[{"data":1,"prerenderedAt":5477},["ShallowReactive",2],{"blog-\u002Fblog\u002Fspherical-excess-large-networks":3,"blog-surround-\u002Fblog\u002Fspherical-excess-large-networks":5466},{"id":4,"title":5,"author":6,"body":7,"category":5433,"description":5434,"extension":5435,"image":5436,"meta":5437,"navigation":5458,"path":5459,"publishedAt":5460,"readTime":5461,"seo":5462,"sitemap":5463,"stem":5464,"__hash__":5465},"blog\u002Fblog\u002F24.spherical-excess-large-networks.md","Spherical Excess: Adjusting Large-Scale Control Networks","Surveyiiing Expert",{"type":8,"value":9,"toc":5408},"minimark",[10,15,333,337,468,472,478,482,487,1101,1317,1321,1486,1490,2232,4214,4218,4654,4658,4871,4875,5142,5146,5323,5327,5396,5400],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,115,116,190,191,252,253,332],"p",{},"In small-scale engineering surveys, we treat the Earth as a flat plane, and the sum of the angles in a triangle equals exactly ",[20,21,24,54],"span",{"className":22},[23],"katex",[20,25,28],{"className":26},[27],"katex-mathml",[29,30,32],"math",{"xmlns":31},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[33,34,35,49],"semantics",{},[36,37,38],"mrow",{},[39,40,41,45],"msup",{},[42,43,44],"mn",{},"180",[46,47,48],"mo",{},"∘",[50,51,53],"annotation",{"encoding":52},"application\u002Fx-tex","180^\\circ",[20,55,59],{"className":56,"ariaHidden":58},[57],"katex-html","true",[20,60,63,68,73],{"className":61},[62],"base",[20,64],{"className":65,"style":67},[66],"strut","height:0.6741em;",[20,69,72],{"className":70},[71],"mord","18",[20,74,76,80],{"className":75},[71],[20,77,79],{"className":78},[71],"0",[20,81,84],{"className":82},[83],"msupsub",[20,85,88],{"className":86},[87],"vlist-t",[20,89,92],{"className":90},[91],"vlist-r",[20,93,96],{"className":94,"style":67},[95],"vlist",[20,97,99,104],{"style":98},"top:-3.063em;margin-right:0.05em;",[20,100],{"className":101,"style":103},[102],"pstrut","height:2.7em;",[20,105,111],{"className":106},[107,108,109,110],"sizing","reset-size6","size3","mtight",[20,112,48],{"className":113},[114,110],"mbin",". However, as the area of the survey grows—typically exceeding ",[20,117,119,143],{"className":118},[23],[20,120,122],{"className":121},[27],[29,123,124],{"xmlns":31},[33,125,126,140],{},[36,127,128,131],{},[42,129,130],{},"200",[39,132,133,137],{},[134,135,136],"mtext",{}," km",[42,138,139],{},"2",[50,141,142],{"encoding":52},"200 \\text{ km}^2",[20,144,146],{"className":145,"ariaHidden":58},[57],[20,147,149,153,156],{"className":148},[62],[20,150],{"className":151,"style":152},[66],"height:0.8984em;",[20,154,130],{"className":155},[71],[20,157,159,166],{"className":158},[71],[20,160,163],{"className":161},[71,162],"text",[20,164,136],{"className":165},[71],[20,167,169],{"className":168},[83],[20,170,172],{"className":171},[87],[20,173,175],{"className":174},[91],[20,176,178],{"className":177,"style":152},[95],[20,179,181,184],{"style":180},"top:-3.1473em;margin-right:0.05em;",[20,182],{"className":183,"style":103},[102],[20,185,187],{"className":186},[107,108,109,110],[20,188,139],{"className":189},[71,110],"—the Earth’s curvature becomes significant. The angles of a triangle observed on the curved surface of the ellipsoid will sum to more than ",[20,192,194,211],{"className":193},[23],[20,195,197],{"className":196},[27],[29,198,199],{"xmlns":31},[33,200,201,209],{},[36,202,203],{},[39,204,205,207],{},[42,206,44],{},[46,208,48],{},[50,210,53],{"encoding":52},[20,212,214],{"className":213,"ariaHidden":58},[57],[20,215,217,220,223],{"className":216},[62],[20,218],{"className":219,"style":67},[66],[20,221,72],{"className":222},[71],[20,224,226,229],{"className":225},[71],[20,227,79],{"className":228},[71],[20,230,232],{"className":231},[83],[20,233,235],{"className":234},[87],[20,236,238],{"className":237},[91],[20,239,241],{"className":240,"style":67},[95],[20,242,243,246],{"style":98},[20,244],{"className":245,"style":103},[102],[20,247,249],{"className":248},[107,108,109,110],[20,250,48],{"className":251},[114,110],". This additional amount is known as ",[254,255,256,257,331],"strong",{},"Spherical Excess (",[20,258,260,286],{"className":259},[23],[20,261,263],{"className":262},[27],[29,264,265],{"xmlns":31},[33,266,267,283],{},[36,268,269],{},[39,270,271,275],{},[272,273,274],"mi",{},"E",[36,276,277,281],{},[46,278,280],{"mathvariant":279},"normal","′",[46,282,280],{"mathvariant":279},[50,284,285],{"encoding":52},"E''",[20,287,289],{"className":288,"ariaHidden":58},[57],[20,290,292,296],{"className":291},[62],[20,293],{"className":294,"style":295},[66],"height:0.7519em;",[20,297,299,304],{"className":298},[71],[20,300,274],{"className":301,"style":303},[71,302],"mathnormal","margin-right:0.0576em;",[20,305,307],{"className":306},[83],[20,308,310],{"className":309},[87],[20,311,313],{"className":312},[91],[20,314,316],{"className":315,"style":295},[95],[20,317,318,321],{"style":98},[20,319],{"className":320,"style":103},[102],[20,322,324],{"className":323},[107,108,109,110],[20,325,327],{"className":326},[71,110],[20,328,330],{"className":329},[71,110],"′′",")"," .",[11,334,336],{"id":335},"why-this-matters","Why This Matters",[16,338,339,340,401,402,467],{},"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 ",[20,341,343,360],{"className":342},[23],[20,344,346],{"className":345},[27],[29,347,348],{"xmlns":31},[33,349,350,358],{},[36,351,352],{},[39,353,354,356],{},[42,355,44],{},[46,357,48],{},[50,359,53],{"encoding":52},[20,361,363],{"className":362,"ariaHidden":58},[57],[20,364,366,369,372],{"className":365},[62],[20,367],{"className":368,"style":67},[66],[20,370,72],{"className":371},[71],[20,373,375,378],{"className":374},[71],[20,376,79],{"className":377},[71],[20,379,381],{"className":380},[83],[20,382,384],{"className":383},[87],[20,385,387],{"className":386},[91],[20,388,390],{"className":389,"style":67},[95],[20,391,392,395],{"style":98},[20,393],{"className":394,"style":103},[102],[20,396,398],{"className":397},[107,108,109,110],[20,399,48],{"className":400},[114,110]," without accounting for ",[20,403,405,426],{"className":404},[23],[20,406,408],{"className":407},[27],[29,409,410],{"xmlns":31},[33,411,412,424],{},[36,413,414],{},[39,415,416,418],{},[272,417,274],{},[36,419,420,422],{},[46,421,280],{"mathvariant":279},[46,423,280],{"mathvariant":279},[50,425,285],{"encoding":52},[20,427,429],{"className":428,"ariaHidden":58},[57],[20,430,432,435],{"className":431},[62],[20,433],{"className":434,"style":295},[66],[20,436,438,441],{"className":437},[71],[20,439,274],{"className":440,"style":303},[71,302],[20,442,444],{"className":443},[83],[20,445,447],{"className":446},[87],[20,448,450],{"className":449},[91],[20,451,453],{"className":452,"style":295},[95],[20,454,455,458],{"style":98},[20,456],{"className":457,"style":103},[102],[20,459,461],{"className":460},[107,108,109,110],[20,462,464],{"className":463},[71,110],[20,465,330],{"className":466},[71,110],", the resulting coordinates will be distorted, leading to misclosures in long-distance traverses or tunnel breakthroughs .",[11,469,471],{"id":470},"theory","Theory",[16,473,474,477],{},[254,475,476],{},"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 .",[11,479,481],{"id":480},"mathematical-principles","Mathematical Principles",[483,484,486],"h3",{"id":485},"_1-the-spherical-excess-formula","1. The Spherical Excess Formula",[16,488,489,490,555,556,1100],{},"The excess (",[20,491,493,514],{"className":492},[23],[20,494,496],{"className":495},[27],[29,497,498],{"xmlns":31},[33,499,500,512],{},[36,501,502],{},[39,503,504,506],{},[272,505,274],{},[36,507,508,510],{},[46,509,280],{"mathvariant":279},[46,511,280],{"mathvariant":279},[50,513,285],{"encoding":52},[20,515,517],{"className":516,"ariaHidden":58},[57],[20,518,520,523],{"className":519},[62],[20,521],{"className":522,"style":295},[66],[20,524,526,529],{"className":525},[71],[20,527,274],{"className":528,"style":303},[71,302],[20,530,532],{"className":531},[83],[20,533,535],{"className":534},[87],[20,536,538],{"className":537},[91],[20,539,541],{"className":540,"style":295},[95],[20,542,543,546],{"style":98},[20,544],{"className":545,"style":103},[102],[20,547,549],{"className":548},[107,108,109,110],[20,550,552],{"className":551},[71,110],[20,553,330],{"className":554},[71,110],") in seconds of arc is proportional to the area of the triangle:\n",[20,557,559,664],{"className":558},[23],[20,560,562],{"className":561},[27],[29,563,564],{"xmlns":31},[33,565,566,661],{},[36,567,568,578,581,613,616,649,652,655,658],{},[39,569,570,572],{},[272,571,274],{},[36,573,574,576],{},[46,575,280],{"mathvariant":279},[46,577,280],{"mathvariant":279},[46,579,580],{},"=",[582,583,584,587],"mfrac",{},[134,585,586],{},"Area of Triangle",[36,588,589,596,599,602],{},[39,590,591,594],{},[272,592,593],{},"R",[42,595,139],{},[272,597,598],{},"sin",[46,600,601],{},"⁡",[39,603,604,607],{},[42,605,606],{},"1",[36,608,609,611],{},[46,610,280],{"mathvariant":279},[46,612,280],{"mathvariant":279},[46,614,615],{},"≈",[582,617,618,639],{},[36,619,620,623,626,628,630],{},[272,621,622],{},"a",[272,624,625],{},"b",[272,627,598],{},[46,629,601],{},[631,632,633,636],"mover",{"accent":58},[272,634,635],{},"C",[46,637,638],{},"^",[36,640,641,643],{},[42,642,139],{},[39,644,645,647],{},[272,646,593],{},[42,648,139],{},[46,650,651],{},"×",[42,653,654],{},"206",[46,656,657],{"separator":58},",",[42,659,660],{},"265",[50,662,663],{"encoding":52},"E'' = \\frac{\\text{Area of Triangle}}{R^2 \\sin 1''} \\approx \\frac{ab \\sin \\hat{C}}{2R^2} \\times 206,265",[20,665,667,717,906,1079],{"className":666,"ariaHidden":58},[57],[20,668,670,673,705,710,714],{"className":669},[62],[20,671],{"className":672,"style":295},[66],[20,674,676,679],{"className":675},[71],[20,677,274],{"className":678,"style":303},[71,302],[20,680,682],{"className":681},[83],[20,683,685],{"className":684},[87],[20,686,688],{"className":687},[91],[20,689,691],{"className":690,"style":295},[95],[20,692,693,696],{"style":98},[20,694],{"className":695,"style":103},[102],[20,697,699],{"className":698},[107,108,109,110],[20,700,702],{"className":701},[71,110],[20,703,330],{"className":704},[71,110],[20,706],{"className":707,"style":709},[708],"mspace","margin-right:0.2778em;",[20,711,580],{"className":712},[713],"mrel",[20,715],{"className":716,"style":709},[708],[20,718,720,724,897,900,903],{"className":719},[62],[20,721],{"className":722,"style":723},[66],"height:1.2772em;vertical-align:-0.345em;",[20,725,727,732,893],{"className":726},[71],[20,728],{"className":729},[730,731],"mopen","nulldelimiter",[20,733,735],{"className":734},[582],[20,736,739,884],{"className":737},[87,738],"vlist-t2",[20,740,742,879],{"className":741},[91],[20,743,746,850,861],{"className":744,"style":745},[95],"height:0.9322em;",[20,747,749,753],{"style":748},"top:-2.655em;",[20,750],{"className":751,"style":752},[102],"height:3em;",[20,754,756],{"className":755},[107,108,109,110],[20,757,759,794,798,814,817],{"className":758},[71,110],[20,760,762,766],{"className":761},[71,110],[20,763,593],{"className":764,"style":765},[71,302,110],"margin-right:0.0077em;",[20,767,769],{"className":768},[83],[20,770,772],{"className":771},[87],[20,773,775],{"className":774},[91],[20,776,779],{"className":777,"style":778},[95],"height:0.7463em;",[20,780,782,786],{"style":781},"top:-2.786em;margin-right:0.0714em;",[20,783],{"className":784,"style":785},[102],"height:2.5em;",[20,787,791],{"className":788},[107,789,790,110],"reset-size3","size1",[20,792,139],{"className":793},[71,110],[20,795],{"className":796,"style":797},[708,110],"margin-right:0.1952em;",[20,799,802,806,810],{"className":800},[801,110],"mop",[20,803,805],{"className":804},[110],"s",[20,807,809],{"className":808},[110],"i",[20,811,813],{"className":812},[110],"n",[20,815],{"className":816,"style":797},[708,110],[20,818,820,823],{"className":819},[71,110],[20,821,606],{"className":822},[71,110],[20,824,826],{"className":825},[83],[20,827,829],{"className":828},[87],[20,830,832],{"className":831},[91],[20,833,836],{"className":834,"style":835},[95],"height:0.6828em;",[20,837,838,841],{"style":781},[20,839],{"className":840,"style":785},[102],[20,842,844],{"className":843},[107,789,790,110],[20,845,847],{"className":846},[71,110],[20,848,330],{"className":849},[71,110],[20,851,853,856],{"style":852},"top:-3.23em;",[20,854],{"className":855,"style":752},[102],[20,857],{"className":858,"style":860},[859],"frac-line","border-bottom-width:0.04em;",[20,862,864,867],{"style":863},"top:-3.4461em;",[20,865],{"className":866,"style":752},[102],[20,868,870],{"className":869},[107,108,109,110],[20,871,873],{"className":872},[71,110],[20,874,876],{"className":875},[71,162,110],[20,877,586],{"className":878},[71,110],[20,880,883],{"className":881},[882],"vlist-s","​",[20,885,887],{"className":886},[91],[20,888,891],{"className":889,"style":890},[95],"height:0.345em;",[20,892],{},[20,894],{"className":895},[896,731],"mclose",[20,898],{"className":899,"style":709},[708],[20,901,615],{"className":902},[713],[20,904],{"className":905,"style":709},[708],[20,907,909,913,1069,1073,1076],{"className":908},[62],[20,910],{"className":911,"style":912},[66],"height:1.4017em;vertical-align:-0.345em;",[20,914,916,919,1066],{"className":915},[71],[20,917],{"className":918},[730,731],[20,920,922],{"className":921},[582],[20,923,925,1058],{"className":924},[87,738],[20,926,928,1055],{"className":927},[91],[20,929,932,975,983],{"className":930,"style":931},[95],"height:1.0567em;",[20,933,934,937],{"style":748},[20,935],{"className":936,"style":752},[102],[20,938,940],{"className":939},[107,108,109,110],[20,941,943,946],{"className":942},[71,110],[20,944,139],{"className":945},[71,110],[20,947,949,952],{"className":948},[71,110],[20,950,593],{"className":951,"style":765},[71,302,110],[20,953,955],{"className":954},[83],[20,956,958],{"className":957},[87],[20,959,961],{"className":960},[91],[20,962,964],{"className":963,"style":778},[95],[20,965,966,969],{"style":781},[20,967],{"className":968,"style":785},[102],[20,970,972],{"className":971},[107,789,790,110],[20,973,139],{"className":974},[71,110],[20,976,977,980],{"style":852},[20,978],{"className":979,"style":752},[102],[20,981],{"className":982,"style":860},[859],[20,984,986,989],{"style":985},"top:-3.394em;",[20,987],{"className":988,"style":752},[102],[20,990,992],{"className":991},[107,108,109,110],[20,993,995,999,1002,1014,1017],{"className":994},[71,110],[20,996,998],{"className":997},[71,302,110],"ab",[20,1000],{"className":1001,"style":79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b",[20,1129,1131],{"className":1130,"ariaHidden":58},[57],[20,1132,1134,1138,1141,1144,1147],{"className":1133},[62],[20,1135],{"className":1136,"style":1137},[66],"height:0.8889em;vertical-align:-0.1944em;",[20,1139,622],{"className":1140},[71,302],[20,1142,657],{"className":1143},[1092],[20,1145],{"className":1146,"style":1096},[708],[20,1148,625],{"className":1149},[71,302]," = Lengths of two sides of the triangle .",[1105,1152,1153,1216],{},[20,1154,1156,1174],{"className":1155},[23],[20,1157,1159],{"className":1158},[27],[29,1160,1161],{"xmlns":31},[33,1162,1163,1171],{},[36,1164,1165],{},[631,1166,1167,1169],{"accent":58},[272,1168,635],{},[46,1170,638],{},[50,1172,1173],{"encoding":52},"\\hat{C}",[20,1175,1177],{"className":1176,"ariaHidden":58},[57],[20,1178,1180,1183],{"className":1179},[62],[20,1181],{"className":1182,"style":1030},[66],[20,1184,1186],{"className":1185},[71,1020],[20,1187,1189],{"className":1188},[87],[20,1190,1192],{"className":1191},[91],[20,1193,1195,1204],{"className":1194,"style":1030},[95],[20,1196,1198,1201],{"style":1197},"top:-3em;",[20,1199],{"className":1200,"style":752},[102],[20,1202,635],{"className":1203,"style":1040},[71,302],[20,1205,1207,1210],{"style":1206},"top:-3.2523em;",[20,1208],{"className":1209,"style":752},[102],[20,1211,1213],{"className":1212,"style":1051},[1050],[20,1214,638],{"className":1215},[71]," = The included observed angle .",[1105,1218,1219,1248,1249,1316],{},[20,1220,1222,1235],{"className":1221},[23],[20,1223,1225],{"className":1224},[27],[29,1226,1227],{"xmlns":31},[33,1228,1229,1233],{},[36,1230,1231],{},[272,1232,593],{},[50,1234,593],{"encoding":52},[20,1236,1238],{"className":1237,"ariaHidden":58},[57],[20,1239,1241,1245],{"className":1240},[62],[20,1242],{"className":1243,"style":1244},[66],"height:0.6833em;",[20,1246,593],{"className":1247,"style":765},[71,302]," = The mean radius of the Earth (",[20,1250,1252,1276],{"className":1251},[23],[20,1253,1255],{"className":1254},[27],[29,1256,1257],{"xmlns":31},[33,1258,1259,1273],{},[36,1260,1261,1263,1266,1268,1271],{},[46,1262,615],{},[42,1264,1265],{},"6",[46,1267,657],{"separator":58},[42,1269,1270],{},"370",[134,1272,136],{},[50,1274,1275],{"encoding":52},"\\approx 6,370 \\text{ km}",[20,1277,1279,1292],{"className":1278,"ariaHidden":58},[57],[20,1280,1282,1286,1289],{"className":1281},[62],[20,1283],{"className":1284,"style":1285},[66],"height:0.4831em;",[20,1287,615],{"className":1288},[713],[20,1290],{"className":1291,"style":709},[708],[20,1293,1295,1298,1301,1304,1307,1310],{"className":1294},[62],[20,1296],{"className":1297,"style":1137},[66],[20,1299,1265],{"className":1300},[71],[20,1302,657],{"className":1303},[1092],[20,1305],{"className":1306,"style":1096},[708],[20,1308,1270],{"className":1309},[71],[20,1311,1313],{"className":1312},[71,162],[20,1314,136],{"className":1315},[71],") .",[483,1318,1320],{"id":1319},"_2-angular-closure-requirement","2. Angular Closure Requirement",[16,1322,1323,1324],{},"In a major triangulation, the sum of the adjusted spheroidal angles should equal:\n",[20,1325,1327,1366],{"className":1326},[23],[20,1328,1330],{"className":1329},[27],[29,1331,1332],{"xmlns":31},[33,1333,1334,1363],{},[36,1335,1336,1339,1342,1344,1350,1353],{},[46,1337,1338],{},"∑",[134,1340,1341],{},"Angles",[46,1343,580],{},[39,1345,1346,1348],{},[42,1347,44],{},[46,1349,48],{},[46,1351,1352],{},"+",[39,1354,1355,1357],{},[272,1356,274],{},[36,1358,1359,1361],{},[46,1360,280],{"mathvariant":279},[46,1362,280],{"mathvariant":279},[50,1364,1365],{"encoding":52},"\\sum \\text{Angles} = 180^\\circ + E''",[20,1367,1369,1400,1448],{"className":1368,"ariaHidden":58},[57],[20,1370,1372,1376,1382,1385,1391,1394,1397],{"className":1371},[62],[20,1373],{"className":1374,"style":1375},[66],"height:1em;vertical-align:-0.25em;",[20,1377,1338],{"className":1378,"style":1381},[801,1379,1380],"op-symbol","small-op","position:relative;top:0em;",[20,1383],{"className":1384,"style":1096},[708],[20,1386,1388],{"className":1387},[71,162],[20,1389,1341],{"className":1390},[71],[20,1392],{"className":1393,"style":709},[708],[20,1395,580],{"className":1396},[713],[20,1398],{"className":1399,"style":709},[708],[20,1401,1403,1407,1410,1439,1442,1445],{"className":1402},[62],[20,1404],{"className":1405,"style":1406},[66],"height:0.7574em;vertical-align:-0.0833em;",[20,1408,72],{"className":1409},[71],[20,1411,1413,1416],{"className":1412},[71],[20,1414,79],{"className":1415},[71],[20,1417,1419],{"className":1418},[83],[20,1420,1422],{"className":1421},[87],[20,1423,1425],{"className":1424},[91],[20,1426,1428],{"className":1427,"style":67},[95],[20,1429,1430,1433],{"style":98},[20,1431],{"className":1432,"style":103},[102],[20,1434,1436],{"className":1435},[107,108,109,110],[20,1437,48],{"className":1438},[114,110],[20,1440],{"className":1441,"style":1072},[708],[20,1443,1352],{"className":1444},[114],[20,1446],{"className":1447,"style":1072},[708],[20,1449,1451,1454],{"className":1450},[62],[20,1452],{"className":1453,"style":295},[66],[20,1455,1457,1460],{"className":1456},[71],[20,1458,274],{"className":1459,"style":303},[71,302],[20,1461,1463],{"className":1462},[83],[20,1464,1466],{"className":1465},[87],[20,1467,1469],{"className":1468},[91],[20,1470,1472],{"className":1471,"style":295},[95],[20,1473,1474,1477],{"style":98},[20,1475],{"className":1476,"style":103},[102],[20,1478,1480],{"className":1479},[107,108,109,110],[20,1481,1483],{"className":1482},[71,110],[20,1484,330],{"className":1485},[71,110],[11,1487,1489],{"id":1488},"step-by-step-example","Step-by-Step Example",[16,1491,1492,1495,1496,1547,1548,1576,1577,1611,1612,1651,1652,1703,1704,1880,1881,1880,2056,2231],{},[254,1493,1494],{},"Problem:"," In a major triangulation, the mean values of the angles ",[20,1497,1499,1521],{"className":1498},[23],[20,1500,1502],{"className":1501},[27],[29,1503,1504],{"xmlns":31},[33,1505,1506,1518],{},[36,1507,1508,1511,1513,1516],{},[272,1509,1510],{},"A",[46,1512,657],{"separator":58},[272,1514,1515],{},"B",[46,1517,657],{"separator":58},[50,1519,1520],{"encoding":52},"A, B,",[20,1522,1524],{"className":1523,"ariaHidden":58},[57],[20,1525,1527,1531,1534,1537,1540,1544],{"className":1526},[62],[20,1528],{"className":1529,"style":1530},[66],"height:0.8778em;vertical-align:-0.1944em;",[20,1532,1510],{"className":1533},[71,302],[20,1535,657],{"className":1536},[1092],[20,1538],{"className":1539,"style":1096},[708],[20,1541,1515],{"className":1542,"style":1543},[71,302],"margin-right:0.0502em;",[20,1545,657],{"className":1546},[1092]," and ",[20,1549,1551,1564],{"className":1550},[23],[20,1552,1554],{"className":1553},[27],[29,1555,1556],{"xmlns":31},[33,1557,1558,1562],{},[36,1559,1560],{},[272,1561,635],{},[50,1563,635],{"encoding":52},[20,1565,1567],{"className":1566,"ariaHidden":58},[57],[20,1568,1570,1573],{"className":1569},[62],[20,1571],{"className":1572,"style":1244},[66],[20,1574,635],{"className":1575,"style":1040},[71,302]," were measured. The length of side ",[20,1578,1580,1596],{"className":1579},[23],[20,1581,1583],{"className":1582},[27],[29,1584,1585],{"xmlns":31},[33,1586,1587,1593],{},[36,1588,1589,1591],{},[272,1590,1515],{},[272,1592,635],{},[50,1594,1595],{"encoding":52},"BC",[20,1597,1599],{"className":1598,"ariaHidden":58},[57],[20,1600,1602,1605,1608],{"className":1601},[62],[20,1603],{"className":1604,"style":1244},[66],[20,1606,1515],{"className":1607,"style":1543},[71,302],[20,1609,635],{"className":1610,"style":1040},[71,302]," was ",[20,1613,1615,1632],{"className":1614},[23],[20,1616,1618],{"className":1617},[27],[29,1619,1620],{"xmlns":31},[33,1621,1622,1629],{},[36,1623,1624,1627],{},[42,1625,1626],{},"37.5",[134,1628,136],{},[50,1630,1631],{"encoding":52},"37.5 \\text{ km}",[20,1633,1635],{"className":1634,"ariaHidden":58},[57],[20,1636,1638,1642,1645],{"className":1637},[62],[20,1639],{"className":1640,"style":1641},[66],"height:0.6944em;",[20,1643,1626],{"className":1644},[71],[20,1646,1648],{"className":1647},[71,162],[20,1649,136],{"className":1650},[71]," and the radius of the Earth is ",[20,1653,1655,1676],{"className":1654},[23],[20,1656,1658],{"className":1657},[27],[29,1659,1660],{"xmlns":31},[33,1661,1662,1673],{},[36,1663,1664,1666,1668,1671],{},[42,1665,1265],{},[46,1667,657],{"separator":58},[42,1669,1670],{},"267",[134,1672,136],{},[50,1674,1675],{"encoding":52},"6,267 \\text{ km}",[20,1677,1679],{"className":1678,"ariaHidden":58},[57],[20,1680,1682,1685,1688,1691,1694,1697],{"className":1681},[62],[20,1683],{"className":1684,"style":1137},[66],[20,1686,1265],{"className":1687},[71],[20,1689,657],{"className":1690},[1092],[20,1692],{"className":1693,"style":1096},[708],[20,1695,1670],{"className":1696},[71],[20,1698,1700],{"className":1699},[71,162],[20,1701,136],{"className":1702},[71],". Observed angles: ",[20,1705,1707,1749],{"className":1706},[23],[20,1708,1710],{"className":1709},[27],[29,1711,1712],{"xmlns":31},[33,1713,1714,1746],{},[36,1715,1716,1718,1720,1727,1735],{},[272,1717,1510],{},[46,1719,580],{},[39,1721,1722,1725],{},[42,1723,1724],{},"50",[46,1726,48],{},[39,1728,1729,1732],{},[42,1730,1731],{},"22",[46,1733,280],{"mathvariant":279,"lspace":1734,"rspace":1734},"0em",[39,1736,1737,1740],{},[42,1738,1739],{},"32.5",[36,1741,1742,1744],{},[46,1743,280],{"mathvariant":279},[46,1745,280],{"mathvariant":279},[50,1747,1748],{"encoding":52},"A=50^\\circ 22' 32.5''",[20,1750,1752,1770],{"className":1751,"ariaHidden":58},[57],[20,1753,1755,1758,1761,1764,1767],{"className":1754},[62],[20,1756],{"className":1757,"style":1244},[66],[20,1759,1510],{"className":1760},[71,302],[20,1762],{"className":1763,"style":709},[708],[20,1765,580],{"className":1766},[713],[20,1768],{"className":1769,"style":709},[708],[20,1771,1773,1776,1780,1809,1812,1844,1848],{"className":1772},[62],[20,1774],{"className":1775,"style":295},[66],[20,1777,1779],{"className":1778},[71],"5",[20,1781,1783,1786],{"className":1782},[71],[20,1784,79],{"className":1785},[71],[20,1787,1789],{"className":1788},[83],[20,1790,1792],{"className":1791},[87],[20,1793,1795],{"className":1794},[91],[20,1796,1798],{"className":1797,"style":67},[95],[20,1799,1800,1803],{"style":98},[20,1801],{"className":1802,"style":103},[102],[20,1804,1806],{"className":1805},[107,108,109,110],[20,1807,48],{"className":1808},[114,110],[20,1810,139],{"className":1811},[71],[20,1813,1815,1818],{"className":1814},[71],[20,1816,139],{"className":1817},[71],[20,1819,1821],{"className":1820},[83],[20,1822,1824],{"className":1823},[87],[20,1825,1827],{"className":1826},[91],[20,1828,1830],{"className":1829,"style":295},[95],[20,1831,1832,1835],{"style":98},[20,1833],{"className":1834,"style":103},[102],[20,1836,1838],{"className":1837},[107,108,109,110],[20,1839,1841],{"className":1840},[71,110],[20,1842,280],{"className":1843},[71,110],[20,1845,1847],{"className":1846},[71],"32.",[20,1849,1851,1854],{"className":1850},[71],[20,1852,1779],{"className":1853},[71],[20,1855,1857],{"className":1856},[83],[20,1858,1860],{"className":1859},[87],[20,1861,1863],{"className":1862},[91],[20,1864,1866],{"className":1865,"style":295},[95],[20,1867,1868,1871],{"style":98},[20,1869],{"className":1870,"style":103},[102],[20,1872,1874],{"className":1873},[107,108,109,110],[20,1875,1877],{"className":1876},[71,110],[20,1878,330],{"className":1879},[71,110],", ",[20,1882,1884,1925],{"className":1883},[23],[20,1885,1887],{"className":1886},[27],[29,1888,1889],{"xmlns":31},[33,1890,1891,1922],{},[36,1892,1893,1895,1897,1904,1911],{},[272,1894,1515],{},[46,1896,580],{},[39,1898,1899,1902],{},[42,1900,1901],{},"65",[46,1903,48],{},[39,1905,1906,1909],{},[42,1907,1908],{},"40",[46,1910,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,1912,1913,1916],{},[42,1914,1915],{},"47.5",[36,1917,1918,1920],{},[46,1919,280],{"mathvariant":279},[46,1921,280],{"mathvariant":279},[50,1923,1924],{"encoding":52},"B=65^\\circ 40' 47.5''",[20,1926,1928,1946],{"className":1927,"ariaHidden":58},[57],[20,1929,1931,1934,1937,1940,1943],{"className":1930},[62],[20,1932],{"className":1933,"style":1244},[66],[20,1935,1515],{"className":1936,"style":1543},[71,302],[20,1938],{"className":1939,"style":709},[708],[20,1941,580],{"className":1942},[713],[20,1944],{"className":1945,"style":709},[708],[20,1947,1949,1952,1955,1984,1988,2020,2024],{"className":1948},[62],[20,1950],{"className":1951,"style":295},[66],[20,1953,1265],{"className":1954},[71],[20,1956,1958,1961],{"className":1957},[71],[20,1959,1779],{"className":1960},[71],[20,1962,1964],{"className":1963},[83],[20,1965,1967],{"className":1966},[87],[20,1968,1970],{"className":1969},[91],[20,1971,1973],{"className":1972,"style":67},[95],[20,1974,1975,1978],{"style":98},[20,1976],{"className":1977,"style":103},[102],[20,1979,1981],{"className":1980},[107,108,109,110],[20,1982,48],{"className":1983},[114,110],[20,1985,1987],{"className":1986},[71],"4",[20,1989,1991,1994],{"className":1990},[71],[20,1992,79],{"className":1993},[71],[20,1995,1997],{"className":1996},[83],[20,1998,2000],{"className":1999},[87],[20,2001,2003],{"className":2002},[91],[20,2004,2006],{"className":2005,"style":295},[95],[20,2007,2008,2011],{"style":98},[20,2009],{"className":2010,"style":103},[102],[20,2012,2014],{"className":2013},[107,108,109,110],[20,2015,2017],{"className":2016},[71,110],[20,2018,280],{"className":2019},[71,110],[20,2021,2023],{"className":2022},[71],"47.",[20,2025,2027,2030],{"className":2026},[71],[20,2028,1779],{"className":2029},[71],[20,2031,2033],{"className":2032},[83],[20,2034,2036],{"className":2035},[87],[20,2037,2039],{"className":2038},[91],[20,2040,2042],{"className":2041,"style":295},[95],[20,2043,2044,2047],{"style":98},[20,2045],{"className":2046,"style":103},[102],[20,2048,2050],{"className":2049},[107,108,109,110],[20,2051,2053],{"className":2052},[71,110],[20,2054,330],{"className":2055},[71,110],[20,2057,2059,2100],{"className":2058},[23],[20,2060,2062],{"className":2061},[27],[29,2063,2064],{"xmlns":31},[33,2065,2066,2097],{},[36,2067,2068,2070,2072,2079,2086],{},[272,2069,635],{},[46,2071,580],{},[39,2073,2074,2077],{},[42,2075,2076],{},"63",[46,2078,48],{},[39,2080,2081,2084],{},[42,2082,2083],{},"56",[46,2085,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,2087,2088,2091],{},[42,2089,2090],{},"46.5",[36,2092,2093,2095],{},[46,2094,280],{"mathvariant":279},[46,2096,280],{"mathvariant":279},[50,2098,2099],{"encoding":52},"C=63^\\circ 56' 46.5''",[20,2101,2103,2121],{"className":2102,"ariaHidden":58},[57],[20,2104,2106,2109,2112,2115,2118],{"className":2105},[62],[20,2107],{"className":2108,"style":1244},[66],[20,2110,635],{"className":2111,"style":1040},[71,302],[20,2113],{"className":2114,"style":709},[708],[20,2116,580],{"className":2117},[713],[20,2119],{"className":2120,"style":709},[708],[20,2122,2124,2127,2130,2160,2163,2195,2199],{"className":2123},[62],[20,2125],{"className":2126,"style":295},[66],[20,2128,1265],{"className":2129},[71],[20,2131,2133,2137],{"className":2132},[71],[20,2134,2136],{"className":2135},[71],"3",[20,2138,2140],{"className":2139},[83],[20,2141,2143],{"className":2142},[87],[20,2144,2146],{"className":2145},[91],[20,2147,2149],{"className":2148,"style":67},[95],[20,2150,2151,2154],{"style":98},[20,2152],{"className":2153,"style":103},[102],[20,2155,2157],{"className":2156},[107,108,109,110],[20,2158,48],{"className":2159},[114,110],[20,2161,1779],{"className":2162},[71],[20,2164,2166,2169],{"className":2165},[71],[20,2167,1265],{"className":2168},[71],[20,2170,2172],{"className":2171},[83],[20,2173,2175],{"className":2174},[87],[20,2176,2178],{"className":2177},[91],[20,2179,2181],{"className":2180,"style":295},[95],[20,2182,2183,2186],{"style":98},[20,2184],{"className":2185,"style":103},[102],[20,2187,2189],{"className":2188},[107,108,109,110],[20,2190,2192],{"className":2191},[71,110],[20,2193,280],{"className":2194},[71,110],[20,2196,2198],{"className":2197},[71],"46.",[20,2200,2202,2205],{"className":2201},[71],[20,2203,1779],{"className":2204},[71],[20,2206,2208],{"className":2207},[83],[20,2209,2211],{"className":2210},[87],[20,2212,2214],{"className":2213},[91],[20,2215,2217],{"className":2216,"style":295},[95],[20,2218,2219,2222],{"style":98},[20,2220],{"className":2221,"style":103},[102],[20,2223,2225],{"className":2224},[107,108,109,110],[20,2226,2228],{"className":2227},[71,110],[20,2229,330],{"className":2230},[71,110],". Calculate the spherical excess and adjusted angles .",[2233,2234,2235,2908,3269,3622],"ol",{},[1105,2236,2237,2306,2307,2335,2336,2429,2430,2776,2777,332],{},[254,2238,2239,2240,2305],{},"Calculate Spherical Excess (",[20,2241,2243,2264],{"className":2242},[23],[20,2244,2246],{"className":2245},[27],[29,2247,2248],{"xmlns":31},[33,2249,2250,2262],{},[36,2251,2252],{},[39,2253,2254,2256],{},[272,2255,274],{},[36,2257,2258,2260],{},[46,2259,280],{"mathvariant":279},[46,2261,280],{"mathvariant":279},[50,2263,285],{"encoding":52},[20,2265,2267],{"className":2266,"ariaHidden":58},[57],[20,2268,2270,2273],{"className":2269},[62],[20,2271],{"className":2272,"style":295},[66],[20,2274,2276,2279],{"className":2275},[71],[20,2277,274],{"className":2278,"style":303},[71,302],[20,2280,2282],{"className":2281},[83],[20,2283,2285],{"className":2284},[87],[20,2286,2288],{"className":2287},[91],[20,2289,2291],{"className":2290,"style":295},[95],[20,2292,2293,2296],{"style":98},[20,2294],{"className":2295,"style":103},[102],[20,2297,2299],{"className":2298},[107,108,109,110],[20,2300,2302],{"className":2301},[71,110],[20,2303,330],{"className":2304},[71,110],"):","\nUsing the sine rule to find side ",[20,2308,2310,2323],{"className":2309},[23],[20,2311,2313],{"className":2312},[27],[29,2314,2315],{"xmlns":31},[33,2316,2317,2321],{},[36,2318,2319],{},[272,2320,625],{},[50,2322,625],{"encoding":52},[20,2324,2326],{"className":2325,"ariaHidden":58},[57],[20,2327,2329,2332],{"className":2328},[62],[20,2330],{"className":2331,"style":1641},[66],[20,2333,625],{"className":2334},[71,302],": ",[20,2337,2339,2372],{"className":2338},[23],[20,2340,2342],{"className":2341},[27],[29,2343,2344],{"xmlns":31},[33,2345,2346,2369],{},[36,2347,2348,2350,2352,2354,2356,2358,2360,2363,2365,2367],{},[272,2349,625],{},[46,2351,580],{},[272,2353,622],{},[272,2355,598],{},[46,2357,601],{},[272,2359,1515],{},[272,2361,2362],{"mathvariant":279},"\u002F",[272,2364,598],{},[46,2366,601],{},[272,2368,1510],{},[50,2370,2371],{"encoding":52},"b = a \\sin B \u002F \\sin A",[20,2373,2375,2393],{"className":2374,"ariaHidden":58},[57],[20,2376,2378,2381,2384,2387,2390],{"className":2377},[62],[20,2379],{"className":2380,"style":1641},[66],[20,2382,625],{"className":2383},[71,302],[20,2385],{"className":2386,"style":709},[708],[20,2388,580],{"className":2389},[713],[20,2391],{"className":2392,"style":709},[708],[20,2394,2396,2399,2402,2405,2408,2411,2414,2417,2420,2423,2426],{"className":2395},[62],[20,2397],{"className":2398,"style":1375},[66],[20,2400,622],{"className":2401},[71,302],[20,2403],{"className":2404,"style":1096},[708],[20,2406,598],{"className":2407},[801],[20,2409],{"className":2410,"style":1096},[708],[20,2412,1515],{"className":2413,"style":1543},[71,302],[20,2415,2362],{"className":2416},[71],[20,2418],{"className":2419,"style":1096},[708],[20,2421,598],{"className":2422},[801],[20,2424],{"className":2425,"style":1096},[708],[20,2427,1510],{"className":2428},[71,302]," .\n",[20,2431,2433,2503],{"className":2432},[23],[20,2434,2436],{"className":2435},[27],[29,2437,2438],{"xmlns":31},[33,2439,2440,2500],{},[36,2441,2442,2452,2454,2492,2494,2496,2498],{},[39,2443,2444,2446],{},[272,2445,274],{},[36,2447,2448,2450],{},[46,2449,280],{"mathvariant":279},[46,2451,280],{"mathvariant":279},[46,2453,580],{},[582,2455,2456,2476],{},[36,2457,2458,2464,2466,2468,2470,2472,2474],{},[39,2459,2460,2462],{},[272,2461,622],{},[42,2463,139],{},[272,2465,598],{},[46,2467,601],{},[272,2469,1515],{},[272,2471,598],{},[46,2473,601],{},[272,2475,635],{},[36,2477,2478,2480,2486,2488,2490],{},[42,2479,139],{},[39,2481,2482,2484],{},[272,2483,593],{},[42,2485,139],{},[272,2487,598],{},[46,2489,601],{},[272,2491,1510],{},[46,2493,651],{},[42,2495,654],{},[46,2497,657],{"separator":58},[42,2499,660],{},[50,2501,2502],{"encoding":52},"E'' = \\frac{a^2 \\sin B \\sin C}{2R^2 \\sin A} \\times 206,265",[20,2504,2506,2553,2758],{"className":2505,"ariaHidden":58},[57],[20,2507,2509,2512,2544,2547,2550],{"className":2508},[62],[20,2510],{"className":2511,"style":295},[66],[20,2513,2515,2518],{"className":2514},[71],[20,2516,274],{"className":2517,"style":303},[71,302],[20,2519,2521],{"className":2520},[83],[20,2522,2524],{"className":2523},[87],[20,2525,2527],{"className":2526},[91],[20,2528,2530],{"className":2529,"style":295},[95],[20,2531,2532,2535],{"style":98},[20,2533],{"className":2534,"style":103},[102],[20,2536,2538],{"className":2537},[107,108,109,110],[20,2539,2541],{"className":2540},[71,110],[20,2542,330],{"className":2543},[71,110],[20,2545],{"className":2546,"style":709},[708],[20,2548,580],{"className":2549},[713],[20,2551],{"className":2552,"style":709},[708],[20,2554,2556,2560,2749,2752,2755],{"className":2555},[62],[20,2557],{"className":2558,"style":2559},[66],"height:1.3629em;vertical-align:-0.345em;",[20,2561,2563,2566,2746],{"className":2562},[71],[20,2564],{"className":2565},[730,731],[20,2567,2569],{"className":2568},[582],[20,2570,2572,2738],{"className":2571},[87,738],[20,2573,2575,2735],{"className":2574},[91],[20,2576,2579,2643,2651],{"className":2577,"style":2578},[95],"height:1.0179em;",[20,2580,2581,2584],{"style":748},[20,2582],{"className":2583,"style":752},[102],[20,2585,2587],{"className":2586},[107,108,109,110],[20,2588,2590,2593,2622,2625,2637,2640],{"className":2589},[71,110],[20,2591,139],{"className":2592},[71,110],[20,2594,2596,2599],{"className":2595},[71,110],[20,2597,593],{"className":2598,"style":765},[71,302,110],[20,2600,2602],{"className":2601},[83],[20,2603,2605],{"className":2604},[87],[20,2606,2608],{"className":2607},[91],[20,2609,2611],{"className":2610,"style":778},[95],[20,2612,2613,2616],{"style":781},[20,2614],{"className":2615,"style":785},[102],[20,2617,2619],{"className":2618},[107,789,790,110],[20,2620,139],{"className":2621},[71,110],[20,2623],{"className":2624,"style":797},[708,110],[20,2626,2628,2631,2634],{"className":2627},[801,110],[20,2629,805],{"className":2630},[110],[20,2632,809],{"className":2633},[110],[20,2635,813],{"className":2636},[110],[20,2638],{"className":2639,"style":797},[708,110],[20,2641,1510],{"className":2642},[71,302,110],[20,2644,2645,2648],{"style":852},[20,2646],{"className":2647,"style":752},[102],[20,2649],{"className":2650,"style":860},[859],[20,2652,2653,2656],{"style":985},[20,2654],{"className":2655,"style":752},[102],[20,2657,2659],{"className":2658},[107,108,109,110],[20,2660,2662,2693,2696,2708,2711,2714,2717,2729,2732],{"className":2661},[71,110],[20,2663,2665,2668],{"className":2664},[71,110],[20,2666,622],{"className":2667},[71,302,110],[20,2669,2671],{"className":2670},[83],[20,2672,2674],{"className":2673},[87],[20,2675,2677],{"className":2676},[91],[20,2678,2681],{"className":2679,"style":2680},[95],"height:0.8913em;",[20,2682,2684,2687],{"style":2683},"top:-2.931em;margin-right:0.0714em;",[20,2685],{"className":2686,"style":785},[102],[20,2688,2690],{"className":2689},[107,789,790,110],[20,2691,139],{"className":2692},[71,110],[20,2694],{"className":2695,"style":797},[708,110],[20,2697,2699,2702,2705],{"className":2698},[801,110],[20,2700,805],{"className":2701},[110],[20,2703,809],{"className":2704},[110],[20,2706,813],{"className":2707},[110],[20,2709],{"className":2710,"style":797},[708,110],[20,2712,1515],{"className":2713,"style":1543},[71,302,110],[20,2715],{"className":2716,"style":797},[708,110],[20,2718,2720,2723,2726],{"className":2719},[801,110],[20,2721,805],{"className":2722},[110],[20,2724,809],{"className":2725},[110],[20,2727,813],{"className":2728},[110],[20,2730],{"className":2731,"style":797},[708,110],[20,2733,635],{"className":2734,"style":1040},[71,302,110],[20,2736,883],{"className":2737},[882],[20,2739,2741],{"className":2740},[91],[20,2742,2744],{"className":2743,"style":890},[95],[20,2745],{},[20,2747],{"className":2748},[896,731],[20,2750],{"className":2751,"style":1072},[708],[20,2753,651],{"className":2754},[114],[20,2756],{"className":2757,"style":1072},[708],[20,2759,2761,2764,2767,2770,2773],{"className":2760},[62],[20,2762],{"className":2763,"style":1085},[66],[20,2765,654],{"className":2766},[71],[20,2768,657],{"className":2769},[1092],[20,2771],{"className":2772,"style":1096},[708],[20,2774,660],{"className":2775},[71],"\nSubstituting the values: ",[20,2778,2780,2815],{"className":2779},[23],[20,2781,2783],{"className":2782},[27],[29,2784,2785],{"xmlns":31},[33,2786,2787,2812],{},[36,2788,2789,2799,2801],{},[39,2790,2791,2793],{},[272,2792,274],{},[36,2794,2795,2797],{},[46,2796,280],{"mathvariant":279},[46,2798,280],{"mathvariant":279},[46,2800,615],{},[39,2802,2803,2806],{},[42,2804,2805],{},"3.9",[36,2807,2808,2810],{},[46,2809,280],{"mathvariant":279},[46,2811,280],{"mathvariant":279},[50,2813,2814],{"encoding":52},"E'' \\approx 3.9''",[20,2816,2818,2865],{"className":2817,"ariaHidden":58},[57],[20,2819,2821,2824,2856,2859,2862],{"className":2820},[62],[20,2822],{"className":2823,"style":295},[66],[20,2825,2827,2830],{"className":2826},[71],[20,2828,274],{"className":2829,"style":303},[71,302],[20,2831,2833],{"className":2832},[83],[20,2834,2836],{"className":2835},[87],[20,2837,2839],{"className":2838},[91],[20,2840,2842],{"className":2841,"style":295},[95],[20,2843,2844,2847],{"style":98},[20,2845],{"className":2846,"style":103},[102],[20,2848,2850],{"className":2849},[107,108,109,110],[20,2851,2853],{"className":2852},[71,110],[20,2854,330],{"className":2855},[71,110],[20,2857],{"className":2858,"style":709},[708],[20,2860,615],{"className":2861},[713],[20,2863],{"className":2864,"style":709},[708],[20,2866,2868,2871,2875],{"className":2867},[62],[20,2869],{"className":2870,"style":295},[66],[20,2872,2874],{"className":2873},[71],"3.",[20,2876,2878,2882],{"className":2877},[71],[20,2879,2881],{"className":2880},[71],"9",[20,2883,2885],{"className":2884},[83],[20,2886,2888],{"className":2887},[87],[20,2889,2891],{"className":2890},[91],[20,2892,2894],{"className":2893,"style":295},[95],[20,2895,2896,2899],{"style":98},[20,2897],{"className":2898,"style":103},[102],[20,2900,2902],{"className":2901},[107,108,109,110],[20,2903,2905],{"className":2904},[71,110],[20,2906,330],{"className":2907},[71,110],[1105,2909,2910,2913,2914,332],{},[254,2911,2912],{},"Target Sum:","\nTarget = ",[20,2915,2917,2989],{"className":2916},[23],[20,2918,2920],{"className":2919},[27],[29,2921,2922],{"xmlns":31},[33,2923,2924,2986],{},[36,2925,2926,2932,2939,2949,2951,2961,2963,2969,2975],{},[39,2927,2928,2930],{},[42,2929,44],{},[46,2931,48],{},[39,2933,2934,2937],{},[42,2935,2936],{},"00",[46,2938,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,2940,2941,2943],{},[42,2942,2936],{},[36,2944,2945,2947],{},[46,2946,280],{"mathvariant":279},[46,2948,280],{"mathvariant":279},[46,2950,1352],{},[39,2952,2953,2955],{},[42,2954,2805],{},[36,2956,2957,2959],{},[46,2958,280],{"mathvariant":279},[46,2960,280],{"mathvariant":279},[46,2962,580],{},[39,2964,2965,2967],{},[42,2966,44],{},[46,2968,48],{},[39,2970,2971,2973],{},[42,2972,2936],{},[46,2974,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,2976,2977,2980],{},[42,2978,2979],{},"03.9",[36,2981,2982,2984],{},[46,2983,280],{"mathvariant":279},[46,2985,280],{"mathvariant":279},[50,2987,2988],{"encoding":52},"180^\\circ 00' 00'' + 3.9'' = 180^\\circ 00' 03.9''",[20,2990,2992,3110,3160],{"className":2991,"ariaHidden":58},[57],[20,2993,2995,2999,3002,3031,3034,3066,3069,3101,3104,3107],{"className":2994},[62],[20,2996],{"className":2997,"style":2998},[66],"height:0.8352em;vertical-align:-0.0833em;",[20,3000,72],{"className":3001},[71],[20,3003,3005,3008],{"className":3004},[71],[20,3006,79],{"className":3007},[71],[20,3009,3011],{"className":3010},[83],[20,3012,3014],{"className":3013},[87],[20,3015,3017],{"className":3016},[91],[20,3018,3020],{"className":3019,"style":67},[95],[20,3021,3022,3025],{"style":98},[20,3023],{"className":3024,"style":103},[102],[20,3026,3028],{"className":3027},[107,108,109,110],[20,3029,48],{"className":3030},[114,110],[20,3032,79],{"className":3033},[71],[20,3035,3037,3040],{"className":3036},[71],[20,3038,79],{"className":3039},[71],[20,3041,3043],{"className":3042},[83],[20,3044,3046],{"className":3045},[87],[20,3047,3049],{"className":3048},[91],[20,3050,3052],{"className":3051,"style":295},[95],[20,3053,3054,3057],{"style":98},[20,3055],{"className":3056,"style":103},[102],[20,3058,3060],{"className":3059},[107,108,109,110],[20,3061,3063],{"className":3062},[71,110],[20,3064,280],{"className":3065},[71,110],[20,3067,79],{"className":3068},[71],[20,3070,3072,3075],{"className":3071},[71],[20,3073,79],{"className":3074},[71],[20,3076,3078],{"className":3077},[83],[20,3079,3081],{"className":3080},[87],[20,3082,3084],{"className":3083},[91],[20,3085,3087],{"className":3086,"style":295},[95],[20,3088,3089,3092],{"style":98},[20,3090],{"className":3091,"style":103},[102],[20,3093,3095],{"className":3094},[107,108,109,110],[20,3096,3098],{"className":3097},[71,110],[20,3099,330],{"className":3100},[71,110],[20,3102],{"className":3103,"style":1072},[708],[20,3105,1352],{"className":3106},[114],[20,3108],{"className":3109,"style":1072},[708],[20,3111,3113,3116,3119,3151,3154,3157],{"className":3112},[62],[20,3114],{"className":3115,"style":295},[66],[20,3117,2874],{"className":3118},[71],[20,3120,3122,3125],{"className":3121},[71],[20,3123,2881],{"className":3124},[71],[20,3126,3128],{"className":3127},[83],[20,3129,3131],{"className":3130},[87],[20,3132,3134],{"className":3133},[91],[20,3135,3137],{"className":3136,"style":295},[95],[20,3138,3139,3142],{"style":98},[20,3140],{"className":3141,"style":103},[102],[20,3143,3145],{"className":3144},[107,108,109,110],[20,3146,3148],{"className":3147},[71,110],[20,3149,330],{"className":3150},[71,110],[20,3152],{"className":3153,"style":709},[708],[20,3155,580],{"className":3156},[713],[20,3158],{"className":3159,"style":709},[708],[20,3161,3163,3166,3169,3198,3201,3233,3237],{"className":3162},[62],[20,3164],{"className":3165,"style":295},[66],[20,3167,72],{"className":3168},[71],[20,3170,3172,3175],{"className":3171},[71],[20,3173,79],{"className":3174},[71],[20,3176,3178],{"className":3177},[83],[20,3179,3181],{"className":3180},[87],[20,3182,3184],{"className":3183},[91],[20,3185,3187],{"className":3186,"style":67},[95],[20,3188,3189,3192],{"style":98},[20,3190],{"className":3191,"style":103},[102],[20,3193,3195],{"className":3194},[107,108,109,110],[20,3196,48],{"className":3197},[114,110],[20,3199,79],{"className":3200},[71],[20,3202,3204,3207],{"className":3203},[71],[20,3205,79],{"className":3206},[71],[20,3208,3210],{"className":3209},[83],[20,3211,3213],{"className":3212},[87],[20,3214,3216],{"className":3215},[91],[20,3217,3219],{"className":3218,"style":295},[95],[20,3220,3221,3224],{"style":98},[20,3222],{"className":3223,"style":103},[102],[20,3225,3227],{"className":3226},[107,108,109,110],[20,3228,3230],{"className":3229},[71,110],[20,3231,280],{"className":3232},[71,110],[20,3234,3236],{"className":3235},[71],"03.",[20,3238,3240,3243],{"className":3239},[71],[20,3241,2881],{"className":3242},[71],[20,3244,3246],{"className":3245},[83],[20,3247,3249],{"className":3248},[87],[20,3250,3252],{"className":3251},[91],[20,3253,3255],{"className":3254,"style":295},[95],[20,3256,3257,3260],{"style":98},[20,3258],{"className":3259,"style":103},[102],[20,3261,3263],{"className":3262},[107,108,109,110],[20,3264,3266],{"className":3265},[71,110],[20,3267,330],{"className":3268},[71,110],[1105,3270,3271,3274,3275,3425,3426,332],{},[254,3272,3273],{},"Find Misclosure:","\nObserved Sum = ",[20,3276,3278,3313],{"className":3277},[23],[20,3279,3281],{"className":3280},[27],[29,3282,3283],{"xmlns":31},[33,3284,3285,3310],{},[36,3286,3287,3293,3299],{},[39,3288,3289,3291],{},[42,3290,44],{},[46,3292,48],{},[39,3294,3295,3297],{},[42,3296,2936],{},[46,3298,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,3300,3301,3304],{},[42,3302,3303],{},"06.5",[36,3305,3306,3308],{},[46,3307,280],{"mathvariant":279},[46,3309,280],{"mathvariant":279},[50,3311,3312],{"encoding":52},"180^\\circ 00' 06.5''",[20,3314,3316],{"className":3315,"ariaHidden":58},[57],[20,3317,3319,3322,3325,3354,3357,3389,3393],{"className":3318},[62],[20,3320],{"className":3321,"style":295},[66],[20,3323,72],{"className":3324},[71],[20,3326,3328,3331],{"className":3327},[71],[20,3329,79],{"className":3330},[71],[20,3332,3334],{"className":3333},[83],[20,3335,3337],{"className":3336},[87],[20,3338,3340],{"className":3339},[91],[20,3341,3343],{"className":3342,"style":67},[95],[20,3344,3345,3348],{"style":98},[20,3346],{"className":3347,"style":103},[102],[20,3349,3351],{"className":3350},[107,108,109,110],[20,3352,48],{"className":3353},[114,110],[20,3355,79],{"className":3356},[71],[20,3358,3360,3363],{"className":3359},[71],[20,3361,79],{"className":3362},[71],[20,3364,3366],{"className":3365},[83],[20,3367,3369],{"className":3368},[87],[20,3370,3372],{"className":3371},[91],[20,3373,3375],{"className":3374,"style":295},[95],[20,3376,3377,3380],{"style":98},[20,3378],{"className":3379,"style":103},[102],[20,3381,3383],{"className":3382},[107,108,109,110],[20,3384,3386],{"className":3385},[71,110],[20,3387,280],{"className":3388},[71,110],[20,3390,3392],{"className":3391},[71],"06.",[20,3394,3396,3399],{"className":3395},[71],[20,3397,1779],{"className":3398},[71],[20,3400,3402],{"className":3401},[83],[20,3403,3405],{"className":3404},[87],[20,3406,3408],{"className":3407},[91],[20,3409,3411],{"className":3410,"style":295},[95],[20,3412,3413,3416],{"style":98},[20,3414],{"className":3415,"style":103},[102],[20,3417,3419],{"className":3418},[107,108,109,110],[20,3420,3422],{"className":3421},[71,110],[20,3423,330],{"className":3424},[71,110],".\nError = ",[20,3427,3429,3477],{"className":3428},[23],[20,3430,3432],{"className":3431},[27],[29,3433,3434],{"xmlns":31},[33,3435,3436,3474],{},[36,3437,3438,3448,3451,3461,3463],{},[39,3439,3440,3442],{},[42,3441,3303],{},[36,3443,3444,3446],{},[46,3445,280],{"mathvariant":279},[46,3447,280],{"mathvariant":279},[46,3449,3450],{},"−",[39,3452,3453,3455],{},[42,3454,2979],{},[36,3456,3457,3459],{},[46,3458,280],{"mathvariant":279},[46,3460,280],{"mathvariant":279},[46,3462,580],{},[39,3464,3465,3468],{},[42,3466,3467],{},"2.6",[36,3469,3470,3472],{},[46,3471,280],{"mathvariant":279},[46,3473,280],{"mathvariant":279},[50,3475,3476],{"encoding":52},"06.5'' - 03.9'' = 2.6''",[20,3478,3480,3530,3580],{"className":3479,"ariaHidden":58},[57],[20,3481,3483,3486,3489,3521,3524,3527],{"className":3482},[62],[20,3484],{"className":3485,"style":2998},[66],[20,3487,3392],{"className":3488},[71],[20,3490,3492,3495],{"className":3491},[71],[20,3493,1779],{"className":3494},[71],[20,3496,3498],{"className":3497},[83],[20,3499,3501],{"className":3500},[87],[20,3502,3504],{"className":3503},[91],[20,3505,3507],{"className":3506,"style":295},[95],[20,3508,3509,3512],{"style":98},[20,3510],{"className":3511,"style":103},[102],[20,3513,3515],{"className":3514},[107,108,109,110],[20,3516,3518],{"className":3517},[71,110],[20,3519,330],{"className":3520},[71,110],[20,3522],{"className":3523,"style":1072},[708],[20,3525,3450],{"className":3526},[114],[20,3528],{"className":3529,"style":1072},[708],[20,3531,3533,3536,3539,3571,3574,3577],{"className":3532},[62],[20,3534],{"className":3535,"style":295},[66],[20,3537,3236],{"className":3538},[71],[20,3540,3542,3545],{"className":3541},[71],[20,3543,2881],{"className":3544},[71],[20,3546,3548],{"className":3547},[83],[20,3549,3551],{"className":3550},[87],[20,3552,3554],{"className":3553},[91],[20,3555,3557],{"className":3556,"style":295},[95],[20,3558,3559,3562],{"style":98},[20,3560],{"className":3561,"style":103},[102],[20,3563,3565],{"className":3564},[107,108,109,110],[20,3566,3568],{"className":3567},[71,110],[20,3569,330],{"className":3570},[71,110],[20,3572],{"className":3573,"style":709},[708],[20,3575,580],{"className":3576},[713],[20,3578],{"className":3579,"style":709},[708],[20,3581,3583,3586,3590],{"className":3582},[62],[20,3584],{"className":3585,"style":295},[66],[20,3587,3589],{"className":3588},[71],"2.",[20,3591,3593,3596],{"className":3592},[71],[20,3594,1265],{"className":3595},[71],[20,3597,3599],{"className":3598},[83],[20,3600,3602],{"className":3601},[87],[20,3603,3605],{"className":3604},[91],[20,3606,3608],{"className":3607,"style":295},[95],[20,3609,3610,3613],{"style":98},[20,3611],{"className":3612,"style":103},[102],[20,3614,3616],{"className":3615},[107,108,109,110],[20,3617,3619],{"className":3618},[71,110],[20,3620,330],{"className":3621},[71,110],[1105,3623,3624,3627,3628,3658,3659,3696,3697,3870,3871,3870,4042,332],{},[254,3625,3626],{},"Distribute Error:","\nIf angles have different weights (",[20,3629,3631,3645],{"className":3630},[23],[20,3632,3634],{"className":3633},[27],[29,3635,3636],{"xmlns":31},[33,3637,3638,3643],{},[36,3639,3640],{},[272,3641,3642],{},"W",[50,3644,3642],{"encoding":52},[20,3646,3648],{"className":3647,"ariaHidden":58},[57],[20,3649,3651,3654],{"className":3650},[62],[20,3652],{"className":3653,"style":1244},[66],[20,3655,3642],{"className":3656,"style":3657},[71,302],"margin-right:0.1389em;","), the correction is applied proportional to the reciprocal weight (",[20,3660,3662,3680],{"className":3661},[23],[20,3663,3665],{"className":3664},[27],[29,3666,3667],{"xmlns":31},[33,3668,3669,3677],{},[36,3670,3671,3673,3675],{},[42,3672,606],{},[272,3674,2362],{"mathvariant":279},[272,3676,3642],{},[50,3678,3679],{"encoding":52},"1\u002FW",[20,3681,3683],{"className":3682,"ariaHidden":58},[57],[20,3684,3686,3689,3693],{"className":3685},[62],[20,3687],{"className":3688,"style":1375},[66],[20,3690,3692],{"className":3691},[71],"1\u002F",[20,3694,3642],{"className":3695,"style":3657},[71,302],") .\nCorrected ",[20,3698,3700,3739],{"className":3699},[23],[20,3701,3703],{"className":3702},[27],[29,3704,3705],{"xmlns":31},[33,3706,3707,3736],{},[36,3708,3709,3711,3713,3719,3725],{},[272,3710,1510],{},[46,3712,580],{},[39,3714,3715,3717],{},[42,3716,1724],{},[46,3718,48],{},[39,3720,3721,3723],{},[42,3722,1731],{},[46,3724,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,3726,3727,3730],{},[42,3728,3729],{},"31.8",[36,3731,3732,3734],{},[46,3733,280],{"mathvariant":279},[46,3735,280],{"mathvariant":279},[50,3737,3738],{"encoding":52},"A = 50^\\circ 22' 31.8''",[20,3740,3742,3760],{"className":3741,"ariaHidden":58},[57],[20,3743,3745,3748,3751,3754,3757],{"className":3744},[62],[20,3746],{"className":3747,"style":1244},[66],[20,3749,1510],{"className":3750},[71,302],[20,3752],{"className":3753,"style":709},[708],[20,3755,580],{"className":3756},[713],[20,3758],{"className":3759,"style":709},[708],[20,3761,3763,3766,3769,3798,3801,3833,3837],{"className":3762},[62],[20,3764],{"className":3765,"style":295},[66],[20,3767,1779],{"className":3768},[71],[20,3770,3772,3775],{"className":3771},[71],[20,3773,79],{"className":3774},[71],[20,3776,3778],{"className":3777},[83],[20,3779,3781],{"className":3780},[87],[20,3782,3784],{"className":3783},[91],[20,3785,3787],{"className":3786,"style":67},[95],[20,3788,3789,3792],{"style":98},[20,3790],{"className":3791,"style":103},[102],[20,3793,3795],{"className":3794},[107,108,109,110],[20,3796,48],{"className":3797},[114,110],[20,3799,139],{"className":3800},[71],[20,3802,3804,3807],{"className":3803},[71],[20,3805,139],{"className":3806},[71],[20,3808,3810],{"className":3809},[83],[20,3811,3813],{"className":3812},[87],[20,3814,3816],{"className":3815},[91],[20,3817,3819],{"className":3818,"style":295},[95],[20,3820,3821,3824],{"style":98},[20,3822],{"className":3823,"style":103},[102],[20,3825,3827],{"className":3826},[107,108,109,110],[20,3828,3830],{"className":3829},[71,110],[20,3831,280],{"className":3832},[71,110],[20,3834,3836],{"className":3835},[71],"31.",[20,3838,3840,3844],{"className":3839},[71],[20,3841,3843],{"className":3842},[71],"8",[20,3845,3847],{"className":3846},[83],[20,3848,3850],{"className":3849},[87],[20,3851,3853],{"className":3852},[91],[20,3854,3856],{"className":3855,"style":295},[95],[20,3857,3858,3861],{"style":98},[20,3859],{"className":3860,"style":103},[102],[20,3862,3864],{"className":3863},[107,108,109,110],[20,3865,3867],{"className":3866},[71,110],[20,3868,330],{"className":3869},[71,110],"\nCorrected ",[20,3872,3874,3913],{"className":3873},[23],[20,3875,3877],{"className":3876},[27],[29,3878,3879],{"xmlns":31},[33,3880,3881,3910],{},[36,3882,3883,3885,3887,3893,3899],{},[272,3884,1515],{},[46,3886,580],{},[39,3888,3889,3891],{},[42,3890,1901],{},[46,3892,48],{},[39,3894,3895,3897],{},[42,3896,1908],{},[46,3898,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,3900,3901,3904],{},[42,3902,3903],{},"46.2",[36,3905,3906,3908],{},[46,3907,280],{"mathvariant":279},[46,3909,280],{"mathvariant":279},[50,3911,3912],{"encoding":52},"B = 65^\\circ 40' 46.2''",[20,3914,3916,3934],{"className":3915,"ariaHidden":58},[57],[20,3917,3919,3922,3925,3928,3931],{"className":3918},[62],[20,3920],{"className":3921,"style":1244},[66],[20,3923,1515],{"className":3924,"style":1543},[71,302],[20,3926],{"className":3927,"style":709},[708],[20,3929,580],{"className":3930},[713],[20,3932],{"className":3933,"style":709},[708],[20,3935,3937,3940,3943,3972,3975,4007,4010],{"className":3936},[62],[20,3938],{"className":3939,"style":295},[66],[20,3941,1265],{"className":3942},[71],[20,3944,3946,3949],{"className":3945},[71],[20,3947,1779],{"className":3948},[71],[20,3950,3952],{"className":3951},[83],[20,3953,3955],{"className":3954},[87],[20,3956,3958],{"className":3957},[91],[20,3959,3961],{"className":3960,"style":67},[95],[20,3962,3963,3966],{"style":98},[20,3964],{"className":3965,"style":103},[102],[20,3967,3969],{"className":3968},[107,108,109,110],[20,3970,48],{"className":3971},[114,110],[20,3973,1987],{"className":3974},[71],[20,3976,3978,3981],{"className":3977},[71],[20,3979,79],{"className":3980},[71],[20,3982,3984],{"className":3983},[83],[20,3985,3987],{"className":3986},[87],[20,3988,3990],{"className":3989},[91],[20,3991,3993],{"className":3992,"style":295},[95],[20,3994,3995,3998],{"style":98},[20,3996],{"className":3997,"style":103},[102],[20,3999,4001],{"className":4000},[107,108,109,110],[20,4002,4004],{"className":4003},[71,110],[20,4005,280],{"className":4006},[71,110],[20,4008,2198],{"className":4009},[71],[20,4011,4013,4016],{"className":4012},[71],[20,4014,139],{"className":4015},[71],[20,4017,4019],{"className":4018},[83],[20,4020,4022],{"className":4021},[87],[20,4023,4025],{"className":4024},[91],[20,4026,4028],{"className":4027,"style":295},[95],[20,4029,4030,4033],{"style":98},[20,4031],{"className":4032,"style":103},[102],[20,4034,4036],{"className":4035},[107,108,109,110],[20,4037,4039],{"className":4038},[71,110],[20,4040,330],{"className":4041},[71,110],[20,4043,4045,4084],{"className":4044},[23],[20,4046,4048],{"className":4047},[27],[29,4049,4050],{"xmlns":31},[33,4051,4052,4081],{},[36,4053,4054,4056,4058,4064,4070],{},[272,4055,635],{},[46,4057,580],{},[39,4059,4060,4062],{},[42,4061,2076],{},[46,4063,48],{},[39,4065,4066,4068],{},[42,4067,2083],{},[46,4069,280],{"mathvariant":279,"lspace":1734,"rspace":1734},[39,4071,4072,4075],{},[42,4073,4074],{},"45.9",[36,4076,4077,4079],{},[46,4078,280],{"mathvariant":279},[46,4080,280],{"mathvariant":279},[50,4082,4083],{"encoding":52},"C = 63^\\circ 56' 45.9''",[20,4085,4087,4105],{"className":4086,"ariaHidden":58},[57],[20,4088,4090,4093,4096,4099,4102],{"className":4089},[62],[20,4091],{"className":4092,"style":1244},[66],[20,4094,635],{"className":4095,"style":1040},[71,302],[20,4097],{"className":4098,"style":709},[708],[20,4100,580],{"className":4101},[713],[20,4103],{"className":4104,"style":709},[708],[20,4106,4108,4111,4114,4143,4146,4178,4182],{"className":4107},[62],[20,4109],{"className":4110,"style":295},[66],[20,4112,1265],{"className":4113},[71],[20,4115,4117,4120],{"className":4116},[71],[20,4118,2136],{"className":4119},[71],[20,4121,4123],{"className":4122},[83],[20,4124,4126],{"className":4125},[87],[20,4127,4129],{"className":4128},[91],[20,4130,4132],{"className":4131,"style":67},[95],[20,4133,4134,4137],{"style":98},[20,4135],{"className":4136,"style":103},[102],[20,4138,4140],{"className":4139},[107,108,109,110],[20,4141,48],{"className":4142},[114,110],[20,4144,1779],{"className":4145},[71],[20,4147,4149,4152],{"className":4148},[71],[20,4150,1265],{"className":4151},[71],[20,4153,4155],{"className":4154},[83],[20,4156,4158],{"className":4157},[87],[20,4159,4161],{"className":4160},[91],[20,4162,4164],{"className":4163,"style":295},[95],[20,4165,4166,4169],{"style":98},[20,4167],{"className":4168,"style":103},[102],[20,4170,4172],{"className":4171},[107,108,109,110],[20,4173,4175],{"className":4174},[71,110],[20,4176,280],{"className":4177},[71,110],[20,4179,4181],{"className":4180},[71],"45.",[20,4183,4185,4188],{"className":4184},[71],[20,4186,2881],{"className":4187},[71],[20,4189,4191],{"className":4190},[83],[20,4192,4194],{"className":4193},[87],[20,4195,4197],{"className":4196},[91],[20,4198,4200],{"className":4199,"style":295},[95],[20,4201,4202,4205],{"style":98},[20,4203],{"className":4204,"style":103},[102],[20,4206,4208],{"className":4207},[107,108,109,110],[20,4209,4211],{"className":4210},[71,110],[20,4212,330],{"className":4213},[71,110],[11,4215,4217],{"id":4216},"field-workflow","Field Workflow",[4219,4220,4221,4225,4355,4359,4429,4433,4436,4440,4509,4513],"steps",{},[483,4222,4224],{"id":4223},"reconnaissance","Reconnaissance",[16,4226,4227,4228,1547,4291,1316],{},"Establish a network where triangles are well-conditioned (angles between ",[20,4229,4231,4250],{"className":4230},[23],[20,4232,4234],{"className":4233},[27],[29,4235,4236],{"xmlns":31},[33,4237,4238,4247],{},[36,4239,4240],{},[39,4241,4242,4245],{},[42,4243,4244],{},"30",[46,4246,48],{},[50,4248,4249],{"encoding":52},"30^\\circ",[20,4251,4253],{"className":4252,"ariaHidden":58},[57],[20,4254,4256,4259,4262],{"className":4255},[62],[20,4257],{"className":4258,"style":67},[66],[20,4260,2136],{"className":4261},[71],[20,4263,4265,4268],{"className":4264},[71],[20,4266,79],{"className":4267},[71],[20,4269,4271],{"className":4270},[83],[20,4272,4274],{"className":4273},[87],[20,4275,4277],{"className":4276},[91],[20,4278,4280],{"className":4279,"style":67},[95],[20,4281,4282,4285],{"style":98},[20,4283],{"className":4284,"style":103},[102],[20,4286,4288],{"className":4287},[107,108,109,110],[20,4289,48],{"className":4290},[114,110],[20,4292,4294,4313],{"className":4293},[23],[20,4295,4297],{"className":4296},[27],[29,4298,4299],{"xmlns":31},[33,4300,4301,4310],{},[36,4302,4303],{},[39,4304,4305,4308],{},[42,4306,4307],{},"120",[46,4309,48],{},[50,4311,4312],{"encoding":52},"120^\\circ",[20,4314,4316],{"className":4315,"ariaHidden":58},[57],[20,4317,4319,4322,4326],{"className":4318},[62],[20,4320],{"className":4321,"style":67},[66],[20,4323,4325],{"className":4324},[71],"12",[20,4327,4329,4332],{"className":4328},[71],[20,4330,79],{"className":4331},[71],[20,4333,4335],{"className":4334},[83],[20,4336,4338],{"className":4337},[87],[20,4339,4341],{"className":4340},[91],[20,4342,4344],{"className":4343,"style":67},[95],[20,4345,4346,4349],{"style":98},[20,4347],{"className":4348,"style":103},[102],[20,4350,4352],{"className":4351},[107,108,109,110],[20,4353,48],{"className":4354},[114,110],[483,4356,4358],{"id":4357},"precise-angular-observation","Precise Angular Observation",[16,4360,4361,4362,4428],{},"Measure all three angles of each triangle using a ",[20,4363,4365,4387],{"className":4364},[23],[20,4366,4368],{"className":4367},[27],[29,4369,4370],{"xmlns":31},[33,4371,4372,4384],{},[36,4373,4374],{},[39,4375,4376,4378],{},[42,4377,606],{},[36,4379,4380,4382],{},[46,4381,280],{"mathvariant":279},[46,4383,280],{"mathvariant":279},[50,4385,4386],{"encoding":52},"1''",[20,4388,4390],{"className":4389,"ariaHidden":58},[57],[20,4391,4393,4396],{"className":4392},[62],[20,4394],{"className":4395,"style":295},[66],[20,4397,4399,4402],{"className":4398},[71],[20,4400,606],{"className":4401},[71],[20,4403,4405],{"className":4404},[83],[20,4406,4408],{"className":4407},[87],[20,4409,4411],{"className":4410},[91],[20,4412,4414],{"className":4413,"style":295},[95],[20,4415,4416,4419],{"style":98},[20,4417],{"className":4418,"style":103},[102],[20,4420,4422],{"className":4421},[107,108,109,110],[20,4423,4425],{"className":4424},[71,110],[20,4426,330],{"className":4427},[71,110]," theodolite, completing multiple sets on both faces to minimize instrumental error .",[483,4430,4432],{"id":4431},"baseline-measurement","Baseline Measurement",[16,4434,4435],{},"Measure at least one side of the network (the baseline) with high-precision EDM or GPS, reducing the length to the ellipsoid .",[483,4437,4439],{"id":4438},"compute-excess","Compute Excess",[16,4441,4442,4443,4508],{},"Calculate ",[20,4444,4446,4467],{"className":4445},[23],[20,4447,4449],{"className":4448},[27],[29,4450,4451],{"xmlns":31},[33,4452,4453,4465],{},[36,4454,4455],{},[39,4456,4457,4459],{},[272,4458,274],{},[36,4460,4461,4463],{},[46,4462,280],{"mathvariant":279},[46,4464,280],{"mathvariant":279},[50,4466,285],{"encoding":52},[20,4468,4470],{"className":4469,"ariaHidden":58},[57],[20,4471,4473,4476],{"className":4472},[62],[20,4474],{"className":4475,"style":295},[66],[20,4477,4479,4482],{"className":4478},[71],[20,4480,274],{"className":4481,"style":303},[71,302],[20,4483,4485],{"className":4484},[83],[20,4486,4488],{"className":4487},[87],[20,4489,4491],{"className":4490},[91],[20,4492,4494],{"className":4493,"style":295},[95],[20,4495,4496,4499],{"style":98},[20,4497],{"className":4498,"style":103},[102],[20,4500,4502],{"className":4501},[107,108,109,110],[20,4503,4505],{"className":4504},[71,110],[20,4506,330],{"className":4507},[71,110]," for every triangle in the network using the approximate area .",[483,4510,4512],{"id":4511},"distribution","Distribution",[16,4514,4515,4516,4591,4592,4653],{},"Apply the angular misclosure correction, ensuring the sum of the plane angles (Observed - ",[20,4517,4519,4545],{"className":4518},[23],[20,4520,4522],{"className":4521},[27],[29,4523,4524],{"xmlns":31},[33,4525,4526,4542],{},[36,4527,4528,4538,4540],{},[39,4529,4530,4532],{},[272,4531,274],{},[36,4533,4534,4536],{},[46,4535,280],{"mathvariant":279},[46,4537,280],{"mathvariant":279},[272,4539,2362],{"mathvariant":279},[42,4541,2136],{},[50,4543,4544],{"encoding":52},"E''\u002F3",[20,4546,4548],{"className":4547,"ariaHidden":58},[57],[20,4549,4551,4555,4587],{"className":4550},[62],[20,4552],{"className":4553,"style":4554},[66],"height:1.0019em;vertical-align:-0.25em;",[20,4556,4558,4561],{"className":4557},[71],[20,4559,274],{"className":4560,"style":303},[71,302],[20,4562,4564],{"className":4563},[83],[20,4565,4567],{"className":4566},[87],[20,4568,4570],{"className":4569},[91],[20,4571,4573],{"className":4572,"style":295},[95],[20,4574,4575,4578],{"style":98},[20,4576],{"className":4577,"style":103},[102],[20,4579,4581],{"className":4580},[107,108,109,110],[20,4582,4584],{"className":4583},[71,110],[20,4585,330],{"className":4586},[71,110],[20,4588,4590],{"className":4589},[71],"\u002F3",") equals ",[20,4593,4595,4612],{"className":4594},[23],[20,4596,4598],{"className":4597},[27],[29,4599,4600],{"xmlns":31},[33,4601,4602,4610],{},[36,4603,4604],{},[39,4605,4606,4608],{},[42,4607,44],{},[46,4609,48],{},[50,4611,53],{"encoding":52},[20,4613,4615],{"className":4614,"ariaHidden":58},[57],[20,4616,4618,4621,4624],{"className":4617},[62],[20,4619],{"className":4620,"style":67},[66],[20,4622,72],{"className":4623},[71],[20,4625,4627,4630],{"className":4626},[71],[20,4628,79],{"className":4629},[71],[20,4631,4633],{"className":4632},[83],[20,4634,4636],{"className":4635},[87],[20,4637,4639],{"className":4638},[91],[20,4640,4642],{"className":4641,"style":67},[95],[20,4643,4644,4647],{"style":98},[20,4645],{"className":4646,"style":103},[102],[20,4648,4650],{"className":4649},[107,108,109,110],[20,4651,48],{"className":4652},[114,110]," for further coordinate calculation .",[11,4655,4657],{"id":4656},"practical-tips","Practical Tips",[1102,4659,4660,4799],{},[1105,4661,4662,4665,4666,4732,4733,4798],{},[254,4663,4664],{},"The 1-Second Rule:"," For a triangle with an area of ",[20,4667,4669,4688],{"className":4668},[23],[20,4670,4672],{"className":4671},[27],[29,4673,4674],{"xmlns":31},[33,4675,4676,4686],{},[36,4677,4678,4680],{},[42,4679,130],{},[39,4681,4682,4684],{},[134,4683,136],{},[42,4685,139],{},[50,4687,142],{"encoding":52},[20,4689,4691],{"className":4690,"ariaHidden":58},[57],[20,4692,4694,4697,4700],{"className":4693},[62],[20,4695],{"className":4696,"style":152},[66],[20,4698,130],{"className":4699},[71],[20,4701,4703,4709],{"className":4702},[71],[20,4704,4706],{"className":4705},[71,162],[20,4707,136],{"className":4708},[71],[20,4710,4712],{"className":4711},[83],[20,4713,4715],{"className":4714},[87],[20,4716,4718],{"className":4717},[91],[20,4719,4721],{"className":4720,"style":152},[95],[20,4722,4723,4726],{"style":180},[20,4724],{"className":4725,"style":103},[102],[20,4727,4729],{"className":4728},[107,108,109,110],[20,4730,139],{"className":4731},[71,110],", the spherical excess is approximately ",[20,4734,4736,4757],{"className":4735},[23],[20,4737,4739],{"className":4738},[27],[29,4740,4741],{"xmlns":31},[33,4742,4743,4755],{},[36,4744,4745],{},[39,4746,4747,4749],{},[42,4748,606],{},[36,4750,4751,4753],{},[46,4752,280],{"mathvariant":279},[46,4754,280],{"mathvariant":279},[50,4756,4386],{"encoding":52},[20,4758,4760],{"className":4759,"ariaHidden":58},[57],[20,4761,4763,4766],{"className":4762},[62],[20,4764],{"className":4765,"style":295},[66],[20,4767,4769,4772],{"className":4768},[71],[20,4770,606],{"className":4771},[71],[20,4773,4775],{"className":4774},[83],[20,4776,4778],{"className":4777},[87],[20,4779,4781],{"className":4780},[91],[20,4782,4784],{"className":4783,"style":295},[95],[20,4785,4786,4789],{"style":98},[20,4787],{"className":4788,"style":103},[102],[20,4790,4792],{"className":4791},[107,108,109,110],[20,4793,4795],{"className":4794},[71,110],[20,4796,330],{"className":4797},[71,110],". If your triangles are smaller than this, the effect is often negligible for tertiary work .",[1105,4800,4801,4804,4805,4870],{},[254,4802,4803],{},"R is Variable:"," Use the mean radius of curvature for the latitude of the survey site for higher precision in the ",[20,4806,4808,4829],{"className":4807},[23],[20,4809,4811],{"className":4810},[27],[29,4812,4813],{"xmlns":31},[33,4814,4815,4827],{},[36,4816,4817],{},[39,4818,4819,4821],{},[272,4820,274],{},[36,4822,4823,4825],{},[46,4824,280],{"mathvariant":279},[46,4826,280],{"mathvariant":279},[50,4828,285],{"encoding":52},[20,4830,4832],{"className":4831,"ariaHidden":58},[57],[20,4833,4835,4838],{"className":4834},[62],[20,4836],{"className":4837,"style":295},[66],[20,4839,4841,4844],{"className":4840},[71],[20,4842,274],{"className":4843,"style":303},[71,302],[20,4845,4847],{"className":4846},[83],[20,4848,4850],{"className":4849},[87],[20,4851,4853],{"className":4852},[91],[20,4854,4856],{"className":4855,"style":295},[95],[20,4857,4858,4861],{"style":98},[20,4859],{"className":4860,"style":103},[102],[20,4862,4864],{"className":4863},[107,108,109,110],[20,4865,4867],{"className":4866},[71,110],[20,4868,330],{"className":4869},[71,110]," calculation .",[11,4872,4874],{"id":4873},"common-mistakes","Common Mistakes",[1102,4876,4877,5136],{},[1105,4878,4879,4882,4883,4980,4981,5040,5041,1547,5074,1316],{},[254,4880,4881],{},"Forgetting Units:"," When using the formula ",[20,4884,4886,4913],{"className":4885},[23],[20,4887,4889],{"className":4888},[27],[29,4890,4891],{"xmlns":31},[33,4892,4893,4910],{},[36,4894,4895,4897,4899,4902,4904],{},[272,4896,274],{},[46,4898,580],{},[134,4900,4901],{},"Area",[272,4903,2362],{"mathvariant":279},[39,4905,4906,4908],{},[272,4907,593],{},[42,4909,139],{},[50,4911,4912],{"encoding":52},"E = \\text{Area}\u002FR^2",[20,4914,4916,4934],{"className":4915,"ariaHidden":58},[57],[20,4917,4919,4922,4925,4928,4931],{"className":4918},[62],[20,4920],{"className":4921,"style":1244},[66],[20,4923,274],{"className":4924,"style":303},[71,302],[20,4926],{"className":4927,"style":709},[708],[20,4929,580],{"className":4930},[713],[20,4932],{"className":4933,"style":709},[708],[20,4935,4937,4941,4947,4950],{"className":4936},[62],[20,4938],{"className":4939,"style":4940},[66],"height:1.0641em;vertical-align:-0.25em;",[20,4942,4944],{"className":4943},[71,162],[20,4945,4901],{"className":4946},[71],[20,4948,2362],{"className":4949},[71],[20,4951,4953,4956],{"className":4952},[71],[20,4954,593],{"className":4955,"style":765},[71,302],[20,4957,4959],{"className":4958},[83],[20,4960,4962],{"className":4961},[87],[20,4963,4965],{"className":4964},[91],[20,4966,4969],{"className":4967,"style":4968},[95],"height:0.8141em;",[20,4970,4971,4974],{"style":98},[20,4972],{"className":4973,"style":103},[102],[20,4975,4977],{"className":4976},[107,108,109,110],[20,4978,139],{"className":4979},[71,110],", ensure Area and ",[20,4982,4984,5002],{"className":4983},[23],[20,4985,4987],{"className":4986},[27],[29,4988,4989],{"xmlns":31},[33,4990,4991,4999],{},[36,4992,4993],{},[39,4994,4995,4997],{},[272,4996,593],{},[42,4998,139],{},[50,5000,5001],{"encoding":52},"R^2",[20,5003,5005],{"className":5004,"ariaHidden":58},[57],[20,5006,5008,5011],{"className":5007},[62],[20,5009],{"className":5010,"style":4968},[66],[20,5012,5014,5017],{"className":5013},[71],[20,5015,593],{"className":5016,"style":765},[71,302],[20,5018,5020],{"className":5019},[83],[20,5021,5023],{"className":5022},[87],[20,5024,5026],{"className":5025},[91],[20,5027,5029],{"className":5028,"style":4968},[95],[20,5030,5031,5034],{"style":98},[20,5032],{"className":5033,"style":103},[102],[20,5035,5037],{"className":5036},[107,108,109,110],[20,5038,139],{"className":5039},[71,110]," are in the same units (e.g., ",[20,5042,5044,5059],{"className":5043},[23],[20,5045,5047],{"className":5046},[27],[29,5048,5049],{"xmlns":31},[33,5050,5051,5056],{},[36,5052,5053],{},[134,5054,5055],{},"km",[50,5057,5058],{"encoding":52},"\\text{km}",[20,5060,5062],{"className":5061,"ariaHidden":58},[57],[20,5063,5065,5068],{"className":5064},[62],[20,5066],{"className":5067,"style":1641},[66],[20,5069,5071],{"className":5070},[71,162],[20,5072,5055],{"className":5073},[71],[20,5075,5077,5095],{"className":5076},[23],[20,5078,5080],{"className":5079},[27],[29,5081,5082],{"xmlns":31},[33,5083,5084,5092],{},[36,5085,5086],{},[39,5087,5088,5090],{},[134,5089,5055],{},[42,5091,139],{},[50,5093,5094],{"encoding":52},"\\text{km}^2",[20,5096,5098],{"className":5097,"ariaHidden":58},[57],[20,5099,5101,5104],{"className":5100},[62],[20,5102],{"className":5103,"style":152},[66],[20,5105,5107,5113],{"className":5106},[71],[20,5108,5110],{"className":5109},[71,162],[20,5111,5055],{"className":5112},[71],[20,5114,5116],{"className":5115},[83],[20,5117,5119],{"className":5118},[87],[20,5120,5122],{"className":5121},[91],[20,5123,5125],{"className":5124,"style":152},[95],[20,5126,5127,5130],{"style":180},[20,5128],{"className":5129,"style":103},[102],[20,5131,5133],{"className":5132},[107,108,109,110],[20,5134,139],{"className":5135},[71,110],[1105,5137,5138,5141],{},[254,5139,5140],{},"Linear Misclosure:"," Applying spherical excess corrections doesn't fix linear errors; it only addresses the geometric contradiction caused by the Earth's shape .",[11,5143,5145],{"id":5144},"faq","FAQ",[5147,5148,5149,5216,5256],"accordion",{},[5150,5151,5153],"accordion-item",{"label":5152},"Do I need to calculate spherical excess for GPS surveys?",[16,5154,5155,5156,5215],{},"GPS coordinates are calculated in 3D Cartesian space (",[20,5157,5159,5184],{"className":5158},[23],[20,5160,5162],{"className":5161},[27],[29,5163,5164],{"xmlns":31},[33,5165,5166,5181],{},[36,5167,5168,5171,5173,5176,5178],{},[272,5169,5170],{},"X",[46,5172,657],{"separator":58},[272,5174,5175],{},"Y",[46,5177,657],{"separator":58},[272,5179,5180],{},"Z",[50,5182,5183],{"encoding":52},"X, Y, Z",[20,5185,5187],{"className":5186,"ariaHidden":58},[57],[20,5188,5190,5193,5197,5200,5203,5206,5209,5212],{"className":5189},[62],[20,5191],{"className":5192,"style":1530},[66],[20,5194,5170],{"className":5195,"style":5196},[71,302],"margin-right:0.0785em;",[20,5198,657],{"className":5199},[1092],[20,5201],{"className":5202,"style":1096},[708],[20,5204,5175],{"className":5205,"style":1072},[71,302],[20,5207,657],{"className":5208},[1092],[20,5210],{"className":5211,"style":1096},[708],[20,5213,5180],{"className":5214,"style":1040},[71,302],") 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 .",[5150,5217,5219],{"label":5218},"What is Legendre’s Theorem exactly?",[16,5220,5221,5222,5255],{},"It states that a spherical triangle behaves like a plane triangle if you subtract ",[20,5223,5225,5243],{"className":5224},[23],[20,5226,5228],{"className":5227},[27],[29,5229,5230],{"xmlns":31},[33,5231,5232,5240],{},[36,5233,5234,5236,5238],{},[42,5235,606],{},[272,5237,2362],{"mathvariant":279},[42,5239,2136],{},[50,5241,5242],{"encoding":52},"1\u002F3",[20,5244,5246],{"className":5245,"ariaHidden":58},[57],[20,5247,5249,5252],{"className":5248},[62],[20,5250],{"className":5251,"style":1375},[66],[20,5253,5242],{"className":5254},[71]," of the spherical excess from each of the spherical angles .",[5150,5257,5259],{"label":5258},"Is spherical excess always positive?",[16,5260,5261,5262,332],{},"Yes. Because the Earth is convex, the observed angles will always sum to more than ",[20,5263,5265,5282],{"className":5264},[23],[20,5266,5268],{"className":5267},[27],[29,5269,5270],{"xmlns":31},[33,5271,5272,5280],{},[36,5273,5274],{},[39,5275,5276,5278],{},[42,5277,44],{},[46,5279,48],{},[50,5281,53],{"encoding":52},[20,5283,5285],{"className":5284,"ariaHidden":58},[57],[20,5286,5288,5291,5294],{"className":5287},[62],[20,5289],{"className":5290,"style":67},[66],[20,5292,72],{"className":5293},[71],[20,5295,5297,5300],{"className":5296},[71],[20,5298,79],{"className":5299},[71],[20,5301,5303],{"className":5302},[83],[20,5304,5306],{"className":5305},[87],[20,5307,5309],{"className":5308},[91],[20,5310,5312],{"className":5311,"style":67},[95],[20,5313,5314,5317],{"style":98},[20,5315],{"className":5316,"style":103},[102],[20,5318,5320],{"className":5319},[107,108,109,110],[20,5321,48],{"className":5322},[114,110],[11,5324,5326],{"id":5325},"conclusion","Conclusion",[16,5328,5329,5330,5395],{},"Spherical excess is the boundary where plane surveying meets geodesy. By understanding how to calculate ",[20,5331,5333,5354],{"className":5332},[23],[20,5334,5336],{"className":5335},[27],[29,5337,5338],{"xmlns":31},[33,5339,5340,5352],{},[36,5341,5342],{},[39,5343,5344,5346],{},[272,5345,274],{},[36,5347,5348,5350],{},[46,5349,280],{"mathvariant":279},[46,5351,280],{"mathvariant":279},[50,5353,285],{"encoding":52},[20,5355,5357],{"className":5356,"ariaHidden":58},[57],[20,5358,5360,5363],{"className":5359},[62],[20,5361],{"className":5362,"style":295},[66],[20,5364,5366,5369],{"className":5365},[71],[20,5367,274],{"className":5368,"style":303},[71,302],[20,5370,5372],{"className":5371},[83],[20,5373,5375],{"className":5374},[87],[20,5376,5378],{"className":5377},[91],[20,5379,5381],{"className":5380,"style":295},[95],[20,5382,5383,5386],{"style":98},[20,5384],{"className":5385,"style":103},[102],[20,5387,5389],{"className":5388},[107,108,109,110],[20,5390,5392],{"className":5391},[71,110],[20,5393,330],{"className":5394},[71,110],", the engineering surveyor ensures that large-scale control networks remain mathematically consistent and capable of supporting high-precision construction across vast distances.",[11,5397,5399],{"id":5398},"references","References",[16,5401,5402,5403,5407],{},"Schofield, W. (2001). ",[5404,5405,5406],"em",{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.",{"title":5409,"searchDepth":5410,"depth":5410,"links":5411},"",2,[5412,5413,5414,5415,5420,5421,5428,5429,5430,5431,5432],{"id":13,"depth":5410,"text":14},{"id":335,"depth":5410,"text":336},{"id":470,"depth":5410,"text":471},{"id":480,"depth":5410,"text":481,"children":5416},[5417,5419],{"id":485,"depth":5418,"text":486},3,{"id":1319,"depth":5418,"text":1320},{"id":1488,"depth":5410,"text":1489},{"id":4216,"depth":5410,"text":4217,"children":5422},[5423,5424,5425,5426,5427],{"id":4223,"depth":5418,"text":4224},{"id":4357,"depth":5418,"text":4358},{"id":4431,"depth":5418,"text":4432},{"id":4438,"depth":5418,"text":4439},{"id":4511,"depth":5418,"text":4512},{"id":4656,"depth":5410,"text":4657},{"id":4873,"depth":5410,"text":4874},{"id":5144,"depth":5410,"text":5145},{"id":5325,"depth":5410,"text":5326},{"id":5398,"depth":5410,"text":5399},"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.","md","\u002Fimages\u002Fspherical-excess.jpg",{"authorCredentials":5438,"authorBio":5439,"commentCount":5440,"tags":5441,"relatedTools":5447,"relatedArticles":5450,"sources":5453},"LS, P.Eng","Specialist in geodetic surveying and primary control network adjustment.",0,[5442,5443,5444,5445,5446],"Geodesy","Spherical Excess","Triangulation","Network Adjustment","Math",[5448,5449],"Theodolite","Least Squares Software",[5451,5452],"wgs84-datum-projections","weighted-observations-mpv",[5454],{"title":5455,"type":5456,"note":5457},"Engineering Surveying (Fifth Edition)","Book","Primary source for spherical excess formulas and Legendre’s theorem.",true,"\u002Fblog\u002Fspherical-excess-large-networks","2024-04-11",16,{"title":5,"description":5434},{"loc":5459},"blog\u002F24.spherical-excess-large-networks","PYxbA57iyyzpqzu7LWCStZnGumqEOPTf5eJG4fRaJTY",[5467,5472],{"title":5468,"path":5469,"stem":5470,"description":5471,"children":-1},"WGS84 and Projections: The Mathematics of Global to Local Mapping","\u002Fblog\u002Fwgs84-datum-and-geodetic-projections","blog\u002F22.wgs84-datum-and-geodetic-projections","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.",{"title":5473,"path":5474,"stem":5475,"description":5476,"children":-1},"Machine Guidance: Integrating RTK GPS with Construction Plant","\u002Fblog\u002Fmachine-guidance-rtk-gps-construction","blog\u002F25.machine-guidance-rtk-gps-construction","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.",1783851944994]