[{"data":1,"prerenderedAt":6381},["ShallowReactive",2],{"blog-\u002Fblog\u002Fgyro-theodolite-underground-orientation":3,"blog-surround-\u002Fblog\u002Fgyro-theodolite-underground-orientation":6370},{"id":4,"title":5,"author":6,"body":7,"category":6339,"description":6340,"extension":6341,"image":6342,"meta":6343,"navigation":6362,"path":6363,"publishedAt":6364,"readTime":6365,"seo":6366,"sitemap":6367,"stem":6368,"__hash__":6369},"blog\u002Fblog\u002F6.gyro-theodolite-underground-orientation.md","The Gyro-Theodolite: Mastering True North Underground","Surveyiiing Expert",{"type":8,"value":9,"toc":6314},"minimark",[10,15,37,41,241,245,314,318,324,351,355,692,697,1290,1864,1868,2126,2130,2453,2458,3360,6161,6165,6222,6226,6253,6257,6289,6293,6299,6303],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,28,29,32,33,36],"p",{},"In the isolation of a tunnel or mine, traditional celestial observations or GPS signals are impossible. To maintain a precise heading, surveyors rely on the ",[20,21,22],"strong",{},"gyro-theodolite",", a north-seeking instrument that integrates a high-speed gyroscope with a precision theodolite ",[25,26,27],"span",{},"1, 2",". Unlike a magnetic compass, which is influenced by local ore bodies and electrical infrastructure, the gyro-theodolite finds ",[20,30,31],{},"True North"," by interacting with the Earth's rotation ",[25,34,35],{},"2",".",[11,38,40],{"id":39},"why-this-matters","Why This Matters",[16,42,43,44,145,146,193,194,233,234,237,238,36],{},"Precision in underground orientation is non-negotiable. A standard error of just ",[25,45,48,83],{"className":46},[47],"katex",[25,49,52],{"className":50},[51],"katex-mathml",[53,54,56],"math",{"xmlns":55},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[57,58,59,78],"semantics",{},[60,61,62,66],"mrow",{},[63,64,65],"mo",{},"±",[67,68,69,73],"msup",{},[70,71,72],"mn",{},"1",[63,74,77],{"mathvariant":75,"lspace":76,"rspace":76},"normal","0em","′",[79,80,82],"annotation",{"encoding":81},"application\u002Fx-tex","\\pm 1'",[25,84,88],{"className":85,"ariaHidden":87},[86],"katex-html","true",[25,89,92,97,101],{"className":90},[91],"base",[25,93],{"className":94,"style":96},[95],"strut","height:0.8352em;vertical-align:-0.0833em;",[25,98,65],{"className":99},[100],"mord",[25,102,104,107],{"className":103},[100],[25,105,72],{"className":106},[100],[25,108,111],{"className":109},[110],"msupsub",[25,112,115],{"className":113},[114],"vlist-t",[25,116,119],{"className":117},[118],"vlist-r",[25,120,124],{"className":121,"style":123},[122],"vlist","height:0.7519em;",[25,125,127,132],{"style":126},"top:-3.063em;margin-right:0.05em;",[25,128],{"className":129,"style":131},[130],"pstrut","height:2.7em;",[25,133,139],{"className":134},[135,136,137,138],"sizing","reset-size6","size3","mtight",[25,140,142],{"className":141},[100,138],[25,143,77],{"className":144},[100,138]," in transferring a bearing through a shaft can result in a positional error of ",[25,147,149,170],{"className":148},[47],[25,150,152],{"className":151},[51],[53,153,154],{"xmlns":55},[57,155,156,167],{},[60,157,158,160,163],{},[63,159,65],{},[70,161,162],{},"300",[164,165,166],"mtext",{}," mm",[79,168,169],{"encoding":81},"\\pm 300\\text{ mm}",[25,171,173],{"className":172,"ariaHidden":87},[86],[25,174,176,180,183,186],{"className":175},[91],[25,177],{"className":178,"style":179},[95],"height:0.7278em;vertical-align:-0.0833em;",[25,181,65],{"className":182},[100],[25,184,162],{"className":185},[100],[25,187,190],{"className":188},[100,189],"text",[25,191,166],{"className":192},[100]," after only ",[25,195,197,214],{"className":196},[47],[25,198,200],{"className":199},[51],[53,201,202],{"xmlns":55},[57,203,204,211],{},[60,205,206,208],{},[70,207,72],{},[164,209,210],{}," km",[79,212,213],{"encoding":81},"1\\text{ km}",[25,215,217],{"className":216,"ariaHidden":87},[86],[25,218,220,224,227],{"className":219},[91],[25,221],{"className":222,"style":223},[95],"height:0.6944em;",[25,225,72],{"className":226},[100],[25,228,230],{"className":229},[100,189],[25,231,210],{"className":232},[100]," of tunnel drivage ",[25,235,236],{},"3",". By using a gyro-theodolite, the surveyor can establish an independent azimuth on any underground baseline, effectively controlling the propagation of angular error in long traverses ",[25,239,240],{},"4",[11,242,244],{"id":243},"background","Background",[16,246,247,248,251,252,310,311,36],{},"The gyro-theodolite typically uses a ",[20,249,250],{},"suspended gyroscope"," (such as the Wild GAK.1). The gyroscope spinner is suspended by a thin metal tape and rotates at high speeds (often over ",[25,253,255,279],{"className":254},[47],[25,256,258],{"className":257},[51],[53,259,260],{"xmlns":55},[57,261,262,276],{},[60,263,264,267,270,273],{},[70,265,266],{},"20",[63,268,269],{"separator":87},",",[70,271,272],{},"000",[164,274,275],{}," RPM",[79,277,278],{"encoding":81},"20,000\\text{ RPM}",[25,280,282],{"className":281,"ariaHidden":87},[86],[25,283,285,289,292,296,301,304],{"className":284},[91],[25,286],{"className":287,"style":288},[95],"height:0.8778em;vertical-align:-0.1944em;",[25,290,266],{"className":291},[100],[25,293,269],{"className":294},[295],"mpunct",[25,297],{"className":298,"style":300},[299],"mspace","margin-right:0.1667em;",[25,302,272],{"className":303},[100],[25,305,307],{"className":306},[100,189],[25,308,275],{"className":309},[100],"). Because the Earth is rotating, the gyroscope experiences a precessional force that causes it to oscillate about the local meridian ",[25,312,313],{},"2, 5",[11,315,317],{"id":316},"theory","Theory",[16,319,320,321,36],{},"The core principle is the interaction between the gyroscope's angular momentum and the Earth’s angular velocity ",[25,322,323],{},"5",[325,326,327,334,343],"ol",{},[328,329,330,333],"li",{},[20,331,332],{},"Precession:"," As the Earth rotates from west to east, the horizontal component of this rotation causes the gyroscope's spin axis to tilt.",[328,335,336,339,340,36],{},[20,337,338],{},"Restoring Torque:"," The suspension tape and gravity create a torque that resists this tilt, forcing the spin axis to swing back toward the north-south plane ",[25,341,342],{},"6",[328,344,345,348,349,36],{},[20,346,347],{},"Damped Harmonic Motion:"," The resulting movement is a damped harmonic oscillation about the meridian. Finding True North involves mathematically identifying the center of this oscillation ",[25,350,342],{},[11,352,354],{"id":353},"mathematical-principles","Mathematical Principles",[16,356,357,358,689,690,36],{},"The motion of the horizontal spin axis about the vertical can be represented by the differential equation:\n",[25,359,361,423],{"className":360},[47],[25,362,364],{"className":363},[51],[53,365,366],{"xmlns":55},[57,367,368,420],{},[60,369,370,379,388,391,397,404,406,412,414,417],{},[371,372,373,377],"msub",{},[374,375,376],"mi",{},"K",[70,378,72],{},[380,381,382,385],"mover",{"accent":87},[374,383,384],{},"θ",[63,386,387],{},"¨",[63,389,390],{},"+",[371,392,393,395],{},[374,394,376],{},[70,396,35],{},[380,398,399,401],{"accent":87},[374,400,384],{},[63,402,403],{},"˙",[63,405,390],{},[371,407,408,410],{},[374,409,376],{},[70,411,236],{},[374,413,384],{},[63,415,416],{},"=",[70,418,419],{},"0",[79,421,422],{"encoding":81},"K_1\\ddot{\\theta} + K_2\\dot{\\theta} + K_3\\theta = 0",[25,424,426,531,618,679],{"className":425,"ariaHidden":87},[86],[25,427,429,433,481,520,524,528],{"className":428},[91],[25,430],{"className":431,"style":432},[95],"height:1.0813em;vertical-align:-0.15em;",[25,434,436,441],{"className":435},[100],[25,437,376],{"className":438,"style":440},[100,439],"mathnormal","margin-right:0.0715em;",[25,442,444],{"className":443},[110],[25,445,448,472],{"className":446},[114,447],"vlist-t2",[25,449,451,467],{"className":450},[118],[25,452,455],{"className":453,"style":454},[122],"height:0.3011em;",[25,456,458,461],{"style":457},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[25,459],{"className":460,"style":131},[130],[25,462,464],{"className":463},[135,136,137,138],[25,465,72],{"className":466},[100,138],[25,468,471],{"className":469},[470],"vlist-s","​",[25,473,475],{"className":474},[118],[25,476,479],{"className":477,"style":478},[122],"height:0.15em;",[25,480],{},[25,482,485],{"className":483},[100,484],"accent",[25,486,488],{"className":487},[114],[25,489,491],{"className":490},[118],[25,492,495,506],{"className":493,"style":494},[122],"height:0.9313em;",[25,496,498,502],{"style":497},"top:-3em;",[25,499],{"className":500,"style":501},[130],"height:3em;",[25,503,384],{"className":504,"style":505},[100,439],"margin-right:0.0278em;",[25,507,509,512],{"style":508},"top:-3.2634em;",[25,510],{"className":511,"style":501},[130],[25,513,517],{"className":514,"style":516},[515],"accent-body","left:-0.1667em;",[25,518,387],{"className":519},[100],[25,521],{"className":522,"style":523},[299],"margin-right:0.2222em;",[25,525,390],{"className":526},[527],"mbin",[25,529],{"className":530,"style":523},[299],[25,532,534,537,577,609,612,615],{"className":533},[91],[25,535],{"className":536,"style":432},[95],[25,538,540,543],{"className":539},[100],[25,541,376],{"className":542,"style":440},[100,439],[25,544,546],{"className":545},[110],[25,547,549,569],{"className":548},[114,447],[25,550,552,566],{"className":551},[118],[25,553,555],{"className":554,"style":454},[122],[25,556,557,560],{"style":457},[25,558],{"className":559,"style":131},[130],[25,561,563],{"className":562},[135,136,137,138],[25,564,35],{"className":565},[100,138],[25,567,471],{"className":568},[470],[25,570,572],{"className":571},[118],[25,573,575],{"className":574,"style":478},[122],[25,576],{},[25,578,580],{"className":579},[100,484],[25,581,583],{"className":582},[114],[25,584,586],{"className":585},[118],[25,587,589,597],{"className":588,"style":494},[122],[25,590,591,594],{"style":497},[25,592],{"className":593,"style":501},[130],[25,595,384],{"className":596,"style":505},[100,439],[25,598,599,602],{"style":508},[25,600],{"className":601,"style":501},[130],[25,603,606],{"className":604,"style":605},[515],"left:-0.0556em;",[25,607,403],{"className":608},[100],[25,610],{"className":611,"style":523},[299],[25,613,390],{"className":614},[527],[25,616],{"className":617,"style":523},[299],[25,619,621,625,665,668,672,676],{"className":620},[91],[25,622],{"className":623,"style":624},[95],"height:0.8444em;vertical-align:-0.15em;",[25,626,628,631],{"className":627},[100],[25,629,376],{"className":630,"style":440},[100,439],[25,632,634],{"className":633},[110],[25,635,637,657],{"className":636},[114,447],[25,638,640,654],{"className":639},[118],[25,641,643],{"className":642,"style":454},[122],[25,644,645,648],{"style":457},[25,646],{"className":647,"style":131},[130],[25,649,651],{"className":650},[135,136,137,138],[25,652,236],{"className":653},[100,138],[25,655,471],{"className":656},[470],[25,658,660],{"className":659},[118],[25,661,663],{"className":662,"style":478},[122],[25,664],{},[25,666,384],{"className":667,"style":505},[100,439],[25,669],{"className":670,"style":671},[299],"margin-right:0.2778em;",[25,673,416],{"className":674},[675],"mrel",[25,677],{"className":678,"style":671},[299],[25,680,682,686],{"className":681},[91],[25,683],{"className":684,"style":685},[95],"height:0.6444em;",[25,687,419],{"className":688},[100]," ",[25,691,342],{},[693,694,696],"h3",{"id":695},"schulers-mean","Schuler’s Mean",[16,698,699,700,702,703,733,734,737,738,918,919,689,1288,36],{},"The most common field calculation is ",[20,701,696],{},", used to find the center of the oscillation from observed reversal points (",[25,704,706,720],{"className":705},[47],[25,707,709],{"className":708},[51],[53,710,711],{"xmlns":55},[57,712,713,718],{},[60,714,715],{},[374,716,717],{},"r",[79,719,717],{"encoding":81},[25,721,723],{"className":722,"ariaHidden":87},[86],[25,724,726,730],{"className":725},[91],[25,727],{"className":728,"style":729},[95],"height:0.4306em;",[25,731,717],{"className":732,"style":505},[100,439],") ",[25,735,736],{},"7",".\nFor three reversal points (",[25,739,741,775],{"className":740},[47],[25,742,744],{"className":743},[51],[53,745,746],{"xmlns":55},[57,747,748,772],{},[60,749,750,756,758,764,766],{},[371,751,752,754],{},[374,753,717],{},[70,755,72],{},[63,757,269],{"separator":87},[371,759,760,762],{},[374,761,717],{},[70,763,35],{},[63,765,269],{"separator":87},[371,767,768,770],{},[374,769,717],{},[70,771,236],{},[79,773,774],{"encoding":81},"r_1, r_2, r_3",[25,776,778],{"className":777,"ariaHidden":87},[86],[25,779,781,785,826,829,832,872,875,878],{"className":780},[91],[25,782],{"className":783,"style":784},[95],"height:0.625em;vertical-align:-0.1944em;",[25,786,788,791],{"className":787},[100],[25,789,717],{"className":790,"style":505},[100,439],[25,792,794],{"className":793},[110],[25,795,797,818],{"className":796},[114,447],[25,798,800,815],{"className":799},[118],[25,801,803],{"className":802,"style":454},[122],[25,804,806,809],{"style":805},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[25,807],{"className":808,"style":131},[130],[25,810,812],{"className":811},[135,136,137,138],[25,813,72],{"className":814},[100,138],[25,816,471],{"className":817},[470],[25,819,821],{"className":820},[118],[25,822,824],{"className":823,"style":478},[122],[25,825],{},[25,827,269],{"className":828},[295],[25,830],{"className":831,"style":300},[299],[25,833,835,838],{"className":834},[100],[25,836,717],{"className":837,"style":505},[100,439],[25,839,841],{"className":840},[110],[25,842,844,864],{"className":843},[114,447],[25,845,847,861],{"className":846},[118],[25,848,850],{"className":849,"style":454},[122],[25,851,852,855],{"style":805},[25,853],{"className":854,"style":131},[130],[25,856,858],{"className":857},[135,136,137,138],[25,859,35],{"className":860},[100,138],[25,862,471],{"className":863},[470],[25,865,867],{"className":866},[118],[25,868,870],{"className":869,"style":478},[122],[25,871],{},[25,873,269],{"className":874},[295],[25,876],{"className":877,"style":300},[299],[25,879,881,884],{"className":880},[100],[25,882,717],{"className":883,"style":505},[100,439],[25,885,887],{"className":886},[110],[25,888,890,910],{"className":889},[114,447],[25,891,893,907],{"className":892},[118],[25,894,896],{"className":895,"style":454},[122],[25,897,898,901],{"style":805},[25,899],{"className":900,"style":131},[130],[25,902,904],{"className":903},[135,136,137,138],[25,905,236],{"className":906},[100,138],[25,908,471],{"className":909},[470],[25,911,913],{"className":912},[118],[25,914,916],{"className":915,"style":478},[122],[25,917],{},"):\n",[25,920,922,981],{"className":921},[47],[25,923,925],{"className":924},[51],[53,926,927],{"xmlns":55},[57,928,929,978],{},[60,930,931,938,940,947,951,957,959,961,967,969,975],{},[371,932,933,936],{},[374,934,935],{},"N",[70,937,72],{},[63,939,416],{},[941,942,943,945],"mfrac",{},[70,944,72],{},[70,946,240],{},[63,948,950],{"stretchy":949},"false","(",[371,952,953,955],{},[374,954,717],{},[70,956,72],{},[63,958,390],{},[70,960,35],{},[371,962,963,965],{},[374,964,717],{},[70,966,35],{},[63,968,390],{},[371,970,971,973],{},[374,972,717],{},[70,974,236],{},[63,976,977],{"stretchy":949},")",[79,979,980],{"encoding":81},"N_1 = \\frac{1}{4}(r_1 + 2r_2 + r_3)",[25,982,984,1042,1179,1238],{"className":983,"ariaHidden":87},[86],[25,985,987,991,1033,1036,1039],{"className":986},[91],[25,988],{"className":989,"style":990},[95],"height:0.8333em;vertical-align:-0.15em;",[25,992,994,998],{"className":993},[100],[25,995,935],{"className":996,"style":997},[100,439],"margin-right:0.109em;",[25,999,1001],{"className":1000},[110],[25,1002,1004,1025],{"className":1003},[114,447],[25,1005,1007,1022],{"className":1006},[118],[25,1008,1010],{"className":1009,"style":454},[122],[25,1011,1013,1016],{"style":1012},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[25,1014],{"className":1015,"style":131},[130],[25,1017,1019],{"className":1018},[135,136,137,138],[25,1020,72],{"className":1021},[100,138],[25,1023,471],{"className":1024},[470],[25,1026,1028],{"className":1027},[118],[25,1029,1031],{"className":1030,"style":478},[122],[25,1032],{},[25,1034],{"className":1035,"style":671},[299],[25,1037,416],{"className":1038},[675],[25,1040],{"className":1041,"style":671},[299],[25,1043,1045,1049,1127,1130,1170,1173,1176],{"className":1044},[91],[25,1046],{"className":1047,"style":1048},[95],"height:1.1901em;vertical-align:-0.345em;",[25,1050,1052,1057,1123],{"className":1051},[100],[25,1053],{"className":1054},[1055,1056],"mopen","nulldelimiter",[25,1058,1060],{"className":1059},[941],[25,1061,1063,1114],{"className":1062},[114,447],[25,1064,1066,1111],{"className":1065},[118],[25,1067,1070,1085,1096],{"className":1068,"style":1069},[122],"height:0.8451em;",[25,1071,1073,1076],{"style":1072},"top:-2.655em;",[25,1074],{"className":1075,"style":501},[130],[25,1077,1079],{"className":1078},[135,136,137,138],[25,1080,1082],{"className":1081},[100,138],[25,1083,240],{"className":1084},[100,138],[25,1086,1088,1091],{"style":1087},"top:-3.23em;",[25,1089],{"className":1090,"style":501},[130],[25,1092],{"className":1093,"style":1095},[1094],"frac-line","border-bottom-width:0.04em;",[25,1097,1099,1102],{"style":1098},"top:-3.394em;",[25,1100],{"className":1101,"style":501},[130],[25,1103,1105],{"className":1104},[135,136,137,138],[25,1106,1108],{"className":1107},[100,138],[25,1109,72],{"className":1110},[100,138],[25,1112,471],{"className":1113},[470],[25,1115,1117],{"className":1116},[118],[25,1118,1121],{"className":1119,"style":1120},[122],"height:0.345em;",[25,1122],{},[25,1124],{"className":1125},[1126,1056],"mclose",[25,1128,950],{"className":1129},[1055],[25,1131,1133,1136],{"className":1132},[100],[25,1134,717],{"className":1135,"style":505},[100,439],[25,1137,1139],{"className":1138},[110],[25,1140,1142,1162],{"className":1141},[114,447],[25,1143,1145,1159],{"className":1144},[118],[25,1146,1148],{"className":1147,"style":454},[122],[25,1149,1150,1153],{"style":805},[25,1151],{"className":1152,"style":131},[130],[25,1154,1156],{"className":1155},[135,136,137,138],[25,1157,72],{"className":1158},[100,138],[25,1160,471],{"className":1161},[470],[25,1163,1165],{"className":1164},[118],[25,1166,1168],{"className":1167,"style":478},[122],[25,1169],{},[25,1171],{"className":1172,"style":523},[299],[25,1174,390],{"className":1175},[527],[25,1177],{"className":1178,"style":523},[299],[25,1180,1182,1186,1189,1229,1232,1235],{"className":1181},[91],[25,1183],{"className":1184,"style":1185},[95],"height:0.7944em;vertical-align:-0.15em;",[25,1187,35],{"className":1188},[100],[25,1190,1192,1195],{"className":1191},[100],[25,1193,717],{"className":1194,"style":505},[100,439],[25,1196,1198],{"className":1197},[110],[25,1199,1201,1221],{"className":1200},[114,447],[25,1202,1204,1218],{"className":1203},[118],[25,1205,1207],{"className":1206,"style":454},[122],[25,1208,1209,1212],{"style":805},[25,1210],{"className":1211,"style":131},[130],[25,1213,1215],{"className":1214},[135,136,137,138],[25,1216,35],{"className":1217},[100,138],[25,1219,471],{"className":1220},[470],[25,1222,1224],{"className":1223},[118],[25,1225,1227],{"className":1226,"style":478},[122],[25,1228],{},[25,1230],{"className":1231,"style":523},[299],[25,1233,390],{"className":1234},[527],[25,1236],{"className":1237,"style":523},[299],[25,1239,1241,1245,1285],{"className":1240},[91],[25,1242],{"className":1243,"style":1244},[95],"height:1em;vertical-align:-0.25em;",[25,1246,1248,1251],{"className":1247},[100],[25,1249,717],{"className":1250,"style":505},[100,439],[25,1252,1254],{"className":1253},[110],[25,1255,1257,1277],{"className":1256},[114,447],[25,1258,1260,1274],{"className":1259},[118],[25,1261,1263],{"className":1262,"style":454},[122],[25,1264,1265,1268],{"style":805},[25,1266],{"className":1267,"style":131},[130],[25,1269,1271],{"className":1270},[135,136,137,138],[25,1272,236],{"className":1273},[100,138],[25,1275,471],{"className":1276},[470],[25,1278,1280],{"className":1279},[118],[25,1281,1283],{"className":1282,"style":478},[122],[25,1284],{},[25,1286,977],{"className":1287},[1126],[25,1289,736],{},[16,1291,1292,1293,918,1525,689,1862,36],{},"For four reversal points (",[25,1294,1296,1338],{"className":1295},[47],[25,1297,1299],{"className":1298},[51],[53,1300,1301],{"xmlns":55},[57,1302,1303,1335],{},[60,1304,1305,1311,1313,1319,1321,1327,1329],{},[371,1306,1307,1309],{},[374,1308,717],{},[70,1310,72],{},[63,1312,269],{"separator":87},[371,1314,1315,1317],{},[374,1316,717],{},[70,1318,35],{},[63,1320,269],{"separator":87},[371,1322,1323,1325],{},[374,1324,717],{},[70,1326,236],{},[63,1328,269],{"separator":87},[371,1330,1331,1333],{},[374,1332,717],{},[70,1334,240],{},[79,1336,1337],{"encoding":81},"r_1, r_2, r_3, r_4",[25,1339,1341],{"className":1340,"ariaHidden":87},[86],[25,1342,1344,1347,1387,1390,1393,1433,1436,1439,1479,1482,1485],{"className":1343},[91],[25,1345],{"className":1346,"style":784},[95],[25,1348,1350,1353],{"className":1349},[100],[25,1351,717],{"className":1352,"style":505},[100,439],[25,1354,1356],{"className":1355},[110],[25,1357,1359,1379],{"className":1358},[114,447],[25,1360,1362,1376],{"className":1361},[118],[25,1363,1365],{"className":1364,"style":454},[122],[25,1366,1367,1370],{"style":805},[25,1368],{"className":1369,"style":131},[130],[25,1371,1373],{"className":1372},[135,136,137,138],[25,1374,72],{"className":1375},[100,138],[25,1377,471],{"className":1378},[470],[25,1380,1382],{"className":1381},[118],[25,1383,1385],{"className":1384,"style":478},[122],[25,1386],{},[25,1388,269],{"className":1389},[295],[25,1391],{"className":1392,"style":300},[299],[25,1394,1396,1399],{"className":1395},[100],[25,1397,717],{"className":1398,"style":505},[100,439],[25,1400,1402],{"className":1401},[110],[25,1403,1405,1425],{"className":1404},[114,447],[25,1406,1408,1422],{"className":1407},[118],[25,1409,1411],{"className":1410,"style":454},[122],[25,1412,1413,1416],{"style":805},[25,1414],{"className":1415,"style":131},[130],[25,1417,1419],{"className":1418},[135,136,137,138],[25,1420,35],{"className":1421},[100,138],[25,1423,471],{"className":1424},[470],[25,1426,1428],{"className":1427},[118],[25,1429,1431],{"className":1430,"style":478},[122],[25,1432],{},[25,1434,269],{"className":1435},[295],[25,1437],{"className":1438,"style":300},[299],[25,1440,1442,1445],{"className":1441},[100],[25,1443,717],{"className":1444,"style":505},[100,439],[25,1446,1448],{"className":1447},[110],[25,1449,1451,1471],{"className":1450},[114,447],[25,1452,1454,1468],{"className":1453},[118],[25,1455,1457],{"className":1456,"style":454},[122],[25,1458,1459,1462],{"style":805},[25,1460],{"className":1461,"style":131},[130],[25,1463,1465],{"className":1464},[135,136,137,138],[25,1466,236],{"className":1467},[100,138],[25,1469,471],{"className":1470},[470],[25,1472,1474],{"className":1473},[118],[25,1475,1477],{"className":1476,"style":478},[122],[25,1478],{},[25,1480,269],{"className":1481},[295],[25,1483],{"className":1484,"style":300},[299],[25,1486,1488,1491],{"className":1487},[100],[25,1489,717],{"className":1490,"style":505},[100,439],[25,1492,1494],{"className":1493},[110],[25,1495,1497,1517],{"className":1496},[114,447],[25,1498,1500,1514],{"className":1499},[118],[25,1501,1503],{"className":1502,"style":454},[122],[25,1504,1505,1508],{"style":805},[25,1506],{"className":1507,"style":131},[130],[25,1509,1511],{"className":1510},[135,136,137,138],[25,1512,240],{"className":1513},[100,138],[25,1515,471],{"className":1516},[470],[25,1518,1520],{"className":1519},[118],[25,1521,1523],{"className":1522,"style":478},[122],[25,1524],{},[25,1526,1528,1585],{"className":1527},[47],[25,1529,1531],{"className":1530},[51],[53,1532,1533],{"xmlns":55},[57,1534,1535,1582],{},[60,1536,1537,1539,1541],{},[374,1538,935],{},[63,1540,416],{},[941,1542,1543,1579],{},[60,1544,1545,1551,1553,1555,1561,1563,1565,1571,1573],{},[371,1546,1547,1549],{},[374,1548,717],{},[70,1550,72],{},[63,1552,390],{},[70,1554,236],{},[371,1556,1557,1559],{},[374,1558,717],{},[70,1560,35],{},[63,1562,390],{},[70,1564,236],{},[371,1566,1567,1569],{},[374,1568,717],{},[70,1570,236],{},[63,1572,390],{},[371,1574,1575,1577],{},[374,1576,717],{},[70,1578,240],{},[70,1580,1581],{},"8",[79,1583,1584],{"encoding":81},"N = \\frac{r_1 + 3r_2 + 3r_3 + r_4}{8}",[25,1586,1588,1607],{"className":1587,"ariaHidden":87},[86],[25,1589,1591,1595,1598,1601,1604],{"className":1590},[91],[25,1592],{"className":1593,"style":1594},[95],"height:0.6833em;",[25,1596,935],{"className":1597,"style":997},[100,439],[25,1599],{"className":1600,"style":671},[299],[25,1602,416],{"className":1603},[675],[25,1605],{"className":1606,"style":671},[299],[25,1608,1610,1614],{"className":1609},[91],[25,1611],{"className":1612,"style":1613},[95],"height:1.2062em;vertical-align:-0.345em;",[25,1615,1617,1620,1859],{"className":1616},[100],[25,1618],{"className":1619},[1055,1056],[25,1621,1623],{"className":1622},[941],[25,1624,1626,1851],{"className":1625},[114,447],[25,1627,1629,1848],{"className":1628},[118],[25,1630,1633,1647,1655],{"className":1631,"style":1632},[122],"height:0.8612em;",[25,1634,1635,1638],{"style":1072},[25,1636],{"className":1637,"style":501},[130],[25,1639,1641],{"className":1640},[135,136,137,138],[25,1642,1644],{"className":1643},[100,138],[25,1645,1581],{"className":1646},[100,138],[25,1648,1649,1652],{"style":1087},[25,1650],{"className":1651,"style":501},[130],[25,1653],{"className":1654,"style":1095},[1094],[25,1656,1658,1661],{"style":1657},"top:-3.4101em;",[25,1659],{"className":1660,"style":501},[130],[25,1662,1664],{"className":1663},[135,136,137,138],[25,1665,1667,1713,1716,1719,1759,1762,1765,1805,1808],{"className":1666},[100,138],[25,1668,1670,1673],{"className":1669},[100,138],[25,1671,717],{"className":1672,"style":505},[100,439,138],[25,1674,1676],{"className":1675},[110],[25,1677,1679,1704],{"className":1678},[114,447],[25,1680,1682,1701],{"className":1681},[118],[25,1683,1686],{"className":1684,"style":1685},[122],"height:0.3173em;",[25,1687,1689,1693],{"style":1688},"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;",[25,1690],{"className":1691,"style":1692},[130],"height:2.5em;",[25,1694,1698],{"className":1695},[135,1696,1697,138],"reset-size3","size1",[25,1699,72],{"className":1700},[100,138],[25,1702,471],{"className":1703},[470],[25,1705,1707],{"className":1706},[118],[25,1708,1711],{"className":1709,"style":1710},[122],"height:0.143em;",[25,1712],{},[25,1714,390],{"className":1715},[527,138],[25,1717,236],{"className":1718},[100,138],[25,1720,1722,1725],{"className":1721},[100,138],[25,1723,717],{"className":1724,"style":505},[100,439,138],[25,1726,1728],{"className":1727},[110],[25,1729,1731,1751],{"className":1730},[114,447],[25,1732,1734,1748],{"className":1733},[118],[25,1735,1737],{"className":1736,"style":1685},[122],[25,1738,1739,1742],{"style":1688},[25,1740],{"className":1741,"style":1692},[130],[25,1743,1745],{"className":1744},[135,1696,1697,138],[25,1746,35],{"className":1747},[100,138],[25,1749,471],{"className":1750},[470],[25,1752,1754],{"className":1753},[118],[25,1755,1757],{"className":1756,"style":1710},[122],[25,1758],{},[25,1760,390],{"className":1761},[527,138],[25,1763,236],{"className":1764},[100,138],[25,1766,1768,1771],{"className":1767},[100,138],[25,1769,717],{"className":1770,"style":505},[100,439,138],[25,1772,1774],{"className":1773},[110],[25,1775,1777,1797],{"className":1776},[114,447],[25,1778,1780,1794],{"className":1779},[118],[25,1781,1783],{"className":1782,"style":1685},[122],[25,1784,1785,1788],{"style":1688},[25,1786],{"className":1787,"style":1692},[130],[25,1789,1791],{"className":1790},[135,1696,1697,138],[25,1792,236],{"className":1793},[100,138],[25,1795,471],{"className":1796},[470],[25,1798,1800],{"className":1799},[118],[25,1801,1803],{"className":1802,"style":1710},[122],[25,1804],{},[25,1806,390],{"className":1807},[527,138],[25,1809,1811,1814],{"className":1810},[100,138],[25,1812,717],{"className":1813,"style":505},[100,439,138],[25,1815,1817],{"className":1816},[110],[25,1818,1820,1840],{"className":1819},[114,447],[25,1821,1823,1837],{"className":1822},[118],[25,1824,1826],{"className":1825,"style":1685},[122],[25,1827,1828,1831],{"style":1688},[25,1829],{"className":1830,"style":1692},[130],[25,1832,1834],{"className":1833},[135,1696,1697,138],[25,1835,240],{"className":1836},[100,138],[25,1838,471],{"className":1839},[470],[25,1841,1843],{"className":1842},[118],[25,1844,1846],{"className":1845,"style":1710},[122],[25,1847],{},[25,1849,471],{"className":1850},[470],[25,1852,1854],{"className":1853},[118],[25,1855,1857],{"className":1856,"style":1120},[122],[25,1858],{},[25,1860],{"className":1861},[1126,1056],[25,1863,342],{},[11,1865,1867],{"id":1866},"field-workflow","Field Workflow",[1869,1870,1871,1875,1996,2000,2005,2009,2015,2019,2116,2120],"steps",{},[693,1872,1874],{"id":1873},"setup-and-pre-orientation","Setup and Pre-Orientation",[16,1876,1877,1878,1993,1994,36],{},"Set up the instrument on the baseline. Perform a \"pre-orientation\" to within ",[25,1879,1881,1909],{"className":1880},[47],[25,1882,1884],{"className":1883},[51],[53,1885,1886],{"xmlns":55},[57,1887,1888,1906],{},[60,1889,1890,1897,1900],{},[67,1891,1892,1894],{},[70,1893,35],{},[63,1895,1896],{},"∘",[63,1898,1899],{},"−",[67,1901,1902,1904],{},[70,1903,236],{},[63,1905,1896],{},[79,1907,1908],{"encoding":81},"2^\\circ-3^\\circ",[25,1910,1912,1958],{"className":1911,"ariaHidden":87},[86],[25,1913,1915,1919,1949,1952,1955],{"className":1914},[91],[25,1916],{"className":1917,"style":1918},[95],"height:0.7574em;vertical-align:-0.0833em;",[25,1920,1922,1925],{"className":1921},[100],[25,1923,35],{"className":1924},[100],[25,1926,1928],{"className":1927},[110],[25,1929,1931],{"className":1930},[114],[25,1932,1934],{"className":1933},[118],[25,1935,1938],{"className":1936,"style":1937},[122],"height:0.6741em;",[25,1939,1940,1943],{"style":126},[25,1941],{"className":1942,"style":131},[130],[25,1944,1946],{"className":1945},[135,136,137,138],[25,1947,1896],{"className":1948},[527,138],[25,1950],{"className":1951,"style":523},[299],[25,1953,1899],{"className":1954},[527],[25,1956],{"className":1957,"style":523},[299],[25,1959,1961,1964],{"className":1960},[91],[25,1962],{"className":1963,"style":1937},[95],[25,1965,1967,1970],{"className":1966},[100],[25,1968,236],{"className":1969},[100],[25,1971,1973],{"className":1972},[110],[25,1974,1976],{"className":1975},[114],[25,1977,1979],{"className":1978},[118],[25,1980,1982],{"className":1981,"style":1937},[122],[25,1983,1984,1987],{"style":126},[25,1985],{"className":1986,"style":131},[130],[25,1988,1990],{"className":1989},[135,136,137,138],[25,1991,1896],{"className":1992},[527,138]," of north using a manual search or known approximate bearing ",[25,1995,736],{},[693,1997,1999],{"id":1998},"spin-up-and-release","Spin-Up and Release",[16,2001,2002,2003,36],{},"Start the gyro spinner. Once it reaches full operating speed, gently uncage the gyro. It will begin to oscillate across the meridian ",[25,2004,736],{},[693,2006,2008],{"id":2007},"tracking-the-oscillation","Tracking the Oscillation",[16,2010,2011,2012,36],{},"Use the theodolite’s slow-motion screws to keep the moving gyro mark centered on the internal gyro scale. Note the horizontal circle readings at the points of maximum excursion (Reversal Points) ",[25,2013,2014],{},"7, 8",[693,2016,2018],{"id":2017},"data-reduction","Data Reduction",[16,2020,2021,2022,2025,2026,2054,2055,2083,2084,733,2113,36],{},"Apply Schuler's Mean to the reversal points to find the circle reading of ",[20,2023,2024],{},"Gyro North"," (",[25,2027,2029,2042],{"className":2028},[47],[25,2030,2032],{"className":2031},[51],[53,2033,2034],{"xmlns":55},[57,2035,2036,2040],{},[60,2037,2038],{},[374,2039,935],{},[79,2041,935],{"encoding":81},[25,2043,2045],{"className":2044,"ariaHidden":87},[86],[25,2046,2048,2051],{"className":2047},[91],[25,2049],{"className":2050,"style":1594},[95],[25,2052,935],{"className":2053,"style":997},[100,439],"). Correct this value for the instrument constant (",[25,2056,2058,2071],{"className":2057},[47],[25,2059,2061],{"className":2060},[51],[53,2062,2063],{"xmlns":55},[57,2064,2065,2069],{},[60,2066,2067],{},[374,2068,376],{},[79,2070,376],{"encoding":81},[25,2072,2074],{"className":2073,"ariaHidden":87},[86],[25,2075,2077,2080],{"className":2076},[91],[25,2078],{"className":2079,"style":1594},[95],[25,2081,376],{"className":2082,"style":440},[100,439],") and the tape zero error (",[25,2085,2087,2101],{"className":2086},[47],[25,2088,2090],{"className":2089},[51],[53,2091,2092],{"xmlns":55},[57,2093,2094,2099],{},[60,2095,2096],{},[374,2097,2098],{},"Z",[79,2100,2098],{"encoding":81},[25,2102,2104],{"className":2103,"ariaHidden":87},[86],[25,2105,2107,2110],{"className":2106},[91],[25,2108],{"className":2109,"style":1594},[95],[25,2111,2098],{"className":2112,"style":440},[100,439],[25,2114,2115],{},"6, 9",[693,2117,2119],{"id":2118},"orientation-to-baseline","Orientation to Baseline",[16,2121,2122,2123,36],{},"After the gyro observations, sight the baseline target to determine its horizontal circle reading. The difference between the baseline reading and the corrected Gyro North gives the True Azimuth ",[25,2124,2125],{},"10",[11,2127,2129],{"id":2128},"step-by-step-example","Step-by-Step Example",[16,2131,2132,2135,2136,2206,2207,2235,2236,2311,2312,2235,2340,36],{},[20,2133,2134],{},"Problem:"," A surveyor records four reversal points with a gyro-theodolite. Find the horizontal circle reading of True North (",[25,2137,2139,2157],{"className":2138},[47],[25,2140,2142],{"className":2141},[51],[53,2143,2144],{"xmlns":55},[57,2145,2146,2154],{},[60,2147,2148],{},[371,2149,2150,2152],{},[374,2151,935],{},[70,2153,419],{},[79,2155,2156],{"encoding":81},"N_0",[25,2158,2160],{"className":2159,"ariaHidden":87},[86],[25,2161,2163,2166],{"className":2162},[91],[25,2164],{"className":2165,"style":990},[95],[25,2167,2169,2172],{"className":2168},[100],[25,2170,935],{"className":2171,"style":997},[100,439],[25,2173,2175],{"className":2174},[110],[25,2176,2178,2198],{"className":2177},[114,447],[25,2179,2181,2195],{"className":2180},[118],[25,2182,2184],{"className":2183,"style":454},[122],[25,2185,2186,2189],{"style":1012},[25,2187],{"className":2188,"style":131},[130],[25,2190,2192],{"className":2191},[135,136,137,138],[25,2193,419],{"className":2194},[100,138],[25,2196,471],{"className":2197},[470],[25,2199,2201],{"className":2200},[118],[25,2202,2204],{"className":2203,"style":478},[122],[25,2205],{},") if the tape zero correction (",[25,2208,2210,2223],{"className":2209},[47],[25,2211,2213],{"className":2212},[51],[53,2214,2215],{"xmlns":55},[57,2216,2217,2221],{},[60,2218,2219],{},[374,2220,2098],{},[79,2222,2098],{"encoding":81},[25,2224,2226],{"className":2225,"ariaHidden":87},[86],[25,2227,2229,2232],{"className":2228},[91],[25,2230],{"className":2231,"style":1594},[95],[25,2233,2098],{"className":2234,"style":440},[100,439],") is ",[25,2237,2239,2263],{"className":2238},[47],[25,2240,2242],{"className":2241},[51],[53,2243,2244],{"xmlns":55},[57,2245,2246,2260],{},[60,2247,2248,2250],{},[63,2249,1899],{},[67,2251,2252,2254],{},[70,2253,2125],{},[60,2255,2256,2258],{},[63,2257,77],{"mathvariant":75},[63,2259,77],{"mathvariant":75},[79,2261,2262],{"encoding":81},"-10''",[25,2264,2266],{"className":2265,"ariaHidden":87},[86],[25,2267,2269,2272,2275,2278],{"className":2268},[91],[25,2270],{"className":2271,"style":96},[95],[25,2273,1899],{"className":2274},[100],[25,2276,72],{"className":2277},[100],[25,2279,2281,2284],{"className":2280},[100],[25,2282,419],{"className":2283},[100],[25,2285,2287],{"className":2286},[110],[25,2288,2290],{"className":2289},[114],[25,2291,2293],{"className":2292},[118],[25,2294,2296],{"className":2295,"style":123},[122],[25,2297,2298,2301],{"style":126},[25,2299],{"className":2300,"style":131},[130],[25,2302,2304],{"className":2303},[135,136,137,138],[25,2305,2307],{"className":2306},[100,138],[25,2308,2310],{"className":2309},[100,138],"′′"," and the instrument constant (",[25,2313,2315,2328],{"className":2314},[47],[25,2316,2318],{"className":2317},[51],[53,2319,2320],{"xmlns":55},[57,2321,2322,2326],{},[60,2323,2324],{},[374,2325,376],{},[79,2327,376],{"encoding":81},[25,2329,2331],{"className":2330,"ariaHidden":87},[86],[25,2332,2334,2337],{"className":2333},[91],[25,2335],{"className":2336,"style":1594},[95],[25,2338,376],{"className":2339,"style":440},[100,439],[25,2341,2343,2374],{"className":2342},[47],[25,2344,2346],{"className":2345},[51],[53,2347,2348],{"xmlns":55},[57,2349,2350,2371],{},[60,2351,2352,2354,2360],{},[63,2353,1899],{},[67,2355,2356,2358],{},[70,2357,236],{},[63,2359,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,2361,2362,2365],{},[70,2363,2364],{},"00",[60,2366,2367,2369],{},[63,2368,77],{"mathvariant":75},[63,2370,77],{"mathvariant":75},[79,2372,2373],{"encoding":81},"-3'00''",[25,2375,2377],{"className":2376,"ariaHidden":87},[86],[25,2378,2380,2383,2386,2418,2421],{"className":2379},[91],[25,2381],{"className":2382,"style":96},[95],[25,2384,1899],{"className":2385},[100],[25,2387,2389,2392],{"className":2388},[100],[25,2390,236],{"className":2391},[100],[25,2393,2395],{"className":2394},[110],[25,2396,2398],{"className":2397},[114],[25,2399,2401],{"className":2400},[118],[25,2402,2404],{"className":2403,"style":123},[122],[25,2405,2406,2409],{"style":126},[25,2407],{"className":2408,"style":131},[130],[25,2410,2412],{"className":2411},[135,136,137,138],[25,2413,2415],{"className":2414},[100,138],[25,2416,77],{"className":2417},[100,138],[25,2419,419],{"className":2420},[100],[25,2422,2424,2427],{"className":2423},[100],[25,2425,419],{"className":2426},[100],[25,2428,2430],{"className":2429},[110],[25,2431,2433],{"className":2432},[114],[25,2434,2436],{"className":2435},[118],[25,2437,2439],{"className":2438,"style":123},[122],[25,2440,2441,2444],{"style":126},[25,2442],{"className":2443,"style":131},[130],[25,2445,2447],{"className":2446},[135,136,137,138],[25,2448,2450],{"className":2449},[100,138],[25,2451,2310],{"className":2452},[100,138],[16,2454,2455],{},[20,2456,2457],{},"Observed Reversal Points:",[2459,2460,2461,2685,2911,3134],"ul",{},[328,2462,2463],{},[25,2464,2466,2513],{"className":2465},[47],[25,2467,2469],{"className":2468},[51],[53,2470,2471],{"xmlns":55},[57,2472,2473,2510],{},[60,2474,2475,2481,2484,2486,2493,2499],{},[371,2476,2477,2479],{},[374,2478,717],{},[70,2480,72],{},[164,2482,2483],{}," (left)",[63,2485,416],{},[67,2487,2488,2491],{},[70,2489,2490],{},"42",[63,2492,1896],{},[67,2494,2495,2497],{},[70,2496,2364],{},[63,2498,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,2500,2501,2504],{},[70,2502,2503],{},"31",[60,2505,2506,2508],{},[63,2507,77],{"mathvariant":75},[63,2509,77],{"mathvariant":75},[79,2511,2512],{"encoding":81},"r_1\\text{ (left)} = 42^\\circ 00' 31''",[25,2514,2516,2577],{"className":2515,"ariaHidden":87},[86],[25,2517,2519,2522,2562,2568,2571,2574],{"className":2518},[91],[25,2520],{"className":2521,"style":1244},[95],[25,2523,2525,2528],{"className":2524},[100],[25,2526,717],{"className":2527,"style":505},[100,439],[25,2529,2531],{"className":2530},[110],[25,2532,2534,2554],{"className":2533},[114,447],[25,2535,2537,2551],{"className":2536},[118],[25,2538,2540],{"className":2539,"style":454},[122],[25,2541,2542,2545],{"style":805},[25,2543],{"className":2544,"style":131},[130],[25,2546,2548],{"className":2547},[135,136,137,138],[25,2549,72],{"className":2550},[100,138],[25,2552,471],{"className":2553},[470],[25,2555,2557],{"className":2556},[118],[25,2558,2560],{"className":2559,"style":478},[122],[25,2561],{},[25,2563,2565],{"className":2564},[100,189],[25,2566,2483],{"className":2567},[100],[25,2569],{"className":2570,"style":671},[299],[25,2572,416],{"className":2573},[675],[25,2575],{"className":2576,"style":671},[299],[25,2578,2580,2583,2586,2615,2618,2650,2653],{"className":2579},[91],[25,2581],{"className":2582,"style":123},[95],[25,2584,240],{"className":2585},[100],[25,2587,2589,2592],{"className":2588},[100],[25,2590,35],{"className":2591},[100],[25,2593,2595],{"className":2594},[110],[25,2596,2598],{"className":2597},[114],[25,2599,2601],{"className":2600},[118],[25,2602,2604],{"className":2603,"style":1937},[122],[25,2605,2606,2609],{"style":126},[25,2607],{"className":2608,"style":131},[130],[25,2610,2612],{"className":2611},[135,136,137,138],[25,2613,1896],{"className":2614},[527,138],[25,2616,419],{"className":2617},[100],[25,2619,2621,2624],{"className":2620},[100],[25,2622,419],{"className":2623},[100],[25,2625,2627],{"className":2626},[110],[25,2628,2630],{"className":2629},[114],[25,2631,2633],{"className":2632},[118],[25,2634,2636],{"className":2635,"style":123},[122],[25,2637,2638,2641],{"style":126},[25,2639],{"className":2640,"style":131},[130],[25,2642,2644],{"className":2643},[135,136,137,138],[25,2645,2647],{"className":2646},[100,138],[25,2648,77],{"className":2649},[100,138],[25,2651,236],{"className":2652},[100],[25,2654,2656,2659],{"className":2655},[100],[25,2657,72],{"className":2658},[100],[25,2660,2662],{"className":2661},[110],[25,2663,2665],{"className":2664},[114],[25,2666,2668],{"className":2667},[118],[25,2669,2671],{"className":2670,"style":123},[122],[25,2672,2673,2676],{"style":126},[25,2674],{"className":2675,"style":131},[130],[25,2677,2679],{"className":2678},[135,136,137,138],[25,2680,2682],{"className":2681},[100,138],[25,2683,2310],{"className":2684},[100,138],[328,2686,2687],{},[25,2688,2690,2738],{"className":2689},[47],[25,2691,2693],{"className":2692},[51],[53,2694,2695],{"xmlns":55},[57,2696,2697,2735],{},[60,2698,2699,2705,2708,2710,2717,2724],{},[371,2700,2701,2703],{},[374,2702,717],{},[70,2704,35],{},[164,2706,2707],{}," (right)",[63,2709,416],{},[67,2711,2712,2715],{},[70,2713,2714],{},"49",[63,2716,1896],{},[67,2718,2719,2722],{},[70,2720,2721],{},"40",[63,2723,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,2725,2726,2729],{},[70,2727,2728],{},"32",[60,2730,2731,2733],{},[63,2732,77],{"mathvariant":75},[63,2734,77],{"mathvariant":75},[79,2736,2737],{"encoding":81},"r_2\\text{ (right)} = 49^\\circ 40' 32''",[25,2739,2741,2802],{"className":2740,"ariaHidden":87},[86],[25,2742,2744,2747,2787,2793,2796,2799],{"className":2743},[91],[25,2745],{"className":2746,"style":1244},[95],[25,2748,2750,2753],{"className":2749},[100],[25,2751,717],{"className":2752,"style":505},[100,439],[25,2754,2756],{"className":2755},[110],[25,2757,2759,2779],{"className":2758},[114,447],[25,2760,2762,2776],{"className":2761},[118],[25,2763,2765],{"className":2764,"style":454},[122],[25,2766,2767,2770],{"style":805},[25,2768],{"className":2769,"style":131},[130],[25,2771,2773],{"className":2772},[135,136,137,138],[25,2774,35],{"className":2775},[100,138],[25,2777,471],{"className":2778},[470],[25,2780,2782],{"className":2781},[118],[25,2783,2785],{"className":2784,"style":478},[122],[25,2786],{},[25,2788,2790],{"className":2789},[100,189],[25,2791,2707],{"className":2792},[100],[25,2794],{"className":2795,"style":671},[299],[25,2797,416],{"className":2798},[675],[25,2800],{"className":2801,"style":671},[299],[25,2803,2805,2808,2811,2841,2844,2876,2879],{"className":2804},[91],[25,2806],{"className":2807,"style":123},[95],[25,2809,240],{"className":2810},[100],[25,2812,2814,2818],{"className":2813},[100],[25,2815,2817],{"className":2816},[100],"9",[25,2819,2821],{"className":2820},[110],[25,2822,2824],{"className":2823},[114],[25,2825,2827],{"className":2826},[118],[25,2828,2830],{"className":2829,"style":1937},[122],[25,2831,2832,2835],{"style":126},[25,2833],{"className":2834,"style":131},[130],[25,2836,2838],{"className":2837},[135,136,137,138],[25,2839,1896],{"className":2840},[527,138],[25,2842,240],{"className":2843},[100],[25,2845,2847,2850],{"className":2846},[100],[25,2848,419],{"className":2849},[100],[25,2851,2853],{"className":2852},[110],[25,2854,2856],{"className":2855},[114],[25,2857,2859],{"className":2858},[118],[25,2860,2862],{"className":2861,"style":123},[122],[25,2863,2864,2867],{"style":126},[25,2865],{"className":2866,"style":131},[130],[25,2868,2870],{"className":2869},[135,136,137,138],[25,2871,2873],{"className":2872},[100,138],[25,2874,77],{"className":2875},[100,138],[25,2877,236],{"className":2878},[100],[25,2880,2882,2885],{"className":2881},[100],[25,2883,35],{"className":2884},[100],[25,2886,2888],{"className":2887},[110],[25,2889,2891],{"className":2890},[114],[25,2892,2894],{"className":2893},[118],[25,2895,2897],{"className":2896,"style":123},[122],[25,2898,2899,2902],{"style":126},[25,2900],{"className":2901,"style":131},[130],[25,2903,2905],{"className":2904},[135,136,137,138],[25,2906,2908],{"className":2907},[100,138],[25,2909,2310],{"className":2910},[100,138],[328,2912,2913],{},[25,2914,2916,2962],{"className":2915},[47],[25,2917,2919],{"className":2918},[51],[53,2920,2921],{"xmlns":55},[57,2922,2923,2959],{},[60,2924,2925,2931,2933,2935,2941,2948],{},[371,2926,2927,2929],{},[374,2928,717],{},[70,2930,236],{},[164,2932,2483],{},[63,2934,416],{},[67,2936,2937,2939],{},[70,2938,2490],{},[63,2940,1896],{},[67,2942,2943,2946],{},[70,2944,2945],{},"04",[63,2947,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,2949,2950,2953],{},[70,2951,2952],{},"02",[60,2954,2955,2957],{},[63,2956,77],{"mathvariant":75},[63,2958,77],{"mathvariant":75},[79,2960,2961],{"encoding":81},"r_3\\text{ (left)} = 42^\\circ 04' 02''",[25,2963,2965,3026],{"className":2964,"ariaHidden":87},[86],[25,2966,2968,2971,3011,3017,3020,3023],{"className":2967},[91],[25,2969],{"className":2970,"style":1244},[95],[25,2972,2974,2977],{"className":2973},[100],[25,2975,717],{"className":2976,"style":505},[100,439],[25,2978,2980],{"className":2979},[110],[25,2981,2983,3003],{"className":2982},[114,447],[25,2984,2986,3000],{"className":2985},[118],[25,2987,2989],{"className":2988,"style":454},[122],[25,2990,2991,2994],{"style":805},[25,2992],{"className":2993,"style":131},[130],[25,2995,2997],{"className":2996},[135,136,137,138],[25,2998,236],{"className":2999},[100,138],[25,3001,471],{"className":3002},[470],[25,3004,3006],{"className":3005},[118],[25,3007,3009],{"className":3008,"style":478},[122],[25,3010],{},[25,3012,3014],{"className":3013},[100,189],[25,3015,2483],{"className":3016},[100],[25,3018],{"className":3019,"style":671},[299],[25,3021,416],{"className":3022},[675],[25,3024],{"className":3025,"style":671},[299],[25,3027,3029,3032,3035,3064,3067,3099,3102],{"className":3028},[91],[25,3030],{"className":3031,"style":123},[95],[25,3033,240],{"className":3034},[100],[25,3036,3038,3041],{"className":3037},[100],[25,3039,35],{"className":3040},[100],[25,3042,3044],{"className":3043},[110],[25,3045,3047],{"className":3046},[114],[25,3048,3050],{"className":3049},[118],[25,3051,3053],{"className":3052,"style":1937},[122],[25,3054,3055,3058],{"style":126},[25,3056],{"className":3057,"style":131},[130],[25,3059,3061],{"className":3060},[135,136,137,138],[25,3062,1896],{"className":3063},[527,138],[25,3065,419],{"className":3066},[100],[25,3068,3070,3073],{"className":3069},[100],[25,3071,240],{"className":3072},[100],[25,3074,3076],{"className":3075},[110],[25,3077,3079],{"className":3078},[114],[25,3080,3082],{"className":3081},[118],[25,3083,3085],{"className":3084,"style":123},[122],[25,3086,3087,3090],{"style":126},[25,3088],{"className":3089,"style":131},[130],[25,3091,3093],{"className":3092},[135,136,137,138],[25,3094,3096],{"className":3095},[100,138],[25,3097,77],{"className":3098},[100,138],[25,3100,419],{"className":3101},[100],[25,3103,3105,3108],{"className":3104},[100],[25,3106,35],{"className":3107},[100],[25,3109,3111],{"className":3110},[110],[25,3112,3114],{"className":3113},[114],[25,3115,3117],{"className":3116},[118],[25,3118,3120],{"className":3119,"style":123},[122],[25,3121,3122,3125],{"style":126},[25,3123],{"className":3124,"style":131},[130],[25,3126,3128],{"className":3127},[135,136,137,138],[25,3129,3131],{"className":3130},[100,138],[25,3132,2310],{"className":3133},[100,138],[328,3135,3136,689,3357,36],{},[25,3137,3139,3185],{"className":3138},[47],[25,3140,3142],{"className":3141},[51],[53,3143,3144],{"xmlns":55},[57,3145,3146,3182],{},[60,3147,3148,3154,3156,3158,3164,3171],{},[371,3149,3150,3152],{},[374,3151,717],{},[70,3153,240],{},[164,3155,2707],{},[63,3157,416],{},[67,3159,3160,3162],{},[70,3161,2714],{},[63,3163,1896],{},[67,3165,3166,3169],{},[70,3167,3168],{},"37",[63,3170,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,3172,3173,3176],{},[70,3174,3175],{},"21",[60,3177,3178,3180],{},[63,3179,77],{"mathvariant":75},[63,3181,77],{"mathvariant":75},[79,3183,3184],{"encoding":81},"r_4\\text{ (right)} = 49^\\circ 37' 21''",[25,3186,3188,3249],{"className":3187,"ariaHidden":87},[86],[25,3189,3191,3194,3234,3240,3243,3246],{"className":3190},[91],[25,3192],{"className":3193,"style":1244},[95],[25,3195,3197,3200],{"className":3196},[100],[25,3198,717],{"className":3199,"style":505},[100,439],[25,3201,3203],{"className":3202},[110],[25,3204,3206,3226],{"className":3205},[114,447],[25,3207,3209,3223],{"className":3208},[118],[25,3210,3212],{"className":3211,"style":454},[122],[25,3213,3214,3217],{"style":805},[25,3215],{"className":3216,"style":131},[130],[25,3218,3220],{"className":3219},[135,136,137,138],[25,3221,240],{"className":3222},[100,138],[25,3224,471],{"className":3225},[470],[25,3227,3229],{"className":3228},[118],[25,3230,3232],{"className":3231,"style":478},[122],[25,3233],{},[25,3235,3237],{"className":3236},[100,189],[25,3238,2707],{"className":3239},[100],[25,3241],{"className":3242,"style":671},[299],[25,3244,416],{"className":3245},[675],[25,3247],{"className":3248,"style":671},[299],[25,3250,3252,3255,3258,3287,3290,3322,3325],{"className":3251},[91],[25,3253],{"className":3254,"style":123},[95],[25,3256,240],{"className":3257},[100],[25,3259,3261,3264],{"className":3260},[100],[25,3262,2817],{"className":3263},[100],[25,3265,3267],{"className":3266},[110],[25,3268,3270],{"className":3269},[114],[25,3271,3273],{"className":3272},[118],[25,3274,3276],{"className":3275,"style":1937},[122],[25,3277,3278,3281],{"style":126},[25,3279],{"className":3280,"style":131},[130],[25,3282,3284],{"className":3283},[135,136,137,138],[25,3285,1896],{"className":3286},[527,138],[25,3288,236],{"className":3289},[100],[25,3291,3293,3296],{"className":3292},[100],[25,3294,736],{"className":3295},[100],[25,3297,3299],{"className":3298},[110],[25,3300,3302],{"className":3301},[114],[25,3303,3305],{"className":3304},[118],[25,3306,3308],{"className":3307,"style":123},[122],[25,3309,3310,3313],{"style":126},[25,3311],{"className":3312,"style":131},[130],[25,3314,3316],{"className":3315},[135,136,137,138],[25,3317,3319],{"className":3318},[100,138],[25,3320,77],{"className":3321},[100,138],[25,3323,35],{"className":3324},[100],[25,3326,3328,3331],{"className":3327},[100],[25,3329,72],{"className":3330},[100],[25,3332,3334],{"className":3333},[110],[25,3335,3337],{"className":3336},[114],[25,3338,3340],{"className":3339},[118],[25,3341,3343],{"className":3342,"style":123},[122],[25,3344,3345,3348],{"style":126},[25,3346],{"className":3347,"style":131},[130],[25,3349,3351],{"className":3350},[135,136,137,138],[25,3352,3354],{"className":3353},[100,138],[25,3355,2310],{"className":3356},[100,138],[25,3358,3359],{},"8, 10",[325,3361,3362,4145,4919,5469],{},[328,3363,3364,3438,689,4143,36],{},[20,3365,3366,3367,3437],{},"Calculate first mean (",[25,3368,3370,3388],{"className":3369},[47],[25,3371,3373],{"className":3372},[51],[53,3374,3375],{"xmlns":55},[57,3376,3377,3385],{},[60,3378,3379],{},[371,3380,3381,3383],{},[374,3382,935],{},[70,3384,72],{},[79,3386,3387],{"encoding":81},"N_1",[25,3389,3391],{"className":3390,"ariaHidden":87},[86],[25,3392,3394,3397],{"className":3393},[91],[25,3395],{"className":3396,"style":990},[95],[25,3398,3400,3403],{"className":3399},[100],[25,3401,935],{"className":3402,"style":997},[100,439],[25,3404,3406],{"className":3405},[110],[25,3407,3409,3429],{"className":3408},[114,447],[25,3410,3412,3426],{"className":3411},[118],[25,3413,3415],{"className":3414,"style":454},[122],[25,3416,3417,3420],{"style":1012},[25,3418],{"className":3419,"style":131},[130],[25,3421,3423],{"className":3422},[135,136,137,138],[25,3424,72],{"className":3425},[100,138],[25,3427,471],{"className":3428},[470],[25,3430,3432],{"className":3431},[118],[25,3433,3435],{"className":3434,"style":478},[122],[25,3436],{},"):",[25,3439,3441,3570],{"className":3440},[47],[25,3442,3444],{"className":3443},[51],[53,3445,3446],{"xmlns":55},[57,3447,3448,3567],{},[60,3449,3450,3456,3458,3540,3542,3549,3556],{},[371,3451,3452,3454],{},[374,3453,935],{},[70,3455,72],{},[63,3457,416],{},[941,3459,3460,3538],{},[60,3461,3462,3468,3474,3484,3486,3488,3490,3496,3502,3512,3514,3516,3522,3528],{},[67,3463,3464,3466],{},[70,3465,2490],{},[63,3467,1896],{},[67,3469,3470,3472],{},[70,3471,2364],{},[63,3473,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,3475,3476,3478],{},[70,3477,2503],{},[60,3479,3480,3482],{},[63,3481,77],{"mathvariant":75},[63,3483,77],{"mathvariant":75},[63,3485,390],{},[70,3487,35],{},[63,3489,950],{"stretchy":949},[67,3491,3492,3494],{},[70,3493,2714],{},[63,3495,1896],{},[67,3497,3498,3500],{},[70,3499,2721],{},[63,3501,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,3503,3504,3506],{},[70,3505,2728],{},[60,3507,3508,3510],{},[63,3509,77],{"mathvariant":75},[63,3511,77],{"mathvariant":75},[63,3513,977],{"stretchy":949},[63,3515,390],{},[67,3517,3518,3520],{},[70,3519,2490],{},[63,3521,1896],{},[67,3523,3524,3526],{},[70,3525,2945],{},[63,3527,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,3529,3530,3532],{},[70,3531,2952],{},[60,3533,3534,3536],{},[63,3535,77],{"mathvariant":75},[63,3537,77],{"mathvariant":75},[70,3539,240],{},[63,3541,416],{},[67,3543,3544,3547],{},[70,3545,3546],{},"45",[63,3548,1896],{},[67,3550,3551,3554],{},[70,3552,3553],{},"51",[63,3555,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,3557,3558,3561],{},[70,3559,3560],{},"24",[60,3562,3563,3565],{},[63,3564,77],{"mathvariant":75},[63,3566,77],{"mathvariant":75},[79,3568,3569],{"encoding":81},"N_1 = \\frac{42^\\circ 00' 31'' + 2(49^\\circ 40' 32'') + 42^\\circ 04' 02''}{4} = 45^\\circ 51' 24''",[25,3571,3573,3628,4035],{"className":3572,"ariaHidden":87},[86],[25,3574,3576,3579,3619,3622,3625],{"className":3575},[91],[25,3577],{"className":3578,"style":990},[95],[25,3580,3582,3585],{"className":3581},[100],[25,3583,935],{"className":3584,"style":997},[100,439],[25,3586,3588],{"className":3587},[110],[25,3589,3591,3611],{"className":3590},[114,447],[25,3592,3594,3608],{"className":3593},[118],[25,3595,3597],{"className":3596,"style":454},[122],[25,3598,3599,3602],{"style":1012},[25,3600],{"className":3601,"style":131},[130],[25,3603,3605],{"className":3604},[135,136,137,138],[25,3606,72],{"className":3607},[100,138],[25,3609,471],{"className":3610},[470],[25,3612,3614],{"className":3613},[118],[25,3615,3617],{"className":3616,"style":478},[122],[25,3618],{},[25,3620],{"className":3621,"style":671},[299],[25,3623,416],{"className":3624},[675],[25,3626],{"className":3627,"style":671},[299],[25,3629,3631,3635,4026,4029,4032],{"className":3630},[91],[25,3632],{"className":3633,"style":3634},[95],"height:1.4095em;vertical-align:-0.345em;",[25,3636,3638,3641,4023],{"className":3637},[100],[25,3639],{"className":3640},[1055,1056],[25,3642,3644],{"className":3643},[941],[25,3645,3647,4015],{"className":3646},[114,447],[25,3648,3650,4012],{"className":3649},[118],[25,3651,3654,3668,3676],{"className":3652,"style":3653},[122],"height:1.0645em;",[25,3655,3656,3659],{"style":1072},[25,3657],{"className":3658,"style":501},[130],[25,3660,3662],{"className":3661},[135,136,137,138],[25,3663,3665],{"className":3664},[100,138],[25,3666,240],{"className":3667},[100,138],[25,3669,3670,3673],{"style":1087},[25,3671],{"className":3672,"style":501},[130],[25,3674],{"className":3675,"style":1095},[1094],[25,3677,3679,3682],{"style":3678},"top:-3.485em;",[25,3680],{"className":3681,"style":501},[130],[25,3683,3685],{"className":3684},[135,136,137,138],[25,3686,3688,3691,3722,3725,3758,3761,3793,3796,3799,3802,3805,3834,3837,3869,3872,3904,3907,3910,3913,3942,3945,3977,3980],{"className":3687},[100,138],[25,3689,240],{"className":3690},[100,138],[25,3692,3694,3697],{"className":3693},[100,138],[25,3695,35],{"className":3696},[100,138],[25,3698,3700],{"className":3699},[110],[25,3701,3703],{"className":3702},[114],[25,3704,3706],{"className":3705},[118],[25,3707,3710],{"className":3708,"style":3709},[122],"height:0.7485em;",[25,3711,3713,3716],{"style":3712},"top:-2.931em;margin-right:0.0714em;",[25,3714],{"className":3715,"style":1692},[130],[25,3717,3719],{"className":3718},[135,1696,1697,138],[25,3720,1896],{"className":3721},[527,138],[25,3723,419],{"className":3724},[100,138],[25,3726,3728,3731],{"className":3727},[100,138],[25,3729,419],{"className":3730},[100,138],[25,3732,3734],{"className":3733},[110],[25,3735,3737],{"className":3736},[114],[25,3738,3740],{"className":3739},[118],[25,3741,3744],{"className":3742,"style":3743},[122],"height:0.8278em;",[25,3745,3746,3749],{"style":3712},[25,3747],{"className":3748,"style":1692},[130],[25,3750,3752],{"className":3751},[135,1696,1697,138],[25,3753,3755],{"className":3754},[100,138],[25,3756,77],{"className":3757},[100,138],[25,3759,236],{"className":3760},[100,138],[25,3762,3764,3767],{"className":3763},[100,138],[25,3765,72],{"className":3766},[100,138],[25,3768,3770],{"className":3769},[110],[25,3771,3773],{"className":3772},[114],[25,3774,3776],{"className":3775},[118],[25,3777,3779],{"className":3778,"style":3743},[122],[25,3780,3781,3784],{"style":3712},[25,3782],{"className":3783,"style":1692},[130],[25,3785,3787],{"className":3786},[135,1696,1697,138],[25,3788,3790],{"className":3789},[100,138],[25,3791,2310],{"className":3792},[100,138],[25,3794,390],{"className":3795},[527,138],[25,3797,35],{"className":3798},[100,138],[25,3800,950],{"className":3801},[1055,138],[25,3803,240],{"className":3804},[100,138],[25,3806,3808,3811],{"className":3807},[100,138],[25,3809,2817],{"className":3810},[100,138],[25,3812,3814],{"className":3813},[110],[25,3815,3817],{"className":3816},[114],[25,3818,3820],{"className":3819},[118],[25,3821,3823],{"className":3822,"style":3709},[122],[25,3824,3825,3828],{"style":3712},[25,3826],{"className":3827,"style":1692},[130],[25,3829,3831],{"className":3830},[135,1696,1697,138],[25,3832,1896],{"className":3833},[527,138],[25,3835,240],{"className":3836},[100,138],[25,3838,3840,3843],{"className":3839},[100,138],[25,3841,419],{"className":3842},[100,138],[25,3844,3846],{"className":3845},[110],[25,3847,3849],{"className":3848},[114],[25,3850,3852],{"className":3851},[118],[25,3853,3855],{"className":3854,"style":3743},[122],[25,3856,3857,3860],{"style":3712},[25,3858],{"className":3859,"style":1692},[130],[25,3861,3863],{"className":3862},[135,1696,1697,138],[25,3864,3866],{"className":3865},[100,138],[25,3867,77],{"className":3868},[100,138],[25,3870,236],{"className":3871},[100,138],[25,3873,3875,3878],{"className":3874},[100,138],[25,3876,35],{"className":3877},[100,138],[25,3879,3881],{"className":3880},[110],[25,3882,3884],{"className":3883},[114],[25,3885,3887],{"className":3886},[118],[25,3888,3890],{"className":3889,"style":3743},[122],[25,3891,3892,3895],{"style":3712},[25,3893],{"className":3894,"style":1692},[130],[25,3896,3898],{"className":3897},[135,1696,1697,138],[25,3899,3901],{"className":3900},[100,138],[25,3902,2310],{"className":3903},[100,138],[25,3905,977],{"className":3906},[1126,138],[25,3908,390],{"className":3909},[527,138],[25,3911,240],{"className":3912},[100,138],[25,3914,3916,3919],{"className":3915},[100,138],[25,3917,35],{"className":3918},[100,138],[25,3920,3922],{"className":3921},[110],[25,3923,3925],{"className":3924},[114],[25,3926,3928],{"className":3927},[118],[25,3929,3931],{"className":3930,"style":3709},[122],[25,3932,3933,3936],{"style":3712},[25,3934],{"className":3935,"style":1692},[130],[25,3937,3939],{"className":3938},[135,1696,1697,138],[25,3940,1896],{"className":3941},[527,138],[25,3943,419],{"className":3944},[100,138],[25,3946,3948,3951],{"className":3947},[100,138],[25,3949,240],{"className":3950},[100,138],[25,3952,3954],{"className":3953},[110],[25,3955,3957],{"className":3956},[114],[25,3958,3960],{"className":3959},[118],[25,3961,3963],{"className":3962,"style":3743},[122],[25,3964,3965,3968],{"style":3712},[25,3966],{"className":3967,"style":1692},[130],[25,3969,3971],{"className":3970},[135,1696,1697,138],[25,3972,3974],{"className":3973},[100,138],[25,3975,77],{"className":3976},[100,138],[25,3978,419],{"className":3979},[100,138],[25,3981,3983,3986],{"className":3982},[100,138],[25,3984,35],{"className":3985},[100,138],[25,3987,3989],{"className":3988},[110],[25,3990,3992],{"className":3991},[114],[25,3993,3995],{"className":3994},[118],[25,3996,3998],{"className":3997,"style":3743},[122],[25,3999,4000,4003],{"style":3712},[25,4001],{"className":4002,"style":1692},[130],[25,4004,4006],{"className":4005},[135,1696,1697,138],[25,4007,4009],{"className":4008},[100,138],[25,4010,2310],{"className":4011},[100,138],[25,4013,471],{"className":4014},[470],[25,4016,4018],{"className":4017},[118],[25,4019,4021],{"className":4020,"style":1120},[122],[25,4022],{},[25,4024],{"className":4025},[1126,1056],[25,4027],{"className":4028,"style":671},[299],[25,4030,416],{"className":4031},[675],[25,4033],{"className":4034,"style":671},[299],[25,4036,4038,4041,4044,4073,4076,4108,4111],{"className":4037},[91],[25,4039],{"className":4040,"style":123},[95],[25,4042,240],{"className":4043},[100],[25,4045,4047,4050],{"className":4046},[100],[25,4048,323],{"className":4049},[100],[25,4051,4053],{"className":4052},[110],[25,4054,4056],{"className":4055},[114],[25,4057,4059],{"className":4058},[118],[25,4060,4062],{"className":4061,"style":1937},[122],[25,4063,4064,4067],{"style":126},[25,4065],{"className":4066,"style":131},[130],[25,4068,4070],{"className":4069},[135,136,137,138],[25,4071,1896],{"className":4072},[527,138],[25,4074,323],{"className":4075},[100],[25,4077,4079,4082],{"className":4078},[100],[25,4080,72],{"className":4081},[100],[25,4083,4085],{"className":4084},[110],[25,4086,4088],{"className":4087},[114],[25,4089,4091],{"className":4090},[118],[25,4092,4094],{"className":4093,"style":123},[122],[25,4095,4096,4099],{"style":126},[25,4097],{"className":4098,"style":131},[130],[25,4100,4102],{"className":4101},[135,136,137,138],[25,4103,4105],{"className":4104},[100,138],[25,4106,77],{"className":4107},[100,138],[25,4109,35],{"className":4110},[100],[25,4112,4114,4117],{"className":4113},[100],[25,4115,240],{"className":4116},[100],[25,4118,4120],{"className":4119},[110],[25,4121,4123],{"className":4122},[114],[25,4124,4126],{"className":4125},[118],[25,4127,4129],{"className":4128,"style":123},[122],[25,4130,4131,4134],{"style":126},[25,4132],{"className":4133,"style":131},[130],[25,4135,4137],{"className":4136},[135,136,137,138],[25,4138,4140],{"className":4139},[100,138],[25,4141,2310],{"className":4142},[100,138],[25,4144,1581],{},[328,4146,4147,4220,689,4917,36],{},[20,4148,4149,4150,3437],{},"Calculate second mean (",[25,4151,4153,4171],{"className":4152},[47],[25,4154,4156],{"className":4155},[51],[53,4157,4158],{"xmlns":55},[57,4159,4160,4168],{},[60,4161,4162],{},[371,4163,4164,4166],{},[374,4165,935],{},[70,4167,35],{},[79,4169,4170],{"encoding":81},"N_2",[25,4172,4174],{"className":4173,"ariaHidden":87},[86],[25,4175,4177,4180],{"className":4176},[91],[25,4178],{"className":4179,"style":990},[95],[25,4181,4183,4186],{"className":4182},[100],[25,4184,935],{"className":4185,"style":997},[100,439],[25,4187,4189],{"className":4188},[110],[25,4190,4192,4212],{"className":4191},[114,447],[25,4193,4195,4209],{"className":4194},[118],[25,4196,4198],{"className":4197,"style":454},[122],[25,4199,4200,4203],{"style":1012},[25,4201],{"className":4202,"style":131},[130],[25,4204,4206],{"className":4205},[135,136,137,138],[25,4207,35],{"className":4208},[100,138],[25,4210,471],{"className":4211},[470],[25,4213,4215],{"className":4214},[118],[25,4216,4218],{"className":4217,"style":478},[122],[25,4219],{},[25,4221,4223,4350],{"className":4222},[47],[25,4224,4226],{"className":4225},[51],[53,4227,4228],{"xmlns":55},[57,4229,4230,4347],{},[60,4231,4232,4238,4240,4322,4324,4330,4336],{},[371,4233,4234,4236],{},[374,4235,935],{},[70,4237,35],{},[63,4239,416],{},[941,4241,4242,4320],{},[60,4243,4244,4250,4256,4266,4268,4270,4272,4278,4284,4294,4296,4298,4304,4310],{},[67,4245,4246,4248],{},[70,4247,2714],{},[63,4249,1896],{},[67,4251,4252,4254],{},[70,4253,2721],{},[63,4255,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,4257,4258,4260],{},[70,4259,2728],{},[60,4261,4262,4264],{},[63,4263,77],{"mathvariant":75},[63,4265,77],{"mathvariant":75},[63,4267,390],{},[70,4269,35],{},[63,4271,950],{"stretchy":949},[67,4273,4274,4276],{},[70,4275,2490],{},[63,4277,1896],{},[67,4279,4280,4282],{},[70,4281,2945],{},[63,4283,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,4285,4286,4288],{},[70,4287,2952],{},[60,4289,4290,4292],{},[63,4291,77],{"mathvariant":75},[63,4293,77],{"mathvariant":75},[63,4295,977],{"stretchy":949},[63,4297,390],{},[67,4299,4300,4302],{},[70,4301,2714],{},[63,4303,1896],{},[67,4305,4306,4308],{},[70,4307,3168],{},[63,4309,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,4311,4312,4314],{},[70,4313,3175],{},[60,4315,4316,4318],{},[63,4317,77],{"mathvariant":75},[63,4319,77],{"mathvariant":75},[70,4321,240],{},[63,4323,416],{},[67,4325,4326,4328],{},[70,4327,3546],{},[63,4329,1896],{},[67,4331,4332,4334],{},[70,4333,3553],{},[63,4335,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,4337,4338,4341],{},[70,4339,4340],{},"29",[60,4342,4343,4345],{},[63,4344,77],{"mathvariant":75},[63,4346,77],{"mathvariant":75},[79,4348,4349],{"encoding":81},"N_2 = \\frac{49^\\circ 40' 32'' + 2(42^\\circ 04' 02'') + 49^\\circ 37' 21''}{4} = 45^\\circ 51' 29''",[25,4351,4353,4408,4809],{"className":4352,"ariaHidden":87},[86],[25,4354,4356,4359,4399,4402,4405],{"className":4355},[91],[25,4357],{"className":4358,"style":990},[95],[25,4360,4362,4365],{"className":4361},[100],[25,4363,935],{"className":4364,"style":997},[100,439],[25,4366,4368],{"className":4367},[110],[25,4369,4371,4391],{"className":4370},[114,447],[25,4372,4374,4388],{"className":4373},[118],[25,4375,4377],{"className":4376,"style":454},[122],[25,4378,4379,4382],{"style":1012},[25,4380],{"className":4381,"style":131},[130],[25,4383,4385],{"className":4384},[135,136,137,138],[25,4386,35],{"className":4387},[100,138],[25,4389,471],{"className":4390},[470],[25,4392,4394],{"className":4393},[118],[25,4395,4397],{"className":4396,"style":478},[122],[25,4398],{},[25,4400],{"className":4401,"style":671},[299],[25,4403,416],{"className":4404},[675],[25,4406],{"className":4407,"style":671},[299],[25,4409,4411,4414,4800,4803,4806],{"className":4410},[91],[25,4412],{"className":4413,"style":3634},[95],[25,4415,4417,4420,4797],{"className":4416},[100],[25,4418],{"className":4419},[1055,1056],[25,4421,4423],{"className":4422},[941],[25,4424,4426,4789],{"className":4425},[114,447],[25,4427,4429,4786],{"className":4428},[118],[25,4430,4432,4446,4454],{"className":4431,"style":3653},[122],[25,4433,4434,4437],{"style":1072},[25,4435],{"className":4436,"style":501},[130],[25,4438,4440],{"className":4439},[135,136,137,138],[25,4441,4443],{"className":4442},[100,138],[25,4444,240],{"className":4445},[100,138],[25,4447,4448,4451],{"style":1087},[25,4449],{"className":4450,"style":501},[130],[25,4452],{"className":4453,"style":1095},[1094],[25,4455,4456,4459],{"style":3678},[25,4457],{"className":4458,"style":501},[130],[25,4460,4462],{"className":4461},[135,136,137,138],[25,4463,4465,4468,4497,4500,4532,4535,4567,4570,4573,4576,4579,4608,4611,4643,4646,4678,4681,4684,4687,4716,4719,4751,4754],{"className":4464},[100,138],[25,4466,240],{"className":4467},[100,138],[25,4469,4471,4474],{"className":4470},[100,138],[25,4472,2817],{"className":4473},[100,138],[25,4475,4477],{"className":4476},[110],[25,4478,4480],{"className":4479},[114],[25,4481,4483],{"className":4482},[118],[25,4484,4486],{"className":4485,"style":3709},[122],[25,4487,4488,4491],{"style":3712},[25,4489],{"className":4490,"style":1692},[130],[25,4492,4494],{"className":4493},[135,1696,1697,138],[25,4495,1896],{"className":4496},[527,138],[25,4498,240],{"className":4499},[100,138],[25,4501,4503,4506],{"className":4502},[100,138],[25,4504,419],{"className":4505},[100,138],[25,4507,4509],{"className":4508},[110],[25,4510,4512],{"className":4511},[114],[25,4513,4515],{"className":4514},[118],[25,4516,4518],{"className":4517,"style":3743},[122],[25,4519,4520,4523],{"style":3712},[25,4521],{"className":4522,"style":1692},[130],[25,4524,4526],{"className":4525},[135,1696,1697,138],[25,4527,4529],{"className":4528},[100,138],[25,4530,77],{"className":4531},[100,138],[25,4533,236],{"className":4534},[100,138],[25,4536,4538,4541],{"className":4537},[100,138],[25,4539,35],{"className":4540},[100,138],[25,4542,4544],{"className":4543},[110],[25,4545,4547],{"className":4546},[114],[25,4548,4550],{"className":4549},[118],[25,4551,4553],{"className":4552,"style":3743},[122],[25,4554,4555,4558],{"style":3712},[25,4556],{"className":4557,"style":1692},[130],[25,4559,4561],{"className":4560},[135,1696,1697,138],[25,4562,4564],{"className":4563},[100,138],[25,4565,2310],{"className":4566},[100,138],[25,4568,390],{"className":4569},[527,138],[25,4571,35],{"className":4572},[100,138],[25,4574,950],{"className":4575},[1055,138],[25,4577,240],{"className":4578},[100,138],[25,4580,4582,4585],{"className":4581},[100,138],[25,4583,35],{"className":4584},[100,138],[25,4586,4588],{"className":4587},[110],[25,4589,4591],{"className":4590},[114],[25,4592,4594],{"className":4593},[118],[25,4595,4597],{"className":4596,"style":3709},[122],[25,4598,4599,4602],{"style":3712},[25,4600],{"className":4601,"style":1692},[130],[25,4603,4605],{"className":4604},[135,1696,1697,138],[25,4606,1896],{"className":4607},[527,138],[25,4609,419],{"className":4610},[100,138],[25,4612,4614,4617],{"className":4613},[100,138],[25,4615,240],{"className":4616},[100,138],[25,4618,4620],{"className":4619},[110],[25,4621,4623],{"className":4622},[114],[25,4624,4626],{"className":4625},[118],[25,4627,4629],{"className":4628,"style":3743},[122],[25,4630,4631,4634],{"style":3712},[25,4632],{"className":4633,"style":1692},[130],[25,4635,4637],{"className":4636},[135,1696,1697,138],[25,4638,4640],{"className":4639},[100,138],[25,4641,77],{"className":4642},[100,138],[25,4644,419],{"className":4645},[100,138],[25,4647,4649,4652],{"className":4648},[100,138],[25,4650,35],{"className":4651},[100,138],[25,4653,4655],{"className":4654},[110],[25,4656,4658],{"className":4657},[114],[25,4659,4661],{"className":4660},[118],[25,4662,4664],{"className":4663,"style":3743},[122],[25,4665,4666,4669],{"style":3712},[25,4667],{"className":4668,"style":1692},[130],[25,4670,4672],{"className":4671},[135,1696,1697,138],[25,4673,4675],{"className":4674},[100,138],[25,4676,2310],{"className":4677},[100,138],[25,4679,977],{"className":4680},[1126,138],[25,4682,390],{"className":4683},[527,138],[25,4685,240],{"className":4686},[100,138],[25,4688,4690,4693],{"className":4689},[100,138],[25,4691,2817],{"className":4692},[100,138],[25,4694,4696],{"className":4695},[110],[25,4697,4699],{"className":4698},[114],[25,4700,4702],{"className":4701},[118],[25,4703,4705],{"className":4704,"style":3709},[122],[25,4706,4707,4710],{"style":3712},[25,4708],{"className":4709,"style":1692},[130],[25,4711,4713],{"className":4712},[135,1696,1697,138],[25,4714,1896],{"className":4715},[527,138],[25,4717,236],{"className":4718},[100,138],[25,4720,4722,4725],{"className":4721},[100,138],[25,4723,736],{"className":4724},[100,138],[25,4726,4728],{"className":4727},[110],[25,4729,4731],{"className":4730},[114],[25,4732,4734],{"className":4733},[118],[25,4735,4737],{"className":4736,"style":3743},[122],[25,4738,4739,4742],{"style":3712},[25,4740],{"className":4741,"style":1692},[130],[25,4743,4745],{"className":4744},[135,1696,1697,138],[25,4746,4748],{"className":4747},[100,138],[25,4749,77],{"className":4750},[100,138],[25,4752,35],{"className":4753},[100,138],[25,4755,4757,4760],{"className":4756},[100,138],[25,4758,72],{"className":4759},[100,138],[25,4761,4763],{"className":4762},[110],[25,4764,4766],{"className":4765},[114],[25,4767,4769],{"className":4768},[118],[25,4770,4772],{"className":4771,"style":3743},[122],[25,4773,4774,4777],{"style":3712},[25,4775],{"className":4776,"style":1692},[130],[25,4778,4780],{"className":4779},[135,1696,1697,138],[25,4781,4783],{"className":4782},[100,138],[25,4784,2310],{"className":4785},[100,138],[25,4787,471],{"className":4788},[470],[25,4790,4792],{"className":4791},[118],[25,4793,4795],{"className":4794,"style":1120},[122],[25,4796],{},[25,4798],{"className":4799},[1126,1056],[25,4801],{"className":4802,"style":671},[299],[25,4804,416],{"className":4805},[675],[25,4807],{"className":4808,"style":671},[299],[25,4810,4812,4815,4818,4847,4850,4882,4885],{"className":4811},[91],[25,4813],{"className":4814,"style":123},[95],[25,4816,240],{"className":4817},[100],[25,4819,4821,4824],{"className":4820},[100],[25,4822,323],{"className":4823},[100],[25,4825,4827],{"className":4826},[110],[25,4828,4830],{"className":4829},[114],[25,4831,4833],{"className":4832},[118],[25,4834,4836],{"className":4835,"style":1937},[122],[25,4837,4838,4841],{"style":126},[25,4839],{"className":4840,"style":131},[130],[25,4842,4844],{"className":4843},[135,136,137,138],[25,4845,1896],{"className":4846},[527,138],[25,4848,323],{"className":4849},[100],[25,4851,4853,4856],{"className":4852},[100],[25,4854,72],{"className":4855},[100],[25,4857,4859],{"className":4858},[110],[25,4860,4862],{"className":4861},[114],[25,4863,4865],{"className":4864},[118],[25,4866,4868],{"className":4867,"style":123},[122],[25,4869,4870,4873],{"style":126},[25,4871],{"className":4872,"style":131},[130],[25,4874,4876],{"className":4875},[135,136,137,138],[25,4877,4879],{"className":4878},[100,138],[25,4880,77],{"className":4881},[100,138],[25,4883,35],{"className":4884},[100],[25,4886,4888,4891],{"className":4887},[100],[25,4889,2817],{"className":4890},[100],[25,4892,4894],{"className":4893},[110],[25,4895,4897],{"className":4896},[114],[25,4898,4900],{"className":4899},[118],[25,4901,4903],{"className":4902,"style":123},[122],[25,4904,4905,4908],{"style":126},[25,4906],{"className":4907,"style":131},[130],[25,4909,4911],{"className":4910},[135,136,137,138],[25,4912,4914],{"className":4913},[100,138],[25,4915,2310],{"className":4916},[100,138],[25,4918,1581],{},[328,4920,4921,4952,689,5467,36],{},[20,4922,4923,4924,3437],{},"Find Gyro North (",[25,4925,4927,4940],{"className":4926},[47],[25,4928,4930],{"className":4929},[51],[53,4931,4932],{"xmlns":55},[57,4933,4934,4938],{},[60,4935,4936],{},[374,4937,935],{},[79,4939,935],{"encoding":81},[25,4941,4943],{"className":4942,"ariaHidden":87},[86],[25,4944,4946,4949],{"className":4945},[91],[25,4947],{"className":4948,"style":1594},[95],[25,4950,935],{"className":4951,"style":997},[100,439],[25,4953,4955,5048],{"className":4954},[47],[25,4956,4958],{"className":4957},[51],[53,4959,4960],{"xmlns":55},[57,4961,4962,5045],{},[60,4963,4964,4966,4968,5020,5022,5028,5034],{},[374,4965,935],{},[63,4967,416],{},[941,4969,4970,5018],{},[60,4971,4972,4978,4984,4994,4996,5002,5008],{},[67,4973,4974,4976],{},[70,4975,3546],{},[63,4977,1896],{},[67,4979,4980,4982],{},[70,4981,3553],{},[63,4983,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,4985,4986,4988],{},[70,4987,3560],{},[60,4989,4990,4992],{},[63,4991,77],{"mathvariant":75},[63,4993,77],{"mathvariant":75},[63,4995,390],{},[67,4997,4998,5000],{},[70,4999,3546],{},[63,5001,1896],{},[67,5003,5004,5006],{},[70,5005,3553],{},[63,5007,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,5009,5010,5012],{},[70,5011,4340],{},[60,5013,5014,5016],{},[63,5015,77],{"mathvariant":75},[63,5017,77],{"mathvariant":75},[70,5019,35],{},[63,5021,416],{},[67,5023,5024,5026],{},[70,5025,3546],{},[63,5027,1896],{},[67,5029,5030,5032],{},[70,5031,3553],{},[63,5033,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,5035,5036,5039],{},[70,5037,5038],{},"26.5",[60,5040,5041,5043],{},[63,5042,77],{"mathvariant":75},[63,5044,77],{"mathvariant":75},[79,5046,5047],{"encoding":81},"N = \\frac{45^\\circ 51' 24'' + 45^\\circ 51' 29''}{2} = 45^\\circ 51' 26.5''",[25,5049,5051,5069,5358],{"className":5050,"ariaHidden":87},[86],[25,5052,5054,5057,5060,5063,5066],{"className":5053},[91],[25,5055],{"className":5056,"style":1594},[95],[25,5058,935],{"className":5059,"style":997},[100,439],[25,5061],{"className":5062,"style":671},[299],[25,5064,416],{"className":5065},[675],[25,5067],{"className":5068,"style":671},[299],[25,5070,5072,5076,5349,5352,5355],{"className":5071},[91],[25,5073],{"className":5074,"style":5075},[95],"height:1.3185em;vertical-align:-0.345em;",[25,5077,5079,5082,5346],{"className":5078},[100],[25,5080],{"className":5081},[1055,1056],[25,5083,5085],{"className":5084},[941],[25,5086,5088,5338],{"className":5087},[114,447],[25,5089,5091,5335],{"className":5090},[118],[25,5092,5095,5109,5117],{"className":5093,"style":5094},[122],"height:0.9735em;",[25,5096,5097,5100],{"style":1072},[25,5098],{"className":5099,"style":501},[130],[25,5101,5103],{"className":5102},[135,136,137,138],[25,5104,5106],{"className":5105},[100,138],[25,5107,35],{"className":5108},[100,138],[25,5110,5111,5114],{"style":1087},[25,5112],{"className":5113,"style":501},[130],[25,5115],{"className":5116,"style":1095},[1094],[25,5118,5119,5122],{"style":1098},[25,5120],{"className":5121,"style":501},[130],[25,5123,5125],{"className":5124},[135,136,137,138],[25,5126,5128,5131,5160,5163,5195,5198,5230,5233,5236,5265,5268,5300,5303],{"className":5127},[100,138],[25,5129,240],{"className":5130},[100,138],[25,5132,5134,5137],{"className":5133},[100,138],[25,5135,323],{"className":5136},[100,138],[25,5138,5140],{"className":5139},[110],[25,5141,5143],{"className":5142},[114],[25,5144,5146],{"className":5145},[118],[25,5147,5149],{"className":5148,"style":3709},[122],[25,5150,5151,5154],{"style":3712},[25,5152],{"className":5153,"style":1692},[130],[25,5155,5157],{"className":5156},[135,1696,1697,138],[25,5158,1896],{"className":5159},[527,138],[25,5161,323],{"className":5162},[100,138],[25,5164,5166,5169],{"className":5165},[100,138],[25,5167,72],{"className":5168},[100,138],[25,5170,5172],{"className":5171},[110],[25,5173,5175],{"className":5174},[114],[25,5176,5178],{"className":5177},[118],[25,5179,5181],{"className":5180,"style":3743},[122],[25,5182,5183,5186],{"style":3712},[25,5184],{"className":5185,"style":1692},[130],[25,5187,5189],{"className":5188},[135,1696,1697,138],[25,5190,5192],{"className":5191},[100,138],[25,5193,77],{"className":5194},[100,138],[25,5196,35],{"className":5197},[100,138],[25,5199,5201,5204],{"className":5200},[100,138],[25,5202,240],{"className":5203},[100,138],[25,5205,5207],{"className":5206},[110],[25,5208,5210],{"className":5209},[114],[25,5211,5213],{"className":5212},[118],[25,5214,5216],{"className":5215,"style":3743},[122],[25,5217,5218,5221],{"style":3712},[25,5219],{"className":5220,"style":1692},[130],[25,5222,5224],{"className":5223},[135,1696,1697,138],[25,5225,5227],{"className":5226},[100,138],[25,5228,2310],{"className":5229},[100,138],[25,5231,390],{"className":5232},[527,138],[25,5234,240],{"className":5235},[100,138],[25,5237,5239,5242],{"className":5238},[100,138],[25,5240,323],{"className":5241},[100,138],[25,5243,5245],{"className":5244},[110],[25,5246,5248],{"className":5247},[114],[25,5249,5251],{"className":5250},[118],[25,5252,5254],{"className":5253,"style":3709},[122],[25,5255,5256,5259],{"style":3712},[25,5257],{"className":5258,"style":1692},[130],[25,5260,5262],{"className":5261},[135,1696,1697,138],[25,5263,1896],{"className":5264},[527,138],[25,5266,323],{"className":5267},[100,138],[25,5269,5271,5274],{"className":5270},[100,138],[25,5272,72],{"className":5273},[100,138],[25,5275,5277],{"className":5276},[110],[25,5278,5280],{"className":5279},[114],[25,5281,5283],{"className":5282},[118],[25,5284,5286],{"className":5285,"style":3743},[122],[25,5287,5288,5291],{"style":3712},[25,5289],{"className":5290,"style":1692},[130],[25,5292,5294],{"className":5293},[135,1696,1697,138],[25,5295,5297],{"className":5296},[100,138],[25,5298,77],{"className":5299},[100,138],[25,5301,35],{"className":5302},[100,138],[25,5304,5306,5309],{"className":5305},[100,138],[25,5307,2817],{"className":5308},[100,138],[25,5310,5312],{"className":5311},[110],[25,5313,5315],{"className":5314},[114],[25,5316,5318],{"className":5317},[118],[25,5319,5321],{"className":5320,"style":3743},[122],[25,5322,5323,5326],{"style":3712},[25,5324],{"className":5325,"style":1692},[130],[25,5327,5329],{"className":5328},[135,1696,1697,138],[25,5330,5332],{"className":5331},[100,138],[25,5333,2310],{"className":5334},[100,138],[25,5336,471],{"className":5337},[470],[25,5339,5341],{"className":5340},[118],[25,5342,5344],{"className":5343,"style":1120},[122],[25,5345],{},[25,5347],{"className":5348},[1126,1056],[25,5350],{"className":5351,"style":671},[299],[25,5353,416],{"className":5354},[675],[25,5356],{"className":5357,"style":671},[299],[25,5359,5361,5364,5367,5396,5399,5431,5435],{"className":5360},[91],[25,5362],{"className":5363,"style":123},[95],[25,5365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Corrections (",[25,5475,5477,5494],{"className":5476},[47],[25,5478,5480],{"className":5479},[51],[53,5481,5482],{"xmlns":55},[57,5483,5484,5492],{},[60,5485,5486],{},[371,5487,5488,5490],{},[374,5489,935],{},[70,5491,419],{},[79,5493,2156],{"encoding":81},[25,5495,5497],{"className":5496,"ariaHidden":87},[86],[25,5498,5500,5503],{"className":5499},[91],[25,5501],{"className":5502,"style":990},[95],[25,5504,5506,5509],{"className":5505},[100],[25,5507,935],{"className":5508,"style":997},[100,439],[25,5510,5512],{"className":5511},[110],[25,5513,5515,5535],{"className":5514},[114,447],[25,5516,5518,5532],{"className":5517},[118],[25,5519,5521],{"className":5520,"style":454},[122],[25,5522,5523,5526],{"style":1012},[25,5524],{"className":5525,"style":131},[130],[25,5527,5529],{"className":5528},[135,136,137,138],[25,5530,419],{"className":5531},[100,138],[25,5533,471],{"className":5534},[470],[25,5536,5538],{"className":5537},[118],[25,5539,5541],{"className":5540,"style":478},[122],[25,5542],{},[25,5544,5546,5668],{"className":5545},[47],[25,5547,5549],{"className":5548},[51],[53,5550,5551],{"xmlns":55},[57,5552,5553,5665],{},[60,5554,5555,5561,5563,5565,5567,5569,5571,5573,5575,5581,5587,5597,5599,5601,5603,5613,5615,5617,5619,5621,5627,5637,5639,5641,5647,5654],{},[371,5556,5557,5559],{},[374,5558,935],{},[70,5560,419],{},[63,5562,416],{},[374,5564,935],{},[63,5566,1899],{},[374,5568,2098],{},[63,5570,390],{},[374,5572,376],{},[63,5574,416],{},[67,5576,5577,5579],{},[70,5578,3546],{},[63,5580,1896],{},[67,5582,5583,5585],{},[70,5584,3553],{},[63,5586,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,5588,5589,5591],{},[70,5590,5038],{},[60,5592,5593,5595],{},[63,5594,77],{"mathvariant":75},[63,5596,77],{"mathvariant":75},[63,5598,1899],{},[63,5600,950],{"stretchy":949},[63,5602,1899],{},[67,5604,5605,5607],{},[70,5606,2125],{},[60,5608,5609,5611],{},[63,5610,77],{"mathvariant":75},[63,5612,77],{"mathvariant":75},[63,5614,977],{"stretchy":949},[63,5616,390],{},[63,5618,950],{"stretchy":949},[63,5620,1899],{},[67,5622,5623,5625],{},[70,5624,236],{},[63,5626,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,5628,5629,5631],{},[70,5630,2364],{},[60,5632,5633,5635],{},[63,5634,77],{"mathvariant":75},[63,5636,77],{"mathvariant":75},[63,5638,977],{"stretchy":949},[63,5640,416],{},[67,5642,5643,5645],{},[70,5644,3546],{},[63,5646,1896],{},[67,5648,5649,5652],{},[70,5650,5651],{},"48",[63,5653,77],{"mathvariant":75,"lspace":76,"rspace":76},[67,5655,5656,5659],{},[70,5657,5658],{},"36.5",[60,5660,5661,5663],{},[63,5662,77],{"mathvariant":75},[63,5664,77],{"mathvariant":75},[79,5666,5667],{"encoding":81},"N_0 = N - Z + K = 45^\\circ 51' 26.5'' - (-10'') + (-3' 00'') = 45^\\circ 48' 36.5''",[25,5669,5671,5726,5745,5763,5781,5898,5958,6049],{"className":5670,"ariaHidden":87},[86],[25,5672,5674,5677,5717,5720,5723],{"className":5673},[91],[25,5675],{"className":5676,"style":990},[95],[25,5678,5680,5683],{"className":5679},[100],[25,5681,935],{"className":5682,"style":997},[100,439],[25,5684,5686],{"className":5685},[110],[25,5687,5689,5709],{"className":5688},[114,447],[25,5690,5692,5706],{"className":5691},[118],[25,5693,5695],{"className":5694,"style":454},[122],[25,5696,5697,5700],{"style":1012},[25,5698],{"className":5699,"style":131},[130],[25,5701,5703],{"className":5702},[135,136,137,138],[25,5704,419],{"className":5705},[100,138],[25,5707,471],{"className":5708},[470],[25,5710,5712],{"className":5711},[118],[25,5713,5715],{"className":5714,"style":478},[122],[25,5716],{},[25,5718],{"className":5719,"style":671},[299],[25,5721,416],{"className":5722},[675],[25,5724],{"className":5725,"style":671},[299],[25,5727,5729,5733,5736,5739,5742],{"className":5728},[91],[25,5730],{"className":5731,"style":5732},[95],"height:0.7667em;vertical-align:-0.0833em;",[25,5734,935],{"className":5735,"style":997},[100,439],[25,5737],{"className":5738,"style":523},[299],[25,5740,1899],{"className":5741},[527],[25,5743],{"className":5744,"style":523},[299],[25,5746,5748,5751,5754,5757,5760],{"className":5747},[91],[25,5749],{"className":5750,"style":5732},[95],[25,5752,2098],{"className":5753,"style":440},[100,439],[25,5755],{"className":5756,"style":523},[299],[25,5758,390],{"className":5759},[527],[25,5761],{"className":5762,"style":523},[299],[25,5764,5766,5769,5772,5775,5778],{"className":5765},[91],[25,5767],{"className":5768,"style":1594},[95],[25,5770,376],{"className":5771,"style":440},[100,439],[25,5773],{"className":5774,"style":671},[299],[25,5776,416],{"className":5777},[675],[25,5779],{"className":5780,"style":671},[299],[25,5782,5784,5787,5790,5819,5822,5854,5857,5889,5892,5895],{"className":5783},[91],[25,5785],{"className":5786,"style":96},[95],[25,5788,240],{"className":5789},[100],[25,5791,5793,5796],{"className":5792},[100],[25,5794,323],{"className":5795},[100],[25,5797,5799],{"className":5798},[110],[25,5800,5802],{"className":5801},[114],[25,5803,5805],{"className":5804},[118],[25,5806,5808],{"className":5807,"style":1937},[122],[25,5809,5810,5813],{"style":126},[25,5811],{"className":5812,"style":131},[130],[25,5814,5816],{"className":5815},[135,136,137,138],[25,5817,1896],{"className":5818},[527,138],[25,5820,323],{"className":5821},[100],[25,5823,5825,5828],{"className":5824},[100],[25,5826,72],{"className":5827},[100],[25,5829,5831],{"className":5830},[110],[25,5832,5834],{"className":5833},[114],[25,5835,5837],{"className":5836},[118],[25,5838,5840],{"className":5839,"style":123},[122],[25,5841,5842,5845],{"style":126},[25,5843],{"className":5844,"style":131},[130],[25,5846,5848],{"className":5847},[135,136,137,138],[25,5849,5851],{"className":5850},[100,138],[25,5852,77],{"className":5853},[100,138],[25,5855,5434],{"className":5856},[100],[25,5858,5860,5863],{"className":5859},[100],[25,5861,323],{"className":5862},[100],[25,5864,5866],{"className":5865},[110],[25,5867,5869],{"className":5868},[114],[25,5870,5872],{"className":5871},[118],[25,5873,5875],{"className":5874,"style":123},[122],[25,5876,5877,5880],{"style":126},[25,5878],{"className":5879,"style":131},[130],[25,5881,5883],{"className":5882},[135,136,137,138],[25,5884,5886],{"className":5885},[100,138],[25,5887,2310],{"className":5888},[100,138],[25,5890],{"className":5891,"style":523},[299],[25,5893,1899],{"className":5894},[527],[25,5896],{"className":5897,"style":523},[299],[25,5899,5901,5905,5908,5911,5914,5946,5949,5952,5955],{"className":5900},[91],[25,5902],{"className":5903,"style":5904},[95],"height:1.0019em;vertical-align:-0.25em;",[25,5906,950],{"className":5907},[1055],[25,5909,1899],{"className":5910},[100],[25,5912,72],{"className":5913},[100],[25,5915,5917,5920],{"className":5916},[100],[25,5918,419],{"className":5919},[100],[25,5921,5923],{"className":5922},[110],[25,5924,5926],{"className":5925},[114],[25,5927,5929],{"className":5928},[118],[25,5930,5932],{"className":5931,"style":123},[122],[25,5933,5934,5937],{"style":126},[25,5935],{"className":5936,"style":131},[130],[25,5938,5940],{"className":5939},[135,136,137,138],[25,5941,5943],{"className":5942},[100,138],[25,5944,2310],{"className":5945},[100,138],[25,5947,977],{"className":5948},[1126],[25,5950],{"className":5951,"style":523},[299],[25,5953,390],{"className":5954},[527],[25,5956],{"className":5957,"style":523},[299],[25,5959,5961,5964,5967,5970,6002,6005,6037,6040,6043,6046],{"className":5960},[91],[25,5962],{"className":5963,"style":5904},[95],[25,5965,950],{"className":5966},[1055],[25,5968,1899],{"className":5969},[100],[25,5971,5973,5976],{"className":5972},[100],[25,5974,236],{"className":5975},[100],[25,5977,5979],{"className":5978},[110],[25,5980,5982],{"className":5981},[114],[25,5983,5985],{"className":5984},[118],[25,5986,5988],{"className":5987,"style":123},[122],[25,5989,5990,5993],{"style":126},[25,5991],{"className":5992,"style":131},[130],[25,5994,5996],{"className":5995},[135,136,137,138],[25,5997,5999],{"className":5998},[100,138],[25,6000,77],{"className":6001},[100,138],[25,6003,419],{"className":6004},[100],[25,6006,6008,6011],{"className":6007},[100],[25,6009,419],{"className":6010},[100],[25,6012,6014],{"className":6013},[110],[25,6015,6017],{"className":6016},[114],[25,6018,6020],{"className":6019},[118],[25,6021,6023],{"className":6022,"style":123},[122],[25,6024,6025,6028],{"style":126},[25,6026],{"className":6027,"style":131},[130],[25,6029,6031],{"className":6030},[135,136,137,138],[25,6032,6034],{"className":6033},[100,138],[25,6035,2310],{"className":6036},[100,138],[25,6038,977],{"className":6039},[1126],[25,6041],{"className":6042,"style":671},[299],[25,6044,416],{"className":6045},[675],[25,6047],{"className":6048,"style":671},[299],[25,6050,6052,6055,6058,6087,6090,6122,6126],{"className":6051},[91],[25,6053],{"className":6054,"style":123},[95],[25,6056,240],{"className":6057},[100],[25,6059,6061,6064],{"className":6060},[100],[25,6062,323],{"className":6063},[100],[25,6065,6067],{"className":6066},[110],[25,6068,6070],{"className":6069},[114],[25,6071,6073],{"className":6072},[118],[25,6074,6076],{"className":6075,"style":1937},[122],[25,6077,6078,6081],{"style":126},[25,6079],{"className":6080,"style":131},[130],[25,6082,6084],{"className":6083},[135,136,137,138],[25,6085,1896],{"className":6086},[527,138],[25,6088,240],{"className":6089},[100],[25,6091,6093,6096],{"className":6092},[100],[25,6094,1581],{"className":6095},[100],[25,6097,6099],{"className":6098},[110],[25,6100,6102],{"className":6101},[114],[25,6103,6105],{"className":6104},[118],[25,6106,6108],{"className":6107,"style":123},[122],[25,6109,6110,6113],{"style":126},[25,6111],{"className":6112,"style":131},[130],[25,6114,6116],{"className":6115},[135,136,137,138],[25,6117,6119],{"className":6118},[100,138],[25,6120,77],{"className":6121},[100,138],[25,6123,6125],{"className":6124},[100],"36.",[25,6127,6129,6132],{"className":6128},[100],[25,6130,323],{"className":6131},[100],[25,6133,6135],{"className":6134},[110],[25,6136,6138],{"className":6137},[114],[25,6139,6141],{"className":6140},[118],[25,6142,6144],{"className":6143,"style":123},[122],[25,6145,6146,6149],{"style":126},[25,6147],{"className":6148,"style":131},[130],[25,6150,6152],{"className":6151},[135,136,137,138],[25,6153,6155],{"className":6154},[100,138],[25,6156,2310],{"className":6157},[100,138],[25,6159,6160],{},"9, 10",[11,6162,6164],{"id":6163},"practical-tips","Practical Tips",[2459,6166,6167,6176,6185],{},[328,6168,6169,6172,6173,36],{},[20,6170,6171],{},"Circle Drift:"," The horizontal circle can move due to gyro vibration. Always observe the reference object (RO) before and after gyro observations and take the mean ",[25,6174,6175],{},"11",[328,6177,6178,6181,6182,36],{},[20,6179,6180],{},"Temperature Stability:"," Gyroscopes are sensitive to thermal gradients. Allow the instrument to acclimate to the tunnel temperature before beginning ",[25,6183,6184],{},"12",[328,6186,6187,6190,6191,733,6219,36],{},[20,6188,6189],{},"Tape Zero Check:"," Before spinning up, observe the \"non-spin\" oscillations to determine the torque in the suspension tape (",[25,6192,6194,6207],{"className":6193},[47],[25,6195,6197],{"className":6196},[51],[53,6198,6199],{"xmlns":55},[57,6200,6201,6205],{},[60,6202,6203],{},[374,6204,2098],{},[79,6206,2098],{"encoding":81},[25,6208,6210],{"className":6209,"ariaHidden":87},[86],[25,6211,6213,6216],{"className":6212},[91],[25,6214],{"className":6215,"style":1594},[95],[25,6217,2098],{"className":6218,"style":440},[100,439],[25,6220,6221],{},"8, 9",[11,6223,6225],{"id":6224},"common-mistakes","Common Mistakes",[2459,6227,6228,6237,6245],{},[328,6229,6230,6233,6234,36],{},[20,6231,6232],{},"Hard Clamping:"," Clamping the movements too tightly can affect the pointing. Use gentle pressure ",[25,6235,6236],{},"13, 14",[328,6238,6239,6242,6243,36],{},[20,6240,6241],{},"Ignoring Refraction:"," In tunnels, lateral refraction can bend the line of sight. Combine gyro observations with zig-zag traverses to minimize this ",[25,6244,6184],{},[328,6246,6247,6250,6251,36],{},[20,6248,6249],{},"Fast Uncaging:"," Releasing the gyro too quickly can damage the delicate suspension tape ",[25,6252,736],{},[11,6254,6256],{"id":6255},"faq","FAQ",[6258,6259,6260,6273,6281],"accordion",{},[6261,6262,6264],"accordion-item",{"label":6263},"Does a gyro-theodolite point to Magnetic North?",[16,6265,6266,6267,6270,6271,36],{},"No. It points to ",[20,6268,6269],{},"Geographical (True) North"," because it reacts to the Earth's axis of rotation, not its magnetic field ",[25,6272,35],{},[6261,6274,6276],{"label":6275},"How long does a single observation take?",[16,6277,6278,6279,36],{},"Using the reversal-point method, a set of three to four points typically takes 15–20 minutes, depending on the period of oscillation ",[25,6280,342],{},[6261,6282,6284],{"label":6283},"Can I use it near the North Pole?",[16,6285,6286,6287,36],{},"The north-seeking force is proportional to the cosine of the latitude. As you approach the poles, the force becomes zero, and the instrument becomes unusable ",[25,6288,323],{},[11,6290,6292],{"id":6291},"conclusion","Conclusion",[16,6294,6295,6296,36],{},"The gyro-theodolite is the \"gold standard\" for underground orientation. By removing the dependency on single-shaft plumbing, it provides the engineering surveyor with a robust, independent check on tunnel alignment, ensuring that breakthrough tolerances are met even on multi-kilometer drivages ",[25,6297,6298],{},"4, 15",[11,6300,6302],{"id":6301},"references","References",[16,6304,6305,6306,6310,6311,36],{},"Schofield, W. (2001). ",[6307,6308,6309],"em",{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann. ",[25,6312,6313],{},"16, 17",{"title":6315,"searchDepth":6316,"depth":6316,"links":6317},"",2,[6318,6319,6320,6321,6322,6326,6333,6334,6335,6336,6337,6338],{"id":13,"depth":6316,"text":14},{"id":39,"depth":6316,"text":40},{"id":243,"depth":6316,"text":244},{"id":316,"depth":6316,"text":317},{"id":353,"depth":6316,"text":354,"children":6323},[6324],{"id":695,"depth":6325,"text":696},3,{"id":1866,"depth":6316,"text":1867,"children":6327},[6328,6329,6330,6331,6332],{"id":1873,"depth":6325,"text":1874},{"id":1998,"depth":6325,"text":1999},{"id":2007,"depth":6325,"text":2008},{"id":2017,"depth":6325,"text":2018},{"id":2118,"depth":6325,"text":2119},{"id":2128,"depth":6316,"text":2129},{"id":6163,"depth":6316,"text":6164},{"id":6224,"depth":6316,"text":6225},{"id":6255,"depth":6316,"text":6256},{"id":6291,"depth":6316,"text":6292},{"id":6301,"depth":6316,"text":6302},"Underground Surveying","A comprehensive technical guide to the theory, observation, and calculation of true azimuths using suspended gyroscopes in tunnel and mine surveying.","md","\u002Fimages\u002Fgyro-theodolite.jpg",{"authorCredentials":6344,"authorBio":6345,"commentCount":6346,"tags":6347,"relatedTools":6352,"relatedArticles":6354,"sources":6357},"LS, P.Eng","Specialist in underground dimensional control and high-precision orientation.",0,[6339,6348,6349,6350,6351],"Gyro-Theodolite","Azimuth","Mining","Tunnels",[6348,6353],"Total Station",[6355,6356],"weisbach-triangle-correlation","traverse-adjustment-math",[6358],{"title":6359,"type":6360,"note":6361},"Engineering Surveying (Fifth Edition)","Book","Primary source for gyroscopic theory and observational methods.",true,"\u002Fblog\u002Fgyro-theodolite-underground-orientation","2024-03-25",22,{"title":5,"description":6340},{"loc":6363},"blog\u002F6.gyro-theodolite-underground-orientation","L-UveooX8rHyE9TQ1QZ9XmJhiIsMnEWe9nWO_R4I4Xg",[6371,6376],{"title":6372,"path":6373,"stem":6374,"description":6375,"children":-1},"The Bowditch Method: A Deep Dive into Traverse Adjustment Math","\u002Fblog\u002Fthe-bowditch-method-a-deep-dive-into-traverse-adjustment-math","blog\u002F5.the-bowditch-method-a-deep-dive-into-traverse-adjustment-math","How to balance horizontal control traverses using the Bowditch rule to distribute coordinate misclosures.",{"title":6377,"path":6378,"stem":6379,"description":6380,"children":-1},"The Clothoid Spiral: Engineering Seamless Transitions in Road Design","\u002Fblog\u002Fclothoid-transition-curves","blog\u002F7.clothoid-transition-curves","A technical guide to the geometry, design, and setting out of clothoid transition curves for high-speed road and rail projects.",1783851951532]