className":2737},[67],[24,2739,2741],{"className":2740},[67],"2",[24,2743,2745],{"className":2744},[781],[24,2746,2748],{"className":2747},[785],[24,2749,2751],{"className":2750},[789],[24,2752,2754],{"className":2753,"style":2438},[793],[24,2755,2756,2759],{"style":797},[24,2757],{"className":2758,"style":802},[801],[24,2760,2762],{"className":2761},[806,807,808,809],[24,2763,2765],{"className":2764},[67,809],[24,2766,2396],{"className":2767},[67,809],[24,2769,2771],{"className":2770},[67],"40.",[24,2773,2775,2778],{"className":2774},[67],[24,2776,2741],{"className":2777},[67],[24,2779,2781],{"className":2780},[781],[24,2782,2784],{"className":2783},[785],[24,2785,2787],{"className":2786},[789],[24,2788,2790],{"className":2789,"style":2438},[793],[24,2791,2792,2795],{"style":797},[24,2793],{"className":2794,"style":802},[801],[24,2796,2798],{"className":2797},[806,807,808,809],[24,2799,2801],{"className":2800},[67,809],[24,2802,2545],{"className":2803},[67,809],[24,2805,228],{},[501,2807,2808,2867,2868,2896,2897,2935,2936,496],{},[19,2809,1016,2810,2838,2839,496],{},[24,2811,2813,2826],{"className":2812},[27],[24,2814,2816],{"className":2815},[31],[33,2817,2818],{"xmlns":35},[37,2819,2820,2824],{},[40,2821,2822],{},[43,2823,45],{},[47,2825,45],{"encoding":49},[24,2827,2829],{"className":2828,"ariaHidden":54},[53],[24,2830,2832,2835],{"className":2831},[58],[24,2833],{"className":2834,"style":63},[62],[24,2836,45],{"className":2837,"style":69},[67,68]," using ",[24,2840,2842,2855],{"className":2841},[27],[24,2843,2845],{"className":2844},[31],[33,2846,2847],{"xmlns":35},[37,2848,2849,2853],{},[40,2850,2851],{},[43,2852,1031],{},[47,2854,1034],{"encoding":49},[24,2856,2858],{"className":2857,"ariaHidden":54},[53],[24,2859,2861,2864],{"className":2860},[58],[24,2862],{"className":2863,"style":1044},[62],[24,2865,1031],{"className":2866,"style":1048},[67,68]," If ",[24,2869,2871,2884],{"className":2870},[27],[24,2872,2874],{"className":2873},[31],[33,2875,2876],{"xmlns":35},[37,2877,2878,2882],{},[40,2879,2880],{},[43,2881,1031],{},[47,2883,1034],{"encoding":49},[24,2885,2887],{"className":2886,"ariaHidden":54},[53],[24,2888,2890,2893],{"className":2889},[58],[24,2891],{"className":2892,"style":1044},[62],[24,2894,1031],{"className":2895,"style":1048},[67,68]," (angle ",[24,2898,2900,2917],{"className":2899},[27],[24,2901,2903],{"className":2902},[31],[33,2904,2905],{"xmlns":35},[37,2906,2907,2915],{},[40,2908,2909,2911,2913],{},[43,2910,92],{},[43,2912,85],{},[43,2914,45],{},[47,2916,1070],{"encoding":49},[24,2918,2920],{"className":2919,"ariaHidden":54},[53],[24,2921,2923,2926,2929,2932],{"className":2922},[58],[24,2924],{"className":2925,"style":63},[62],[24,2927,92],{"className":2928,"style":126},[67,68],[24,2930,85],{"className":2931},[67,68],[24,2933,45],{"className":2934,"style":69},[67,68],") is calculated as ",[24,2937,2939,2958],{"className":2938},[27],[24,2940,2942],{"className":2941},[31],[33,2943,2944],{"xmlns":35},[37,2945,2946,2955],{},[40,2947,2948],{},[698,2949,2950,2953],{},[701,2951,2952],{},"52.5545",[87,2954,706],{},[47,2956,2957],{"encoding":49},"52.5545^\\circ",[24,2959,2961],{"className":2960,"ariaHidden":54},[53],[24,2962,2964,2967,2971],{"className":2963},[58],[24,2965],{"className":2966,"style":794},[62],[24,2968,2970],{"className":2969},[67],"52.554",[24,2972,2974,2977],{"className":2973},[67],[24,2975,2708],{"className":2976},[67],[24,2978,2980],{"className":2979},[781],[24,2981,2983],{"className":2982},[785],[24,2984,2986],{"className":2985},[789],[24,2987,2989],{"className":2988,"style":794},[793],[24,2990,2991,2994],{"style":797},[24,2992],{"className":2993,"style":802},[801],[24,2995,2997],{"className":2996},[806,807,808,809],[24,2998,706],{"className":2999},[813,809],[501,3001,3002,3039,146,3253,150],{},[19,3003,3004,3005,496],{},"Length ",[24,3006,3008,3024],{"className":3007},[27],[24,3009,3011],{"className":3010},[31],[33,3012,3013],{"xmlns":35},[37,3014,3015,3021],{},[40,3016,3017,3019],{},[43,3018,92],{},[43,3020,45],{},[47,3022,3023],{"encoding":49},"BP",[24,3025,3027],{"className":3026,"ariaHidden":54},[53],[24,3028,3030,3033,3036],{"className":3029},[58],[24,3031],{"className":3032,"style":63},[62],[24,3034,92],{"className":3035,"style":126},[67,68],[24,3037,45],{"className":3038,"style":69},[67,68],[24,3040,3042,3089],{"className":3041},[27],[24,3043,3045],{"className":3044},[31],[33,3046,3047],{"xmlns":35},[37,3048,3049,3086],{},[40,3050,3051,3053,3055,3057,3079,3081,3084],{},[43,3052,92],{},[43,3054,45],{},[87,3056,696],{},[1113,3058,3059,3071],{},[40,3060,3061,3063,3065,3067,3069],{},[43,3062,85],{},[43,3064,92],{},[43,3066,1129],{},[87,3068,1107],{},[43,3070,1031],{},[40,3072,3073,3075,3077],{},[43,3074,1129],{},[87,3076,1107],{},[43,3078,357],{},[87,3080,696],{},[701,3082,3083],{},"8613.32",[2263,3085,2265],{},[47,3087,3088],{"encoding":49},"BP = \\frac{AB \\sin \\theta}{\\sin \\alpha} = 8613.32\\text{ m}",[24,3090,3092,3113,3238],{"className":3091,"ariaHidden":54},[53],[24,3093,3095,3098,3101,3104,3107,3110],{"className":3094},[58],[24,3096],{"className":3097,"style":63},[62],[24,3099,92],{"className":3100,"style":126},[67,68],[24,3102,45],{"className":3103,"style":69},[67,68],[24,3105],{"className":3106,"style":752},[121],[24,3108,696],{"className":3109},[756],[24,3111],{"className":3112,"style":752},[121],[24,3114,3116,3120,3229,3232,3235],{"className":3115},[58],[24,3117],{"className":3118,"style":3119},[62],"height:1.2251em;vertical-align:-0.345em;",[24,3121,3123,3126,3226],{"className":3122},[67],[24,3124],{"className":3125},[834,1209],[24,3127,3129],{"className":3128},[1113],[24,3130,3132,3218],{"className":3131},[785,1216],[24,3133,3135,3215],{"className":3134},[789],[24,3136,3139,3168,3176],{"className":3137,"style":3138},[793],"height:0.8801em;",[24,3140,3141,3144],{"style":1226},[24,3142],{"className":3143,"style":892},[801],[24,3145,3147],{"className":3146},[806,807,808,809],[24,3148,3150,3162,3165],{"className":3149},[67,809],[24,3151,3153,3156,3159],{"className":3152},[1180,809],[24,3154,1242],{"className":3155},[809],[24,3157,1246],{"className":3158},[809],[24,3160,1250],{"className":3161},[809],[24,3163],{"className":3164,"style":1254},[121,809],[24,3166,357],{"className":3167,"style":374},[67,68,809],[24,3169,3170,3173],{"style":1260},[24,3171],{"className":3172,"style":892},[801],[24,3174],{"className":3175,"style":1268},[1267],[24,3177,3179,3182],{"style":3178},"top:-3.394em;",[24,3180],{"className":3181,"style":892},[801],[24,3183,3185],{"className":3184},[806,807,808,809],[24,3186,3188,3191,3194,3197,3209,3212],{"className":3187},[67,809],[24,3189,85],{"className":3190},[67,68,809],[24,3192,92],{"className":3193,"style":126},[67,68,809],[24,3195],{"className":3196,"style":1254},[121,809],[24,3198,3200,3203,3206],{"className":3199},[1180,809],[24,3201,1242],{"className":3202},[809],[24,3204,1246],{"className":3205},[809],[24,3207,1250],{"className":3208},[809],[24,3210],{"className":3211,"style":1254},[121,809],[24,3213,1031],{"className":3214,"style":1048},[67,68,809],[24,3216,1408],{"className":3217},[1407],[24,3219,3221],{"className":3220},[789],[24,3222,3224],{"className":3223,"style":1459},[793],[24,3225],{},[24,3227],{"className":3228},[913,1209],[24,3230],{"className":3231,"style":752},[121],[24,3233,696],{"className":3234},[756],[24,3236],{"className":3237,"style":752},[121],[24,3239,3241,3244,3247],{"className":3240},[58],[24,3242],{"className":3243,"style":2352},[62],[24,3245,3083],{"className":3246},[67],[24,3248,3250],{"className":3249},[67,2359],[24,3251,2265],{"className":3252},[67],[24,3254,916],{},[501,3256,3257,3288,3289,3322,3323,3356,3357,3385,3386,146,3414,150],{},[19,3258,3259,3260,496],{},"Coordinates of ",[24,3261,3263,3276],{"className":3262},[27],[24,3264,3266],{"className":3265},[31],[33,3267,3268],{"xmlns":35},[37,3269,3270,3274],{},[40,3271,3272],{},[43,3273,45],{},[47,3275,45],{"encoding":49},[24,3277,3279],{"className":3278,"ariaHidden":54},[53],[24,3280,3282,3285],{"className":3281},[58],[24,3283],{"className":3284,"style":63},[62],[24,3286,45],{"className":3287,"style":69},[67,68]," Apply bearing ",[24,3290,3292,3307],{"className":3291},[27],[24,3293,3295],{"className":3294},[31],[33,3296,3297],{"xmlns":35},[37,3298,3299,3305],{},[40,3300,3301,3303],{},[43,3302,92],{},[43,3304,45],{},[47,3306,3023],{"encoding":49},[24,3308,3310],{"className":3309,"ariaHidden":54},[53],[24,3311,3313,3316,3319],{"className":3312},[58],[24,3314],{"className":3315,"style":63},[62],[24,3317,92],{"className":3318,"style":126},[67,68],[24,3320,45],{"className":3321,"style":69},[67,68]," and length ",[24,3324,3326,3341],{"className":3325},[27],[24,3327,3329],{"className":3328},[31],[33,3330,3331],{"xmlns":35},[37,3332,3333,3339],{},[40,3334,3335,3337],{},[43,3336,92],{},[43,3338,45],{},[47,3340,3023],{"encoding":49},[24,3342,3344],{"className":3343,"ariaHidden":54},[53],[24,3345,3347,3350,3353],{"className":3346},[58],[24,3348],{"className":3349,"style":63},[62],[24,3351,92],{"className":3352,"style":126},[67,68],[24,3354,45],{"className":3355,"style":69},[67,68]," to the coordinates of ",[24,3358,3360,3373],{"className":3359},[27],[24,3361,3363],{"className":3362},[31],[33,3364,3365],{"xmlns":35},[37,3366,3367,3371],{},[40,3368,3369],{},[43,3370,92],{},[47,3372,92],{"encoding":49},[24,3374,3376],{"className":3375,"ariaHidden":54},[53],[24,3377,3379,3382],{"className":3378},[58],[24,3380],{"className":3381,"style":63},[62],[24,3383,92],{"className":3384,"style":126},[67,68]," to find ",[24,3387,3389,3402],{"className":3388},[27],[24,3390,3392],{"className":3391},[31],[33,3393,3394],{"xmlns":35},[37,3395,3396,3400],{},[40,3397,3398],{},[43,3399,45],{},[47,3401,45],{"encoding":49},[24,3403,3405],{"className":3404,"ariaHidden":54},[53],[24,3406,3408,3411],{"className":3407},[58],[24,3409],{"className":3410,"style":63},[62],[24,3412,45],{"className":3413,"style":69},[67,68],[24,3415,916],{},[11,3417,3419],{"id":3418},"formula-breakdown","Formula Breakdown",[16,3421,3422,3425,3426,3454,3455,541,3501,3529,3530,3616,3617,3718,3719,3758,3759,917,3762,3765,3766,146,3794,3797,3798,541,3844,3872,3873,150],{},[19,3423,3424],{},"The \"Circle of Danger\":","\nMathematically, if ",[24,3427,3429,3442],{"className":3428},[27],[24,3430,3432],{"className":3431},[31],[33,3433,3434],{"xmlns":35},[37,3435,3436,3440],{},[40,3437,3438],{},[43,3439,45],{},[47,3441,45],{"encoding":49},[24,3443,3445],{"className":3444,"ariaHidden":54},[53],[24,3446,3448,3451],{"className":3447},[58],[24,3449],{"className":3450,"style":63},[62],[24,3452,45],{"className":3453,"style":69},[67,68]," lies on the circle formed by ",[24,3456,3458,3477],{"className":3457},[27],[24,3459,3461],{"className":3460},[31],[33,3462,3463],{"xmlns":35},[37,3464,3465,3475],{},[40,3466,3467,3469,3471,3473],{},[43,3468,85],{},[87,3470,89],{"separator":54},[43,3472,92],{},[87,3474,89],{"separator":54},[47,3476,1677],{"encoding":49},[24,3478,3480],{"className":3479,"ariaHidden":54},[53],[24,3481,3483,3486,3489,3492,3495,3498],{"className":3482},[58],[24,3484],{"className":3485,"style":110},[62],[24,3487,85],{"className":3488},[67,68],[24,3490,89],{"className":3491},[117],[24,3493],{"className":3494,"style":122},[121],[24,3496,92],{"className":3497,"style":126},[67,68],[24,3499,89],{"className":3500},[117],[24,3502,3504,3517],{"className":3503},[27],[24,3505,3507],{"className":3506},[31],[33,3508,3509],{"xmlns":35},[37,3510,3511,3515],{},[40,3512,3513],{},[43,3514,97],{},[47,3516,97],{"encoding":49},[24,3518,3520],{"className":3519,"ariaHidden":54},[53],[24,3521,3523,3526],{"className":3522},[58],[24,3524],{"className":3525,"style":63},[62],[24,3527,97],{"className":3528,"style":136},[67,68],", then ",[24,3531,3533,3556],{"className":3532},[27],[24,3534,3536],{"className":3535},[31],[33,3537,3538],{"xmlns":35},[37,3539,3540,3553],{},[40,3541,3542,3544,3546],{},[43,3543,665],{},[87,3545,696],{},[698,3547,3548,3551],{},[701,3549,3550],{},"180",[87,3552,706],{},[47,3554,3555],{"encoding":49},"S = 180^\\circ",[24,3557,3559,3577],{"className":3558,"ariaHidden":54},[53],[24,3560,3562,3565,3568,3571,3574],{"className":3561},[58],[24,3563],{"className":3564,"style":63},[62],[24,3566,665],{"className":3567,"style":276},[67,68],[24,3569],{"className":3570,"style":752},[121],[24,3572,696],{"className":3573},[756],[24,3575],{"className":3576,"style":752},[121],[24,3578,3580,3583,3587],{"className":3579},[58],[24,3581],{"className":3582,"style":794},[62],[24,3584,3586],{"className":3585},[67],"18",[24,3588,3590,3593],{"className":3589},[67],[24,3591,777],{"className":3592},[67],[24,3594,3596],{"className":3595},[781],[24,3597,3599],{"className":3598},[785],[24,3600,3602],{"className":3601},[789],[24,3603,3605],{"className":3604,"style":794},[793],[24,3606,3607,3610],{"style":797},[24,3608],{"className":3609,"style":802},[801],[24,3611,3613],{"className":3612},[806,807,808,809],[24,3614,706],{"className":3615},[813,809],". Since ",[24,3618,3620,3650],{"className":3619},[27],[24,3621,3623],{"className":3622},[31],[33,3624,3625],{"xmlns":35},[37,3626,3627,3647],{},[40,3628,3629,3631,3633,3635,3641,3643,3645],{},[43,3630,1129],{},[87,3632,1107],{},[87,3634,713],{"stretchy":712},[698,3636,3637,3639],{},[701,3638,3550],{},[87,3640,706],{},[87,3642,733],{"stretchy":712},[87,3644,696],{},[701,3646,777],{},[47,3648,3649],{"encoding":49},"\\sin(180^\\circ) = 0",[24,3651,3653,3709],{"className":3652,"ariaHidden":54},[53],[24,3654,3656,3659,3662,3665,3668,3697,3700,3703,3706],{"className":3655},[58],[24,3657],{"className":3658,"style":830},[62],[24,3660,1129],{"className":3661},[1180],[24,3663,713],{"className":3664},[834],[24,3666,3586],{"className":3667},[67],[24,3669,3671,3674],{"className":3670},[67],[24,3672,777],{"className":3673},[67],[24,3675,3677],{"className":3676},[781],[24,3678,3680],{"className":3679},[785],[24,3681,3683],{"className":3682},[789],[24,3684,3686],{"className":3685,"style":794},[793],[24,3687,3688,3691],{"style":797},[24,3689],{"className":3690,"style":802},[801],[24,3692,3694],{"className":3693},[806,807,808,809],[24,3695,706],{"className":3696},[813,809],[24,3698,733],{"className":3699},[913],[24,3701],{"className":3702,"style":752},[121],[24,3704,696],{"className":3705},[756],[24,3707],{"className":3708,"style":752},[121],[24,3710,3712,3715],{"className":3711},[58],[24,3713],{"className":3714,"style":2352},[62],[24,3716,777],{"className":3717},[67],", the formula for ",[24,3720,3722,3740],{"className":3721},[27],[24,3723,3725],{"className":3724},[31],[33,3726,3727],{"xmlns":35},[37,3728,3729,3737],{},[40,3730,3731,3733,3735],{},[43,3732,1104],{},[87,3734,1107],{},[43,3736,1031],{},[47,3738,3739],{"encoding":49},"\\cot \\theta",[24,3741,3743],{"className":3742,"ariaHidden":54},[53],[24,3744,3746,3749,3752,3755],{"className":3745},[58],[24,3747],{"className":3748,"style":1044},[62],[24,3750,1104],{"className":3751},[1180],[24,3753],{"className":3754,"style":122},[121],[24,3756,1031],{"className":3757,"style":1048},[67,68]," involves division by zero and becomes undefined ",[24,3760,3761],{},"62, 63",[19,3763,3764],{},"Best Geometry:"," Site point ",[24,3767,3769,3782],{"className":3768},[27],[24,3770,3772],{"className":3771},[31],[33,3773,3774],{"xmlns":35},[37,3775,3776,3780],{},[40,3777,3778],{},[43,3779,45],{},[47,3781,45],{"encoding":49},[24,3783,3785],{"className":3784,"ariaHidden":54},[53],[24,3786,3788,3791],{"className":3787},[58],[24,3789],{"className":3790,"style":63},[62],[24,3792,45],{"className":3793,"style":69},[67,68],[139,3795,3796],{},"inside"," the triangle formed by ",[24,3799,3801,3820],{"className":3800},[27],[24,3802,3804],{"className":3803},[31],[33,3805,3806],{"xmlns":35},[37,3807,3808,3818],{},[40,3809,3810,3812,3814,3816],{},[43,3811,85],{},[87,3813,89],{"separator":54},[43,3815,92],{},[87,3817,89],{"separator":54},[47,3819,1677],{"encoding":49},[24,3821,3823],{"className":3822,"ariaHidden":54},[53],[24,3824,3826,3829,3832,3835,3838,3841],{"className":3825},[58],[24,3827],{"className":3828,"style":110},[62],[24,3830,85],{"className":3831},[67,68],[24,3833,89],{"className":3834},[117],[24,3836],{"className":3837,"style":122},[121],[24,3839,92],{"className":3840,"style":126},[67,68],[24,3842,89],{"className":3843},[117],[24,3845,3847,3860],{"className":3846},[27],[24,3848,3850],{"className":3849},[31],[33,3851,3852],{"xmlns":35},[37,3853,3854,3858],{},[40,3855,3856],{},[43,3857,97],{},[47,3859,97],{"encoding":49},[24,3861,3863],{"className":3862,"ariaHidden":54},[53],[24,3864,3866,3869],{"className":3865},[58],[24,3867],{"className":3868,"style":63},[62],[24,3870,97],{"className":3871,"style":136},[67,68]," to avoid this ",[24,3874,3875],{},"63",[11,3877,3879],{"id":3878},"practical-tips","Practical Tips",[3881,3882,3883,4023],"ul",{},[501,3884,3885,3888,3889,493,4020,150],{},[19,3886,3887],{},"Total Station Advantage:"," Modern total stations have a built-in \"Resection\" (or \"Free Station\") program. You simply select the points from the internal database, sight them, and the instrument displays your coordinates and the standard error (",[24,3890,3892,3920],{"className":3891},[27],[24,3893,3895],{"className":3894},[31],[33,3896,3897],{"xmlns":35},[37,3898,3899,3917],{},[40,3900,3901,3908,3910],{},[2241,3902,3903,3905],{},[43,3904,665],{},[43,3906,3907],{},"x",[87,3909,89],{"separator":54},[2241,3911,3912,3914],{},[43,3913,665],{},[43,3915,3916],{},"y",[47,3918,3919],{"encoding":49},"S_x, S_y",[24,3921,3923],{"className":3922,"ariaHidden":54},[53],[24,3924,3926,3930,3972,3975,3978],{"className":3925},[58],[24,3927],{"className":3928,"style":3929},[62],"height:0.9694em;vertical-align:-0.2861em;",[24,3931,3933,3936],{"className":3932},[67],[24,3934,665],{"className":3935,"style":276},[67,68],[24,3937,3939],{"className":3938},[781],[24,3940,3942,3964],{"className":3941},[785,1216],[24,3943,3945,3961],{"className":3944},[789],[24,3946,3949],{"className":3947,"style":3948},[793],"height:0.1514em;",[24,3950,3952,3955],{"style":3951},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[24,3953],{"className":3954,"style":802},[801],[24,3956,3958],{"className":3957},[806,807,808,809],[24,3959,3907],{"className":3960},[67,68,809],[24,3962,1408],{"className":3963},[1407],[24,3965,3967],{"className":3966},[789],[24,3968,3970],{"className":3969,"style":2334},[793],[24,3971],{},[24,3973,89],{"className":3974},[117],[24,3976],{"className":3977,"style":122},[121],[24,3979,3981,3984],{"className":3980},[67],[24,3982,665],{"className":3983,"style":276},[67,68],[24,3985,3987],{"className":3986},[781],[24,3988,3990,4011],{"className":3989},[785,1216],[24,3991,3993,4008],{"className":3992},[789],[24,3994,3996],{"className":3995,"style":3948},[793],[24,3997,3998,4001],{"style":3951},[24,3999],{"className":4000,"style":802},[801],[24,4002,4004],{"className":4003},[806,807,808,809],[24,4005,3916],{"className":4006,"style":4007},[67,68,809],"margin-right:0.0359em;",[24,4009,1408],{"className":4010},[1407],[24,4012,4014],{"className":4013},[789],[24,4015,4018],{"className":4016,"style":4017},[793],"height:0.2861em;",[24,4019],{},[24,4021,4022],{},"64",[501,4024,4025,4028,4029,4067,4068,4106,4107,150],{},[19,4026,4027],{},"Commensurate Accuracy:"," Ensure the points you sight are at a similar distance. Sighting one point at ",[24,4030,4032,4049],{"className":4031},[27],[24,4033,4035],{"className":4034},[31],[33,4036,4037],{"xmlns":35},[37,4038,4039,4046],{},[40,4040,4041,4044],{},[701,4042,4043],{},"50",[2263,4045,2265],{},[47,4047,4048],{"encoding":49},"50\\text{ m}",[24,4050,4052],{"className":4051,"ariaHidden":54},[53],[24,4053,4055,4058,4061],{"className":4054},[58],[24,4056],{"className":4057,"style":2352},[62],[24,4059,4043],{"className":4060},[67],[24,4062,4064],{"className":4063},[67,2359],[24,4065,2265],{"className":4066},[67]," and another at ",[24,4069,4071,4088],{"className":4070},[27],[24,4072,4074],{"className":4073},[31],[33,4075,4076],{"xmlns":35},[37,4077,4078,4085],{},[40,4079,4080,4082],{},[701,4081,2708],{},[2263,4083,4084],{}," km",[47,4086,4087],{"encoding":49},"5\\text{ km}",[24,4089,4091],{"className":4090,"ariaHidden":54},[53],[24,4092,4094,4097,4100],{"className":4093},[58],[24,4095],{"className":4096,"style":1044},[62],[24,4098,2708],{"className":4099},[67],[24,4101,4103],{"className":4102},[67,2359],[24,4104,4084],{"className":4105},[67]," introduces centring errors that degrade the resection ",[24,4108,4109],{},"65",[11,4111,4113],{"id":4112},"common-mistakes","Common Mistakes",[3881,4115,4116,4227],{},[501,4117,4118,2867,4121,541,4167,4195,4196,4224,4225,150],{},[19,4119,4120],{},"Sighting in a Straight Line:",[24,4122,4124,4143],{"className":4123},[27],[24,4125,4127],{"className":4126},[31],[33,4128,4129],{"xmlns":35},[37,4130,4131,4141],{},[40,4132,4133,4135,4137,4139],{},[43,4134,85],{},[87,4136,89],{"separator":54},[43,4138,92],{},[87,4140,89],{"separator":54},[47,4142,1677],{"encoding":49},[24,4144,4146],{"className":4145,"ariaHidden":54},[53],[24,4147,4149,4152,4155,4158,4161,4164],{"className":4148},[58],[24,4150],{"className":4151,"style":110},[62],[24,4153,85],{"className":4154},[67,68],[24,4156,89],{"className":4157},[117],[24,4159],{"className":4160,"style":122},[121],[24,4162,92],{"className":4163,"style":126},[67,68],[24,4165,89],{"className":4166},[117],[24,4168,4170,4183],{"className":4169},[27],[24,4171,4173],{"className":4172},[31],[33,4174,4175],{"xmlns":35},[37,4176,4177,4181],{},[40,4178,4179],{},[43,4180,97],{},[47,4182,97],{"encoding":49},[24,4184,4186],{"className":4185,"ariaHidden":54},[53],[24,4187,4189,4192],{"className":4188},[58],[24,4190],{"className":4191,"style":63},[62],[24,4193,97],{"className":4194,"style":136},[67,68]," lie in a perfectly straight line and ",[24,4197,4199,4212],{"className":4198},[27],[24,4200,4202],{"className":4201},[31],[33,4203,4204],{"xmlns":35},[37,4205,4206,4210],{},[40,4207,4208],{},[43,4209,45],{},[47,4211,45],{"encoding":49},[24,4213,4215],{"className":4214,"ariaHidden":54},[53],[24,4216,4218,4221],{"className":4217},[58],[24,4219],{"className":4220,"style":63},[62],[24,4222,45],{"className":4223,"style":69},[67,68]," is also near that line, the solution becomes very \"weak\" and unreliable ",[24,4226,3761],{},[501,4228,4229,4232,4233,150],{},[19,4230,4231],{},"Poor Point Identification:"," Sighting the \"wrong\" church spire or a nearby station with a similar name will result in a coordinates failure that is often hard to spot without a fourth point ",[24,4234,4235],{},"66, 67",[11,4237,4239],{"id":4238},"faq","FAQ",[4241,4242,4243,4281,4292],"accordion",{},[4244,4245,4247],"accordion-item",{"label":4246},"Is resection less accurate than traversing?",[16,4248,4249,4250,4278,4279,150],{},"Generally, yes. Since only point ",[24,4251,4253,4266],{"className":4252},[27],[24,4254,4256],{"className":4255},[31],[33,4257,4258],{"xmlns":35},[37,4259,4260,4264],{},[40,4261,4262],{},[43,4263,45],{},[47,4265,45],{"encoding":49},[24,4267,4269],{"className":4268,"ariaHidden":54},[53],[24,4270,4272,4275],{"className":4271},[58],[24,4273],{"className":4274,"style":63},[62],[24,4276,45],{"className":4277,"style":69},[67,68]," is occupied, there is no way to cancel out centring errors at the target stations. It is considered a \"weaker\" solution than intersection ",[24,4280,160],{},[4244,4282,4284],{"label":4283},"How many points do I really need?",[16,4285,4286,4287,146,4290,150],{},"Three points is the minimum for a solution. Four points is the minimum for a ",[19,4288,4289],{},"check",[24,4291,2096],{},[4244,4293,4295],{"label":4294},"What is 'Collins Point'?",[16,4296,4297,4298,4328,4329,150],{},"It is a geometric construction used to solve the three-point problem manually before the advent of pocket calculators. It involves creating an auxiliary point (",[24,4299,4301,4315],{"className":4300},[27],[24,4302,4304],{"className":4303},[31],[33,4305,4306],{"xmlns":35},[37,4307,4308,4313],{},[40,4309,4310],{},[43,4311,4312],{},"H",[47,4314,4312],{"encoding":49},[24,4316,4318],{"className":4317,"ariaHidden":54},[53],[24,4319,4321,4324],{"className":4320},[58],[24,4322],{"className":4323,"style":63},[62],[24,4325,4312],{"className":4326,"style":4327},[67,68],"margin-right:0.0813em;",") on the circle ",[24,4330,160],{},[11,4332,4334],{"id":4333},"conclusion","Conclusion",[16,4336,4337],{},"Resection is the surveyor's \"escape hatch\" for difficult site conditions. By mastering the cotangent formula and avoiding the \"Circle of Danger,\" you can turn any clear patch of ground into a high-precision control station.",[11,4339,4341],{"id":4340},"references","References",[16,4343,4344,4345,4348,4349,150],{},"Schofield, W. (2001). ",[139,4346,4347],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann. ",[24,4350,4351],{},"16, 17",{"title":4353,"searchDepth":4354,"depth":4354,"links":4355},"",2,[4356,4357,4358,4359,4363,4369,4370,4371,4372,4373,4374,4375],{"id":13,"depth":4354,"text":14},{"id":153,"depth":4354,"text":154},{"id":164,"depth":4354,"text":165},{"id":231,"depth":4354,"text":232,"children":4360},[4361],{"id":236,"depth":4362,"text":237},3,{"id":1643,"depth":4354,"text":1644,"children":4364},[4365,4366,4367,4368],{"id":1650,"depth":4362,"text":1651},{"id":1735,"depth":4362,"text":1736},{"id":1945,"depth":4362,"text":1946},{"id":2059,"depth":4362,"text":2060},{"id":2099,"depth":4354,"text":2100},{"id":3418,"depth":4354,"text":3419},{"id":3878,"depth":4354,"text":3879},{"id":4112,"depth":4354,"text":4113},{"id":4238,"depth":4354,"text":4239},{"id":4333,"depth":4354,"text":4334},{"id":4340,"depth":4354,"text":4341},"Positioning","A technical guide to coordinate determination using resection, including analytical solutions and the 'Circle of Danger' geometry.","md","\u002Fimages\u002Fresection-surveying.jpg",{"authorCredentials":4381,"commentCount":4382,"tags":4383,"relatedTools":4387,"relatedArticles":4390,"sources":4393},"LS",0,[21,4376,4384,4385,4386],"Control Surveys","Three-Point Problem","Math",[4388,4389],"Total Station","Theodolite",[4391,4392],"traverse-adjustment-math","intersection-methods",[4394],{"title":4395,"type":4396,"note":4397},"Engineering Surveying (Fifth Edition)","Book","Primary source for the analytical and geometric solutions to resection.",true,"\u002Fblog\u002Fresection-three-point-problem","2024-03-29",18,{"title":5,"description":4377},{"loc":4399},"blog\u002F10.resection-three-point-problem","iwwK3sS3RmICaKnj3yoZMfr5Ihh7uQSa1Nb68cJ-eEU",[4407,4412],{"title":4408,"path":4409,"stem":4410,"description":4411,"children":-1},"EDM Calibration: Eliminating Zero, Scale, and Cyclic Errors","\u002Fblog\u002Fedm-errors-and-calibration","blog\u002F9.edm-errors-and-calibration","A technical guide to understanding and correcting the three primary error sources in Electromagnetic Distance Measurement (EDM) equipment.",{"title":4413,"path":4414,"stem":4415,"description":4416,"children":-1},"Mass-Haul Diagrams: The Economics of Earthwork Distribution","\u002Fblog\u002Fmass-haul-diagrams-earthwork-economics","blog\u002F11.mass-haul-diagrams-earthwork-economics","A technical guide to using Mass-Haul Diagrams (MHD) for optimizing earthwork movement, calculating overhaul, and managing project costs.",1783851951008]