[{"data":1,"prerenderedAt":3502},["ShallowReactive",2],{"blog-\u002Fblog\u002Fstadia-tacheometry-optical-measurement":3,"blog-surround-\u002Fblog\u002Fstadia-tacheometry-optical-measurement":3491},{"id":4,"title":5,"author":6,"body":7,"category":3460,"description":3461,"extension":3462,"image":3463,"meta":3464,"navigation":3483,"path":3484,"publishedAt":3485,"readTime":3486,"seo":3487,"sitemap":3488,"stem":3489,"__hash__":3490},"blog\u002Fblog\u002F20.stadia-tacheometry-optical-measurement.md","Stadia Tacheometry: Mastering Optical Distance Measurement","Surveyiiing Expert",{"type":8,"value":9,"toc":3435},"minimark",[10,15,29,33,39,43,100,104,109,309,529,533,572,850,854,1129,1133,1211,1623,2794,2798,3031,3035,3238,3242,3346,3350,3424,3428],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,28],"p",{},"Before the widespread adoption of EDM, ",[20,21,22],"strong",{},"Stadia Tacheometry"," was the primary method for rapid topographic detailing. By observing the interval between two \"stadia hairs\" on a theodolite's reticule, a surveyor can mathematically derive both the horizontal distance to a staff and the difference in elevation ",[25,26,27],"span",{},"18, 19",".",[11,30,32],{"id":31},"why-this-matters","Why This Matters",[16,34,35,36,28],{},"While Total Stations are now standard, tacheometry remains a fundamental skill for understanding how optics relate to geometry. It is still used for \"rough-in\" surveys, verifying electronic distances, and in specialized optical instruments where electronics might fail ",[25,37,38],{},"20",[11,40,42],{"id":41},"theory","Theory",[16,44,45,46,49,50,96,97,28],{},"Tacheometry is based on the properties of ",[20,47,48],{},"Similar Triangles",". In a fixed-hair telescope, the angle subtended by the stadia hairs is constant. As the staff moves further away, the \"Stadia Intercept\" (",[25,51,54,76],{"className":52},[53],"katex",[25,55,58],{"className":56},[57],"katex-mathml",[59,60,62],"math",{"xmlns":61},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[63,64,65,72],"semantics",{},[66,67,68],"mrow",{},[69,70,71],"mi",{},"S",[73,74,71],"annotation",{"encoding":75},"application\u002Fx-tex",[25,77,81],{"className":78,"ariaHidden":80},[79],"katex-html","true",[25,82,85,90],{"className":83},[84],"base",[25,86],{"className":87,"style":89},[88],"strut","height:0.6833em;",[25,91,71],{"className":92,"style":95},[93,94],"mord","mathnormal","margin-right:0.0576em;",")—the distance between the top and bottom hair readings—increases proportionally ",[25,98,99],{},"18",[11,101,103],{"id":102},"mathematical-principles","Mathematical Principles",[105,106,108],"h3",{"id":107},"_1-the-basic-horizontal-formula","1. The Basic Horizontal Formula",[16,110,111,112,114,115,308],{},"For a horizontal line of sight ",[25,113,27],{},":\n",[25,116,118,159],{"className":117},[53],[25,119,121],{"className":120},[57],[59,122,123],{"xmlns":61},[63,124,125,156],{},[66,126,127,130,134,144,146,149],{},[69,128,129],{},"D",[131,132,133],"mo",{},"=",[135,136,137,140],"msub",{},[69,138,139],{},"K",[141,142,143],"mn",{},"1",[69,145,71],{},[131,147,148],{},"+",[135,150,151,153],{},[69,152,139],{},[141,154,155],{},"2",[73,157,158],{"encoding":75},"D = K_1 S + K_2",[25,160,162,184,262],{"className":161,"ariaHidden":80},[79],[25,163,165,168,172,177,181],{"className":164},[84],[25,166],{"className":167,"style":89},[88],[25,169,129],{"className":170,"style":171},[93,94],"margin-right:0.0278em;",[25,173],{"className":174,"style":176},[175],"mspace","margin-right:0.2778em;",[25,178,133],{"className":179},[180],"mrel",[25,182],{"className":183,"style":176},[175],[25,185,187,191,248,251,255,259],{"className":186},[84],[25,188],{"className":189,"style":190},[88],"height:0.8333em;vertical-align:-0.15em;",[25,192,194,198],{"className":193},[93],[25,195,139],{"className":196,"style":197},[93,94],"margin-right:0.0715em;",[25,199,202],{"className":200},[201],"msupsub",[25,203,207,239],{"className":204},[205,206],"vlist-t","vlist-t2",[25,208,211,234],{"className":209},[210],"vlist-r",[25,212,216],{"className":213,"style":215},[214],"vlist","height:0.3011em;",[25,217,219,224],{"style":218},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[25,220],{"className":221,"style":223},[222],"pstrut","height:2.7em;",[25,225,231],{"className":226},[227,228,229,230],"sizing","reset-size6","size3","mtight",[25,232,143],{"className":233},[93,230],[25,235,238],{"className":236},[237],"vlist-s","​",[25,240,242],{"className":241},[210],[25,243,246],{"className":244,"style":245},[214],"height:0.15em;",[25,247],{},[25,249,71],{"className":250,"style":95},[93,94],[25,252],{"className":253,"style":254},[175],"margin-right:0.2222em;",[25,256,148],{"className":257},[258],"mbin",[25,260],{"className":261,"style":254},[175],[25,263,265,268],{"className":264},[84],[25,266],{"className":267,"style":190},[88],[25,269,271,274],{"className":270},[93],[25,272,139],{"className":273,"style":197},[93,94],[25,275,277],{"className":276},[201],[25,278,280,300],{"className":279},[205,206],[25,281,283,297],{"className":282},[210],[25,284,286],{"className":285,"style":215},[214],[25,287,288,291],{"style":218},[25,289],{"className":290,"style":223},[222],[25,292,294],{"className":293},[227,228,229,230],[25,295,155],{"className":296},[93,230],[25,298,238],{"className":299},[237],[25,301,303],{"className":302},[210],[25,304,306],{"className":305,"style":245},[214],[25,307],{},"\nWhere:",[310,311,312,344,375,452],"ul",{},[313,314,315,343],"li",{},[25,316,318,331],{"className":317},[53],[25,319,321],{"className":320},[57],[59,322,323],{"xmlns":61},[63,324,325,329],{},[66,326,327],{},[69,328,129],{},[73,330,129],{"encoding":75},[25,332,334],{"className":333,"ariaHidden":80},[79],[25,335,337,340],{"className":336},[84],[25,338],{"className":339,"style":89},[88],[25,341,129],{"className":342,"style":171},[93,94]," = Horizontal distance.",[313,345,346,374],{},[25,347,349,362],{"className":348},[53],[25,350,352],{"className":351},[57],[59,353,354],{"xmlns":61},[63,355,356,360],{},[66,357,358],{},[69,359,71],{},[73,361,71],{"encoding":75},[25,363,365],{"className":364,"ariaHidden":80},[79],[25,366,368,371],{"className":367},[84],[25,369],{"className":370,"style":89},[88],[25,372,71],{"className":373,"style":95},[93,94]," = Stadia intercept (Top reading - Bottom reading).",[313,376,377,447,448,451],{},[25,378,380,398],{"className":379},[53],[25,381,383],{"className":382},[57],[59,384,385],{"xmlns":61},[63,386,387,395],{},[66,388,389],{},[135,390,391,393],{},[69,392,139],{},[141,394,143],{},[73,396,397],{"encoding":75},"K_1",[25,399,401],{"className":400,"ariaHidden":80},[79],[25,402,404,407],{"className":403},[84],[25,405],{"className":406,"style":190},[88],[25,408,410,413],{"className":409},[93],[25,411,139],{"className":412,"style":197},[93,94],[25,414,416],{"className":415},[201],[25,417,419,439],{"className":418},[205,206],[25,420,422,436],{"className":421},[210],[25,423,425],{"className":424,"style":215},[214],[25,426,427,430],{"style":218},[25,428],{"className":429,"style":223},[222],[25,431,433],{"className":432},[227,228,229,230],[25,434,143],{"className":435},[93,230],[25,437,238],{"className":438},[237],[25,440,442],{"className":441},[210],[25,443,445],{"className":444,"style":245},[214],[25,446],{}," = Multiplying constant (usually ",[20,449,450],{},"100",").",[313,453,454,524,525,528],{},[25,455,457,475],{"className":456},[53],[25,458,460],{"className":459},[57],[59,461,462],{"xmlns":61},[63,463,464,472],{},[66,465,466],{},[135,467,468,470],{},[69,469,139],{},[141,471,155],{},[73,473,474],{"encoding":75},"K_2",[25,476,478],{"className":477,"ariaHidden":80},[79],[25,479,481,484],{"className":480},[84],[25,482],{"className":483,"style":190},[88],[25,485,487,490],{"className":486},[93],[25,488,139],{"className":489,"style":197},[93,94],[25,491,493],{"className":492},[201],[25,494,496,516],{"className":495},[205,206],[25,497,499,513],{"className":498},[210],[25,500,502],{"className":501,"style":215},[214],[25,503,504,507],{"style":218},[25,505],{"className":506,"style":223},[222],[25,508,510],{"className":509},[227,228,229,230],[25,511,155],{"className":512},[93,230],[25,514,238],{"className":515},[237],[25,517,519],{"className":518},[210],[25,520,522],{"className":521,"style":245},[214],[25,523],{}," = Additive constant (usually ",[20,526,527],{},"0"," in modern anallactic telescopes).",[105,530,532],{"id":531},"_2-the-inclined-sight-formula","2. The Inclined Sight Formula",[16,534,535,536,567,568,571],{},"In engineering, sights are rarely horizontal. When the telescope is tilted at an angle ",[25,537,539,554],{"className":538},[53],[25,540,542],{"className":541},[57],[59,543,544],{"xmlns":61},[63,545,546,551],{},[66,547,548],{},[69,549,550],{},"θ",[73,552,553],{"encoding":75},"\\theta",[25,555,557],{"className":556,"ariaHidden":80},[79],[25,558,560,564],{"className":559},[84],[25,561],{"className":562,"style":563},[88],"height:0.6944em;",[25,565,550],{"className":566,"style":171},[93,94],", the formulas must account for the slant ",[25,569,570],{},"21",":",[310,573,574,721],{},[313,575,576,608],{},[20,577,578,579,607],{},"Horizontal Distance (",[25,580,582,595],{"className":581},[53],[25,583,585],{"className":584},[57],[59,586,587],{"xmlns":61},[63,588,589,593],{},[66,590,591],{},[69,592,129],{},[73,594,129],{"encoding":75},[25,596,598],{"className":597,"ariaHidden":80},[79],[25,599,601,604],{"className":600},[84],[25,602],{"className":603,"style":89},[88],[25,605,129],{"className":606,"style":171},[93,94],"):",[25,609,611,646],{"className":610},[53],[25,612,614],{"className":613},[57],[59,615,616],{"xmlns":61},[63,617,618,643],{},[66,619,620,622,624,626,628,641],{},[69,621,129],{},[131,623,133],{},[141,625,450],{},[69,627,71],{},[629,630,631,639],"msup",{},[66,632,633,636],{},[69,634,635],{},"cos",[131,637,638],{},"⁡",[141,640,155],{},[69,642,550],{},[73,644,645],{"encoding":75},"D = 100 S \\cos^2 \\theta",[25,647,649,667],{"className":648,"ariaHidden":80},[79],[25,650,652,655,658,661,664],{"className":651},[84],[25,653],{"className":654,"style":89},[88],[25,656,129],{"className":657,"style":171},[93,94],[25,659],{"className":660,"style":176},[175],[25,662,133],{"className":663},[180],[25,665],{"className":666,"style":176},[175],[25,668,670,674,677,680,684,715,718],{"className":669},[84],[25,671],{"className":672,"style":673},[88],"height:0.8141em;",[25,675,450],{"className":676},[93],[25,678,71],{"className":679,"style":95},[93,94],[25,681],{"className":682,"style":683},[175],"margin-right:0.1667em;",[25,685,688,691],{"className":686},[687],"mop",[25,689,635],{"className":690},[687],[25,692,694],{"className":693},[201],[25,695,697],{"className":696},[205],[25,698,700],{"className":699},[210],[25,701,703],{"className":702,"style":673},[214],[25,704,706,709],{"style":705},"top:-3.063em;margin-right:0.05em;",[25,707],{"className":708,"style":223},[222],[25,710,712],{"className":711},[227,228,229,230],[25,713,155],{"className":714},[93,230],[25,716],{"className":717,"style":683},[175],[25,719,550],{"className":720,"style":171},[93,94],[313,722,723,764],{},[20,724,725,726,607],{},"Vertical Height (",[25,727,729,748],{"className":728},[53],[25,730,732],{"className":731},[57],[59,733,734],{"xmlns":61},[63,735,736,745],{},[66,737,738,742],{},[69,739,741],{"mathvariant":740},"normal","Δ",[69,743,744],{},"H",[73,746,747],{"encoding":75},"\\Delta H",[25,749,751],{"className":750,"ariaHidden":80},[79],[25,752,754,757,760],{"className":753},[84],[25,755],{"className":756,"style":89},[88],[25,758,741],{"className":759},[93],[25,761,744],{"className":762,"style":763},[93,94],"margin-right:0.0813em;",[25,765,767,799],{"className":766},[53],[25,768,770],{"className":769},[57],[59,771,772],{"xmlns":61},[63,773,774,796],{},[66,775,776,778,780,782,785,787,790,792,794],{},[69,777,741],{"mathvariant":740},[69,779,744],{},[131,781,133],{},[141,783,784],{},"50",[69,786,71],{},[69,788,789],{},"sin",[131,791,638],{},[141,793,155],{},[69,795,550],{},[73,797,798],{"encoding":75},"\\Delta H = 50 S \\sin 2\\theta",[25,800,802,823],{"className":801,"ariaHidden":80},[79],[25,803,805,808,811,814,817,820],{"className":804},[84],[25,806],{"className":807,"style":89},[88],[25,809,741],{"className":810},[93],[25,812,744],{"className":813,"style":763},[93,94],[25,815],{"className":816,"style":176},[175],[25,818,133],{"className":819},[180],[25,821],{"className":822,"style":176},[175],[25,824,826,829,832,835,838,841,844,847],{"className":825},[84],[25,827],{"className":828,"style":563},[88],[25,830,784],{"className":831},[93],[25,833,71],{"className":834,"style":95},[93,94],[25,836],{"className":837,"style":683},[175],[25,839,789],{"className":840},[687],[25,842],{"className":843,"style":683},[175],[25,845,155],{"className":846},[93],[25,848,550],{"className":849,"style":171},[93,94],[11,851,853],{"id":852},"field-workflow","Field Workflow",[855,856,857,861,942,946,949,953,964,968,1002,1006],"steps",{},[105,858,860],{"id":859},"instrument-setup","Instrument Setup",[16,862,863,864,939,940,28],{},"Set up the theodolite over a known point. Measure the height of the instrument (",[25,865,867,887],{"className":866},[53],[25,868,870],{"className":869},[57],[59,871,872],{"xmlns":61},[63,873,874,884],{},[66,875,876],{},[135,877,878,881],{},[69,879,880],{},"h",[69,882,883],{},"i",[73,885,886],{"encoding":75},"h_i",[25,888,890],{"className":889,"ariaHidden":80},[79],[25,891,893,897],{"className":892},[84],[25,894],{"className":895,"style":896},[88],"height:0.8444em;vertical-align:-0.15em;",[25,898,900,903],{"className":899},[93],[25,901,880],{"className":902},[93,94],[25,904,906],{"className":905},[201],[25,907,909,931],{"className":908},[205,206],[25,910,912,928],{"className":911},[210],[25,913,916],{"className":914,"style":915},[214],"height:0.3117em;",[25,917,919,922],{"style":918},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[25,920],{"className":921,"style":223},[222],[25,923,925],{"className":924},[227,228,229,230],[25,926,883],{"className":927},[93,94,230],[25,929,238],{"className":930},[237],[25,932,934],{"className":933},[210],[25,935,937],{"className":936,"style":245},[214],[25,938],{},") above the peg ",[25,941,570],{},[105,943,945],{"id":944},"sight-the-staff","Sight the Staff",[16,947,948],{},"Direct the telescope to a staff held vertically at the target point.",[105,950,952],{"id":951},"record-three-hair-readings","Record Three Hair Readings",[16,954,955,956,960,961,28],{},"Note the Top, Middle, and Bottom stadia hair readings.\n",[957,958,959],"em",{},"Check:"," The middle reading should be the exact mean of the top and bottom readings ",[25,962,963],{},"18, 22",[105,965,967],{"id":966},"measure-vertical-angle","Measure Vertical Angle",[16,969,970,971,999,1000,28],{},"Note the vertical circle reading to determine the angle of inclination (",[25,972,974,987],{"className":973},[53],[25,975,977],{"className":976},[57],[59,978,979],{"xmlns":61},[63,980,981,985],{},[66,982,983],{},[69,984,550],{},[73,986,553],{"encoding":75},[25,988,990],{"className":989,"ariaHidden":80},[79],[25,991,993,996],{"className":992},[84],[25,994],{"className":995,"style":563},[88],[25,997,550],{"className":998,"style":171},[93,94],") ",[25,1001,570],{},[105,1003,1005],{"id":1004},"reduction","Reduction",[16,1007,1008,1009,1080,1081,1125,1126,28],{},"Apply the ",[25,1010,1012,1036],{"className":1011},[53],[25,1013,1015],{"className":1014},[57],[59,1016,1017],{"xmlns":61},[63,1018,1019,1033],{},[66,1020,1021,1031],{},[629,1022,1023,1029],{},[66,1024,1025,1027],{},[69,1026,635],{},[131,1028,638],{},[141,1030,155],{},[69,1032,550],{},[73,1034,1035],{"encoding":75},"\\cos^2 \\theta",[25,1037,1039],{"className":1038,"ariaHidden":80},[79],[25,1040,1042,1045,1074,1077],{"className":1041},[84],[25,1043],{"className":1044,"style":673},[88],[25,1046,1048,1051],{"className":1047},[687],[25,1049,635],{"className":1050},[687],[25,1052,1054],{"className":1053},[201],[25,1055,1057],{"className":1056},[205],[25,1058,1060],{"className":1059},[210],[25,1061,1063],{"className":1062,"style":673},[214],[25,1064,1065,1068],{"style":705},[25,1066],{"className":1067,"style":223},[222],[25,1069,1071],{"className":1070},[227,228,229,230],[25,1072,155],{"className":1073},[93,230],[25,1075],{"className":1076,"style":683},[175],[25,1078,550],{"className":1079,"style":171},[93,94]," and ",[25,1082,1084,1104],{"className":1083},[53],[25,1085,1087],{"className":1086},[57],[59,1088,1089],{"xmlns":61},[63,1090,1091,1101],{},[66,1092,1093,1095,1097,1099],{},[69,1094,789],{},[131,1096,638],{},[141,1098,155],{},[69,1100,550],{},[73,1102,1103],{"encoding":75},"\\sin 2\\theta",[25,1105,1107],{"className":1106,"ariaHidden":80},[79],[25,1108,1110,1113,1116,1119,1122],{"className":1109},[84],[25,1111],{"className":1112,"style":563},[88],[25,1114,789],{"className":1115},[687],[25,1117],{"className":1118,"style":683},[175],[25,1120,155],{"className":1121},[93],[25,1123,550],{"className":1124,"style":171},[93,94]," formulas to calculate distance and level ",[25,1127,1128],{},"21, 22",[11,1130,1132],{"id":1131},"step-by-step-example","Step-by-Step Example",[16,1134,1135,1138,1139,451],{},[20,1136,1137],{},"Problem:"," Find the distance and level of point B from station A (",[25,1140,1142,1169],{"className":1141},[53],[25,1143,1145],{"className":1144},[57],[59,1146,1147],{"xmlns":61},[63,1148,1149,1166],{},[66,1150,1151,1154,1157,1159,1162],{},[69,1152,1153],{},"R",[69,1155,1156],{},"L",[131,1158,133],{},[141,1160,1161],{},"100.00",[1163,1164,1165],"mtext",{}," m",[73,1167,1168],{"encoding":75},"RL = 100.00\\text{ m}",[25,1170,1172,1194],{"className":1171,"ariaHidden":80},[79],[25,1173,1175,1178,1182,1185,1188,1191],{"className":1174},[84],[25,1176],{"className":1177,"style":89},[88],[25,1179,1153],{"className":1180,"style":1181},[93,94],"margin-right:0.0077em;",[25,1183,1156],{"className":1184},[93,94],[25,1186],{"className":1187,"style":176},[175],[25,1189,133],{"className":1190},[180],[25,1192],{"className":1193,"style":176},[175],[25,1195,1197,1201,1204],{"className":1196},[84],[25,1198],{"className":1199,"style":1200},[88],"height:0.6444em;",[25,1202,1161],{"className":1203},[93],[25,1205,1208],{"className":1206},[93,1207],"text",[25,1209,1165],{"className":1210},[93],[310,1212,1213,1316,1500],{},[313,1214,1215,28],{},[25,1216,1218,1243],{"className":1217},[53],[25,1219,1221],{"className":1220},[57],[59,1222,1223],{"xmlns":61},[63,1224,1225,1240],{},[66,1226,1227,1233,1235,1238],{},[135,1228,1229,1231],{},[69,1230,880],{},[69,1232,883],{},[131,1234,133],{},[141,1236,1237],{},"1.50",[1163,1239,1165],{},[73,1241,1242],{"encoding":75},"h_i = 1.50\\text{ m}",[25,1244,1246,1301],{"className":1245,"ariaHidden":80},[79],[25,1247,1249,1252,1292,1295,1298],{"className":1248},[84],[25,1250],{"className":1251,"style":896},[88],[25,1253,1255,1258],{"className":1254},[93],[25,1256,880],{"className":1257},[93,94],[25,1259,1261],{"className":1260},[201],[25,1262,1264,1284],{"className":1263},[205,206],[25,1265,1267,1281],{"className":1266},[210],[25,1268,1270],{"className":1269,"style":915},[214],[25,1271,1272,1275],{"style":918},[25,1273],{"className":1274,"style":223},[222],[25,1276,1278],{"className":1277},[227,228,229,230],[25,1279,883],{"className":1280},[93,94,230],[25,1282,238],{"className":1283},[237],[25,1285,1287],{"className":1286},[210],[25,1288,1290],{"className":1289,"style":245},[214],[25,1291],{},[25,1293],{"className":1294,"style":176},[175],[25,1296,133],{"className":1297},[180],[25,1299],{"className":1300,"style":176},[175],[25,1302,1304,1307,1310],{"className":1303},[84],[25,1305],{"className":1306,"style":1200},[88],[25,1308,1237],{"className":1309},[93],[25,1311,1313],{"className":1312},[93,1207],[25,1314,1165],{"className":1315},[93],[313,1317,1318,1319,28],{},"Vertical Angle ",[25,1320,1322,1367],{"className":1321},[53],[25,1323,1325],{"className":1324},[57],[59,1326,1327],{"xmlns":61},[63,1328,1329,1364],{},[66,1330,1331,1333,1335,1337,1345,1354],{},[69,1332,550],{},[131,1334,133],{},[131,1336,148],{},[629,1338,1339,1342],{},[141,1340,1341],{},"5",[131,1343,1344],{},"∘",[629,1346,1347,1350],{},[141,1348,1349],{},"00",[131,1351,1353],{"mathvariant":740,"lspace":1352,"rspace":1352},"0em","′",[629,1355,1356,1358],{},[141,1357,1349],{},[66,1359,1360,1362],{},[131,1361,1353],{"mathvariant":740},[131,1363,1353],{"mathvariant":740},[73,1365,1366],{"encoding":75},"\\theta = +5^\\circ 00' 00''",[25,1368,1370,1388],{"className":1369,"ariaHidden":80},[79],[25,1371,1373,1376,1379,1382,1385],{"className":1372},[84],[25,1374],{"className":1375,"style":563},[88],[25,1377,550],{"className":1378,"style":171},[93,94],[25,1380],{"className":1381,"style":176},[175],[25,1383,133],{"className":1384},[180],[25,1386],{"className":1387,"style":176},[175],[25,1389,1391,1395,1398,1428,1431,1464,1467],{"className":1390},[84],[25,1392],{"className":1393,"style":1394},[88],"height:0.8352em;vertical-align:-0.0833em;",[25,1396,148],{"className":1397},[93],[25,1399,1401,1404],{"className":1400},[93],[25,1402,1341],{"className":1403},[93],[25,1405,1407],{"className":1406},[201],[25,1408,1410],{"className":1409},[205],[25,1411,1413],{"className":1412},[210],[25,1414,1417],{"className":1415,"style":1416},[214],"height:0.6741em;",[25,1418,1419,1422],{"style":705},[25,1420],{"className":1421,"style":223},[222],[25,1423,1425],{"className":1424},[227,228,229,230],[25,1426,1344],{"className":1427},[258,230],[25,1429,527],{"className":1430},[93],[25,1432,1434,1437],{"className":1433},[93],[25,1435,527],{"className":1436},[93],[25,1438,1440],{"className":1439},[201],[25,1441,1443],{"className":1442},[205],[25,1444,1446],{"className":1445},[210],[25,1447,1450],{"className":1448,"style":1449},[214],"height:0.7519em;",[25,1451,1452,1455],{"style":705},[25,1453],{"className":1454,"style":223},[222],[25,1456,1458],{"className":1457},[227,228,229,230],[25,1459,1461],{"className":1460},[93,230],[25,1462,1353],{"className":1463},[93,230],[25,1465,527],{"className":1466},[93],[25,1468,1470,1473],{"className":1469},[93],[25,1471,527],{"className":1472},[93],[25,1474,1476],{"className":1475},[201],[25,1477,1479],{"className":1478},[205],[25,1480,1482],{"className":1481},[210],[25,1483,1485],{"className":1484,"style":1449},[214],[25,1486,1487,1490],{"style":705},[25,1488],{"className":1489,"style":223},[222],[25,1491,1493],{"className":1492},[227,228,229,230],[25,1494,1496],{"className":1495},[93,230],[25,1497,1499],{"className":1498},[93,230],"′′",[313,1501,1502,1503,1541,1542,1580,1581,1619,1620,28],{},"Stadia Readings: Top = ",[25,1504,1506,1523],{"className":1505},[53],[25,1507,1509],{"className":1508},[57],[59,1510,1511],{"xmlns":61},[63,1512,1513,1520],{},[66,1514,1515,1518],{},[141,1516,1517],{},"2.412",[1163,1519,1165],{},[73,1521,1522],{"encoding":75},"2.412\\text{ m}",[25,1524,1526],{"className":1525,"ariaHidden":80},[79],[25,1527,1529,1532,1535],{"className":1528},[84],[25,1530],{"className":1531,"style":1200},[88],[25,1533,1517],{"className":1534},[93],[25,1536,1538],{"className":1537},[93,1207],[25,1539,1165],{"className":1540},[93],", Mid = ",[25,1543,1545,1562],{"className":1544},[53],[25,1546,1548],{"className":1547},[57],[59,1549,1550],{"xmlns":61},[63,1551,1552,1559],{},[66,1553,1554,1557],{},[141,1555,1556],{},"1.926",[1163,1558,1165],{},[73,1560,1561],{"encoding":75},"1.926\\text{ m}",[25,1563,1565],{"className":1564,"ariaHidden":80},[79],[25,1566,1568,1571,1574],{"className":1567},[84],[25,1569],{"className":1570,"style":1200},[88],[25,1572,1556],{"className":1573},[93],[25,1575,1577],{"className":1576},[93,1207],[25,1578,1165],{"className":1579},[93],", Bottom = ",[25,1582,1584,1601],{"className":1583},[53],[25,1585,1587],{"className":1586},[57],[59,1588,1589],{"xmlns":61},[63,1590,1591,1598],{},[66,1592,1593,1596],{},[141,1594,1595],{},"1.440",[1163,1597,1165],{},[73,1599,1600],{"encoding":75},"1.440\\text{ m}",[25,1602,1604],{"className":1603,"ariaHidden":80},[79],[25,1605,1607,1610,1613],{"className":1606},[84],[25,1608],{"className":1609,"style":1200},[88],[25,1611,1595],{"className":1612},[93],[25,1614,1616],{"className":1615},[93,1207],[25,1617,1165],{"className":1618},[93]," ",[25,1621,1622],{},"22",[1624,1625,1626,1765,2056,2328],"ol",{},[313,1627,1628,1659],{},[20,1629,1630,1631,607],{},"Calculate Intercept (",[25,1632,1634,1647],{"className":1633},[53],[25,1635,1637],{"className":1636},[57],[59,1638,1639],{"xmlns":61},[63,1640,1641,1645],{},[66,1642,1643],{},[69,1644,71],{},[73,1646,71],{"encoding":75},[25,1648,1650],{"className":1649,"ariaHidden":80},[79],[25,1651,1653,1656],{"className":1652},[84],[25,1654],{"className":1655,"style":89},[88],[25,1657,71],{"className":1658,"style":95},[93,94],[25,1660,1662,1692],{"className":1661},[53],[25,1663,1665],{"className":1664},[57],[59,1666,1667],{"xmlns":61},[63,1668,1669,1689],{},[66,1670,1671,1673,1675,1677,1680,1682,1684,1687],{},[69,1672,71],{},[131,1674,133],{},[141,1676,1517],{},[131,1678,1679],{},"−",[141,1681,1595],{},[131,1683,133],{},[141,1685,1686],{},"0.972",[1163,1688,1165],{},[73,1690,1691],{"encoding":75},"S = 2.412 - 1.440 = 0.972\\text{ m}",[25,1693,1695,1713,1732,1750],{"className":1694,"ariaHidden":80},[79],[25,1696,1698,1701,1704,1707,1710],{"className":1697},[84],[25,1699],{"className":1700,"style":89},[88],[25,1702,71],{"className":1703,"style":95},[93,94],[25,1705],{"className":1706,"style":176},[175],[25,1708,133],{"className":1709},[180],[25,1711],{"className":1712,"style":176},[175],[25,1714,1716,1720,1723,1726,1729],{"className":1715},[84],[25,1717],{"className":1718,"style":1719},[88],"height:0.7278em;vertical-align:-0.0833em;",[25,1721,1517],{"className":1722},[93],[25,1724],{"className":1725,"style":254},[175],[25,1727,1679],{"className":1728},[258],[25,1730],{"className":1731,"style":254},[175],[25,1733,1735,1738,1741,1744,1747],{"className":1734},[84],[25,1736],{"className":1737,"style":1200},[88],[25,1739,1595],{"className":1740},[93],[25,1742],{"className":1743,"style":176},[175],[25,1745,133],{"className":1746},[180],[25,1748],{"className":1749,"style":176},[175],[25,1751,1753,1756,1759],{"className":1752},[84],[25,1754],{"className":1755,"style":1200},[88],[25,1757,1686],{"className":1758},[93],[25,1760,1762],{"className":1761},[93,1207],[25,1763,1165],{"className":1764},[93],[313,1766,1767,1798],{},[20,1768,1769,1770,607],{},"Calculate Horizontal Distance (",[25,1771,1773,1786],{"className":1772},[53],[25,1774,1776],{"className":1775},[57],[59,1777,1778],{"xmlns":61},[63,1779,1780,1784],{},[66,1781,1782],{},[69,1783,129],{},[73,1785,129],{"encoding":75},[25,1787,1789],{"className":1788,"ariaHidden":80},[79],[25,1790,1792,1795],{"className":1791},[84],[25,1793],{"className":1794,"style":89},[88],[25,1796,129],{"className":1797,"style":171},[93,94],[25,1799,1801,1866],{"className":1800},[53],[25,1802,1804],{"className":1803},[57],[59,1805,1806],{"xmlns":61},[63,1807,1808,1863],{},[66,1809,1810,1812,1814,1816,1819,1821,1823,1833,1837,1843,1846,1848,1851,1853,1856,1858,1861],{},[69,1811,129],{},[131,1813,133],{},[141,1815,450],{},[131,1817,1818],{},"×",[141,1820,1686],{},[131,1822,1818],{},[629,1824,1825,1831],{},[66,1826,1827,1829],{},[69,1828,635],{},[131,1830,638],{},[141,1832,155],{},[131,1834,1836],{"stretchy":1835},"false","(",[629,1838,1839,1841],{},[141,1840,1341],{},[131,1842,1344],{},[131,1844,1845],{"stretchy":1835},")",[131,1847,133],{},[141,1849,1850],{},"97.2",[131,1852,1818],{},[141,1854,1855],{},"0.9924",[131,1857,133],{},[141,1859,1860],{},"96.46",[1163,1862,1165],{},[73,1864,1865],{"encoding":75},"D = 100 \\times 0.972 \\times \\cos^2(5^\\circ) = 97.2 \\times 0.9924 = 96.46\\text{ m}",[25,1867,1869,1887,1905,1923,2005,2023,2041],{"className":1868,"ariaHidden":80},[79],[25,1870,1872,1875,1878,1881,1884],{"className":1871},[84],[25,1873],{"className":1874,"style":89},[88],[25,1876,129],{"className":1877,"style":171},[93,94],[25,1879],{"className":1880,"style":176},[175],[25,1882,133],{"className":1883},[180],[25,1885],{"className":1886,"style":176},[175],[25,1888,1890,1893,1896,1899,1902],{"className":1889},[84],[25,1891],{"className":1892,"style":1719},[88],[25,1894,450],{"className":1895},[93],[25,1897],{"className":1898,"style":254},[175],[25,1900,1818],{"className":1901},[258],[25,1903],{"className":1904,"style":254},[175],[25,1906,1908,1911,1914,1917,1920],{"className":1907},[84],[25,1909],{"className":1910,"style":1719},[88],[25,1912,1686],{"className":1913},[93],[25,1915],{"className":1916,"style":254},[175],[25,1918,1818],{"className":1919},[258],[25,1921],{"className":1922,"style":254},[175],[25,1924,1926,1930,1959,1963,1992,1996,1999,2002],{"className":1925},[84],[25,1927],{"className":1928,"style":1929},[88],"height:1.0641em;vertical-align:-0.25em;",[25,1931,1933,1936],{"className":1932},[687],[25,1934,635],{"className":1935},[687],[25,1937,1939],{"className":1938},[201],[25,1940,1942],{"className":1941},[205],[25,1943,1945],{"className":1944},[210],[25,1946,1948],{"className":1947,"style":673},[214],[25,1949,1950,1953],{"style":705},[25,1951],{"className":1952,"style":223},[222],[25,1954,1956],{"className":1955},[227,228,229,230],[25,1957,155],{"className":1958},[93,230],[25,1960,1836],{"className":1961},[1962],"mopen",[25,1964,1966,1969],{"className":1965},[93],[25,1967,1341],{"className":1968},[93],[25,1970,1972],{"className":1971},[201],[25,1973,1975],{"className":1974},[205],[25,1976,1978],{"className":1977},[210],[25,1979,1981],{"className":1980,"style":1416},[214],[25,1982,1983,1986],{"style":705},[25,1984],{"className":1985,"style":223},[222],[25,1987,1989],{"className":1988},[227,228,229,230],[25,1990,1344],{"className":1991},[258,230],[25,1993,1845],{"className":1994},[1995],"mclose",[25,1997],{"className":1998,"style":176},[175],[25,2000,133],{"className":2001},[180],[25,2003],{"className":2004,"style":176},[175],[25,2006,2008,2011,2014,2017,2020],{"className":2007},[84],[25,2009],{"className":2010,"style":1719},[88],[25,2012,1850],{"className":2013},[93],[25,2015],{"className":2016,"style":254},[175],[25,2018,1818],{"className":2019},[258],[25,2021],{"className":2022,"style":254},[175],[25,2024,2026,2029,2032,2035,2038],{"className":2025},[84],[25,2027],{"className":2028,"style":1200},[88],[25,2030,1855],{"className":2031},[93],[25,2033],{"className":2034,"style":176},[175],[25,2036,133],{"className":2037},[180],[25,2039],{"className":2040,"style":176},[175],[25,2042,2044,2047,2050],{"className":2043},[84],[25,2045],{"className":2046,"style":1200},[88],[25,2048,1860],{"className":2049},[93],[25,2051,2053],{"className":2052},[93,1207],[25,2054,1165],{"className":2055},[93],[313,2057,2058,2094],{},[20,2059,2060,2061,607],{},"Calculate Vertical Component (",[25,2062,2064,2079],{"className":2063},[53],[25,2065,2067],{"className":2066},[57],[59,2068,2069],{"xmlns":61},[63,2070,2071,2077],{},[66,2072,2073,2075],{},[69,2074,741],{"mathvariant":740},[69,2076,744],{},[73,2078,747],{"encoding":75},[25,2080,2082],{"className":2081,"ariaHidden":80},[79],[25,2083,2085,2088,2091],{"className":2084},[84],[25,2086],{"className":2087,"style":89},[88],[25,2089,741],{"className":2090},[93],[25,2092,744],{"className":2093,"style":763},[93,94],[25,2095,2097,2158],{"className":2096},[53],[25,2098,2100],{"className":2099},[57],[59,2101,2102],{"xmlns":61},[63,2103,2104,2155],{},[66,2105,2106,2108,2110,2112,2114,2116,2118,2120,2122,2124,2126,2132,2134,2136,2138,2140,2146,2148,2150,2153],{},[69,2107,741],{"mathvariant":740},[69,2109,744],{},[131,2111,133],{},[141,2113,450],{},[131,2115,1818],{},[141,2117,1686],{},[131,2119,1818],{},[69,2121,635],{},[131,2123,638],{},[131,2125,1836],{"stretchy":1835},[629,2127,2128,2130],{},[141,2129,1341],{},[131,2131,1344],{},[131,2133,1845],{"stretchy":1835},[69,2135,789],{},[131,2137,638],{},[131,2139,1836],{"stretchy":1835},[629,2141,2142,2144],{},[141,2143,1341],{},[131,2145,1344],{},[131,2147,1845],{"stretchy":1835},[131,2149,133],{},[141,2151,2152],{},"8.47",[1163,2154,1165],{},[73,2156,2157],{"encoding":75},"\\Delta H = 100 \\times 0.972 \\times \\cos(5^\\circ) \\sin(5^\\circ) = 8.47\\text{ m}",[25,2159,2161,2182,2200,2218,2313],{"className":2160,"ariaHidden":80},[79],[25,2162,2164,2167,2170,2173,2176,2179],{"className":2163},[84],[25,2165],{"className":2166,"style":89},[88],[25,2168,741],{"className":2169},[93],[25,2171,744],{"className":2172,"style":763},[93,94],[25,2174],{"className":2175,"style":176},[175],[25,2177,133],{"className":2178},[180],[25,2180],{"className":2181,"style":176},[175],[25,2183,2185,2188,2191,2194,2197],{"className":2184},[84],[25,2186],{"className":2187,"style":1719},[88],[25,2189,450],{"className":2190},[93],[25,2192],{"className":2193,"style":254},[175],[25,2195,1818],{"className":2196},[258],[25,2198],{"className":2199,"style":254},[175],[25,2201,2203,2206,2209,2212,2215],{"className":2202},[84],[25,2204],{"className":2205,"style":1719},[88],[25,2207,1686],{"className":2208},[93],[25,2210],{"className":2211,"style":254},[175],[25,2213,1818],{"className":2214},[258],[25,2216],{"className":2217,"style":254},[175],[25,2219,2221,2225,2228,2231,2260,2263,2266,2269,2272,2301,2304,2307,2310],{"className":2220},[84],[25,2222],{"className":2223,"style":2224},[88],"height:1em;vertical-align:-0.25em;",[25,2226,635],{"className":2227},[687],[25,2229,1836],{"className":2230},[1962],[25,2232,2234,2237],{"className":2233},[93],[25,2235,1341],{"className":2236},[93],[25,2238,2240],{"className":2239},[201],[25,2241,2243],{"className":2242},[205],[25,2244,2246],{"className":2245},[210],[25,2247,2249],{"className":2248,"style":1416},[214],[25,2250,2251,2254],{"style":705},[25,2252],{"className":2253,"style":223},[222],[25,2255,2257],{"className":2256},[227,228,229,230],[25,2258,1344],{"className":2259},[258,230],[25,2261,1845],{"className":2262},[1995],[25,2264],{"className":2265,"style":683},[175],[25,2267,789],{"className":2268},[687],[25,2270,1836],{"className":2271},[1962],[25,2273,2275,2278],{"className":2274},[93],[25,2276,1341],{"className":2277},[93],[25,2279,2281],{"className":2280},[201],[25,2282,2284],{"className":2283},[205],[25,2285,2287],{"className":2286},[210],[25,2288,2290],{"className":2289,"style":1416},[214],[25,2291,2292,2295],{"style":705},[25,2293],{"className":2294,"style":223},[222],[25,2296,2298],{"className":2297},[227,228,229,230],[25,2299,1344],{"className":2300},[258,230],[25,2302,1845],{"className":2303},[1995],[25,2305],{"className":2306,"style":176},[175],[25,2308,133],{"className":2309},[180],[25,2311],{"className":2312,"style":176},[175],[25,2314,2316,2319,2322],{"className":2315},[84],[25,2317],{"className":2318,"style":1200},[88],[25,2320,2152],{"className":2321},[93],[25,2323,2325],{"className":2324},[93,1207],[25,2326,1165],{"className":2327},[93],[313,2329,2330,2333,2598,1619,2792,28],{},[20,2331,2332],{},"Final RL of B:",[25,2334,2336,2387],{"className":2335},[53],[25,2337,2339],{"className":2338},[57],[59,2340,2341],{"xmlns":61},[63,2342,2343,2384],{},[66,2344,2345,2347,2354,2356,2358,2365,2367,2373,2375,2377,2379,2381],{},[69,2346,1153],{},[135,2348,2349,2351],{},[69,2350,1156],{},[69,2352,2353],{},"B",[131,2355,133],{},[69,2357,1153],{},[135,2359,2360,2362],{},[69,2361,1156],{},[69,2363,2364],{},"A",[131,2366,148],{},[135,2368,2369,2371],{},[69,2370,880],{},[69,2372,883],{},[131,2374,148],{},[69,2376,741],{"mathvariant":740},[69,2378,744],{},[131,2380,1679],{},[1163,2382,2383],{},"Mid Reading",[73,2385,2386],{"encoding":75},"RL_B = RL_A + h_i + \\Delta H - \\text{Mid Reading}",[25,2388,2390,2450,2508,2563,2585],{"className":2389,"ariaHidden":80},[79],[25,2391,2393,2396,2399,2441,2444,2447],{"className":2392},[84],[25,2394],{"className":2395,"style":190},[88],[25,2397,1153],{"className":2398,"style":1181},[93,94],[25,2400,2402,2405],{"className":2401},[93],[25,2403,1156],{"className":2404},[93,94],[25,2406,2408],{"className":2407},[201],[25,2409,2411,2433],{"className":2410},[205,206],[25,2412,2414,2430],{"className":2413},[210],[25,2415,2418],{"className":2416,"style":2417},[214],"height:0.3283em;",[25,2419,2420,2423],{"style":918},[25,2421],{"className":2422,"style":223},[222],[25,2424,2426],{"className":2425},[227,228,229,230],[25,2427,2353],{"className":2428,"style":2429},[93,94,230],"margin-right:0.0502em;",[25,2431,238],{"className":2432},[237],[25,2434,2436],{"className":2435},[210],[25,2437,2439],{"className":2438,"style":245},[214],[25,2440],{},[25,2442],{"className":2443,"style":176},[175],[25,2445,133],{"className":2446},[180],[25,2448],{"className":2449,"style":176},[175],[25,2451,2453,2456,2459,2499,2502,2505],{"className":2452},[84],[25,2454],{"className":2455,"style":190},[88],[25,2457,1153],{"className":2458,"style":1181},[93,94],[25,2460,2462,2465],{"className":2461},[93],[25,2463,1156],{"className":2464},[93,94],[25,2466,2468],{"className":2467},[201],[25,2469,2471,2491],{"className":2470},[205,206],[25,2472,2474,2488],{"className":2473},[210],[25,2475,2477],{"className":2476,"style":2417},[214],[25,2478,2479,2482],{"style":918},[25,2480],{"className":2481,"style":223},[222],[25,2483,2485],{"className":2484},[227,228,229,230],[25,2486,2364],{"className":2487},[93,94,230],[25,2489,238],{"className":2490},[237],[25,2492,2494],{"className":2493},[210],[25,2495,2497],{"className":2496,"style":245},[214],[25,2498],{},[25,2500],{"className":2501,"style":254},[175],[25,2503,148],{"className":2504},[258],[25,2506],{"className":2507,"style":254},[175],[25,2509,2511,2514,2554,2557,2560],{"className":2510},[84],[25,2512],{"className":2513,"style":896},[88],[25,2515,2517,2520],{"className":2516},[93],[25,2518,880],{"className":2519},[93,94],[25,2521,2523],{"className":2522},[201],[25,2524,2526,2546],{"className":2525},[205,206],[25,2527,2529,2543],{"className":2528},[210],[25,2530,2532],{"className":2531,"style":915},[214],[25,2533,2534,2537],{"style":918},[25,2535],{"className":2536,"style":223},[222],[25,2538,2540],{"className":2539},[227,228,229,230],[25,2541,883],{"className":2542},[93,94,230],[25,2544,238],{"className":2545},[237],[25,2547,2549],{"className":2548},[210],[25,2550,2552],{"className":2551,"style":245},[214],[25,2553],{},[25,2555],{"className":2556,"style":254},[175],[25,2558,148],{"className":2559},[258],[25,2561],{"className":2562,"style":254},[175],[25,2564,2566,2570,2573,2576,2579,2582],{"className":2565},[84],[25,2567],{"className":2568,"style":2569},[88],"height:0.7667em;vertical-align:-0.0833em;",[25,2571,741],{"className":2572},[93],[25,2574,744],{"className":2575,"style":763},[93,94],[25,2577],{"className":2578,"style":254},[175],[25,2580,1679],{"className":2581},[258],[25,2583],{"className":2584,"style":254},[175],[25,2586,2588,2592],{"className":2587},[84],[25,2589],{"className":2590,"style":2591},[88],"height:0.8889em;vertical-align:-0.1944em;",[25,2593,2595],{"className":2594},[93,1207],[25,2596,2383],{"className":2597},[93],[25,2599,2601,2644],{"className":2600},[53],[25,2602,2604],{"className":2603},[57],[59,2605,2606],{"xmlns":61},[63,2607,2608,2641],{},[66,2609,2610,2612,2618,2620,2622,2624,2626,2628,2630,2632,2634,2636,2639],{},[69,2611,1153],{},[135,2613,2614,2616],{},[69,2615,1156],{},[69,2617,2353],{},[131,2619,133],{},[141,2621,1161],{},[131,2623,148],{},[141,2625,1237],{},[131,2627,148],{},[141,2629,2152],{},[131,2631,1679],{},[141,2633,1556],{},[131,2635,133],{},[141,2637,2638],{},"108.044",[1163,2640,1165],{},[73,2642,2643],{"encoding":75},"RL_B = 100.00 + 1.50 + 8.47 - 1.926 = 108.044\\text{ m}",[25,2645,2647,2705,2723,2741,2759,2777],{"className":2646,"ariaHidden":80},[79],[25,2648,2650,2653,2656,2696,2699,2702],{"className":2649},[84],[25,2651],{"className":2652,"style":190},[88],[25,2654,1153],{"className":2655,"style":1181},[93,94],[25,2657,2659,2662],{"className":2658},[93],[25,2660,1156],{"className":2661},[93,94],[25,2663,2665],{"className":2664},[201],[25,2666,2668,2688],{"className":2667},[205,206],[25,2669,2671,2685],{"className":2670},[210],[25,2672,2674],{"className":2673,"style":2417},[214],[25,2675,2676,2679],{"style":918},[25,2677],{"className":2678,"style":223},[222],[25,2680,2682],{"className":2681},[227,228,229,230],[25,2683,2353],{"className":2684,"style":2429},[93,94,230],[25,2686,238],{"className":2687},[237],[25,2689,2691],{"className":2690},[210],[25,2692,2694],{"className":2693,"style":245},[214],[25,2695],{},[25,2697],{"className":2698,"style":176},[175],[25,2700,133],{"className":2701},[180],[25,2703],{"className":2704,"style":176},[175],[25,2706,2708,2711,2714,2717,2720],{"className":2707},[84],[25,2709],{"className":2710,"style":1719},[88],[25,2712,1161],{"className":2713},[93],[25,2715],{"className":2716,"style":254},[175],[25,2718,148],{"className":2719},[258],[25,2721],{"className":2722,"style":254},[175],[25,2724,2726,2729,2732,2735,2738],{"className":2725},[84],[25,2727],{"className":2728,"style":1719},[88],[25,2730,1237],{"className":2731},[93],[25,2733],{"className":2734,"style":254},[175],[25,2736,148],{"className":2737},[258],[25,2739],{"className":2740,"style":254},[175],[25,2742,2744,2747,2750,2753,2756],{"className":2743},[84],[25,2745],{"className":2746,"style":1719},[88],[25,2748,2152],{"className":2749},[93],[25,2751],{"className":2752,"style":254},[175],[25,2754,1679],{"className":2755},[258],[25,2757],{"className":2758,"style":254},[175],[25,2760,2762,2765,2768,2771,2774],{"className":2761},[84],[25,2763],{"className":2764,"style":1200},[88],[25,2766,1556],{"className":2767},[93],[25,2769],{"className":2770,"style":176},[175],[25,2772,133],{"className":2773},[180],[25,2775],{"className":2776,"style":176},[175],[25,2778,2780,2783,2786],{"className":2779},[84],[25,2781],{"className":2782,"style":1200},[88],[25,2784,2638],{"className":2785},[93],[25,2787,2789],{"className":2788},[93,1207],[25,2790,1165],{"className":2791},[93],[25,2793,1128],{},[11,2795,2797],{"id":2796},"practical-tips","Practical Tips",[310,2799,2800,2918],{},[313,2801,2802,2805,2806,2875,2876,2914,2915,28],{},[20,2803,2804],{},"Constant Checks:"," In modern telescopes, the additive constant (",[25,2807,2809,2826],{"className":2808},[53],[25,2810,2812],{"className":2811},[57],[59,2813,2814],{"xmlns":61},[63,2815,2816,2824],{},[66,2817,2818],{},[135,2819,2820,2822],{},[69,2821,139],{},[141,2823,155],{},[73,2825,474],{"encoding":75},[25,2827,2829],{"className":2828,"ariaHidden":80},[79],[25,2830,2832,2835],{"className":2831},[84],[25,2833],{"className":2834,"style":190},[88],[25,2836,2838,2841],{"className":2837},[93],[25,2839,139],{"className":2840,"style":197},[93,94],[25,2842,2844],{"className":2843},[201],[25,2845,2847,2867],{"className":2846},[205,206],[25,2848,2850,2864],{"className":2849},[210],[25,2851,2853],{"className":2852,"style":215},[214],[25,2854,2855,2858],{"style":218},[25,2856],{"className":2857,"style":223},[222],[25,2859,2861],{"className":2860},[227,228,229,230],[25,2862,155],{"className":2863},[93,230],[25,2865,238],{"className":2866},[237],[25,2868,2870],{"className":2869},[210],[25,2871,2873],{"className":2872,"style":245},[214],[25,2874],{},") is zero because of an internal \"anallactic lens.\" Always check your instrument specifications to confirm if you need to add a value (typically ",[25,2877,2879,2896],{"className":2878},[53],[25,2880,2882],{"className":2881},[57],[59,2883,2884],{"xmlns":61},[63,2885,2886,2893],{},[66,2887,2888,2891],{},[141,2889,2890],{},"0.3",[1163,2892,1165],{},[73,2894,2895],{"encoding":75},"0.3\\text{ m}",[25,2897,2899],{"className":2898,"ariaHidden":80},[79],[25,2900,2902,2905,2908],{"className":2901},[84],[25,2903],{"className":2904,"style":1200},[88],[25,2906,2890],{"className":2907},[93],[25,2909,2911],{"className":2910},[93,1207],[25,2912,1165],{"className":2913},[93]," for older external-focusing telescopes) ",[25,2916,2917],{},"19, 21",[313,2919,2920,2923,2924,2961,2962,2990,2991,1619,3029,28],{},[20,2921,2922],{},"The 1% Rule:"," A handy field check: at ",[25,2925,2927,2943],{"className":2926},[53],[25,2928,2930],{"className":2929},[57],[59,2931,2932],{"xmlns":61},[63,2933,2934,2940],{},[66,2935,2936,2938],{},[141,2937,450],{},[1163,2939,1165],{},[73,2941,2942],{"encoding":75},"100\\text{ m}",[25,2944,2946],{"className":2945,"ariaHidden":80},[79],[25,2947,2949,2952,2955],{"className":2948},[84],[25,2950],{"className":2951,"style":1200},[88],[25,2953,450],{"className":2954},[93],[25,2956,2958],{"className":2957},[93,1207],[25,2959,1165],{"className":2960},[93],", the stadia intercept (",[25,2963,2965,2978],{"className":2964},[53],[25,2966,2968],{"className":2967},[57],[59,2969,2970],{"xmlns":61},[63,2971,2972,2976],{},[66,2973,2974],{},[69,2975,71],{},[73,2977,71],{"encoding":75},[25,2979,2981],{"className":2980,"ariaHidden":80},[79],[25,2982,2984,2987],{"className":2983},[84],[25,2985],{"className":2986,"style":89},[88],[25,2988,71],{"className":2989,"style":95},[93,94],") should be exactly ",[25,2992,2994,3011],{"className":2993},[53],[25,2995,2997],{"className":2996},[57],[59,2998,2999],{"xmlns":61},[63,3000,3001,3008],{},[66,3002,3003,3006],{},[141,3004,3005],{},"1.000",[1163,3007,1165],{},[73,3009,3010],{"encoding":75},"1.000\\text{ m}",[25,3012,3014],{"className":3013,"ariaHidden":80},[79],[25,3015,3017,3020,3023],{"className":3016},[84],[25,3018],{"className":3019,"style":1200},[88],[25,3021,3005],{"className":3022},[93],[25,3024,3026],{"className":3025},[93,1207],[25,3027,1165],{"className":3028},[93],[25,3030,99],{},[11,3032,3034],{"id":3033},"common-mistakes","Common Mistakes",[310,3036,3037,3075],{},[313,3038,3039,3042,3043,3071,3072,28],{},[20,3040,3041],{},"Staff Not Vertical:"," If the staff leans, the intercept (",[25,3044,3046,3059],{"className":3045},[53],[25,3047,3049],{"className":3048},[57],[59,3050,3051],{"xmlns":61},[63,3052,3053,3057],{},[66,3054,3055],{},[69,3056,71],{},[73,3058,71],{"encoding":75},[25,3060,3062],{"className":3061,"ariaHidden":80},[79],[25,3063,3065,3068],{"className":3064},[84],[25,3066],{"className":3067,"style":89},[88],[25,3069,71],{"className":3070,"style":95},[93,94],") will be too large, resulting in a distance that is too long. Use a staff level bubble ",[25,3073,3074],{},"23",[313,3076,3077,3080,3081,3109,3110,3169,3170,3234,3235,28],{},[20,3078,3079],{},"Confusing Zenith with Vertical Angle:"," Surveying formulas use the angle from the horizontal (",[25,3082,3084,3097],{"className":3083},[53],[25,3085,3087],{"className":3086},[57],[59,3088,3089],{"xmlns":61},[63,3090,3091,3095],{},[66,3092,3093],{},[69,3094,550],{},[73,3096,553],{"encoding":75},[25,3098,3100],{"className":3099,"ariaHidden":80},[79],[25,3101,3103,3106],{"className":3102},[84],[25,3104],{"className":3105,"style":563},[88],[25,3107,550],{"className":3108,"style":171},[93,94],"). If your instrument reads Zenith (",[25,3111,3113,3131],{"className":3112},[53],[25,3114,3116],{"className":3115},[57],[59,3117,3118],{"xmlns":61},[63,3119,3120,3128],{},[66,3121,3122],{},[629,3123,3124,3126],{},[141,3125,527],{},[131,3127,1344],{},[73,3129,3130],{"encoding":75},"0^\\circ",[25,3132,3134],{"className":3133,"ariaHidden":80},[79],[25,3135,3137,3140],{"className":3136},[84],[25,3138],{"className":3139,"style":1416},[88],[25,3141,3143,3146],{"className":3142},[93],[25,3144,527],{"className":3145},[93],[25,3147,3149],{"className":3148},[201],[25,3150,3152],{"className":3151},[205],[25,3153,3155],{"className":3154},[210],[25,3156,3158],{"className":3157,"style":1416},[214],[25,3159,3160,3163],{"style":705},[25,3161],{"className":3162,"style":223},[222],[25,3164,3166],{"className":3165},[227,228,229,230],[25,3167,1344],{"className":3168},[258,230]," at the top), you must subtract it from ",[25,3171,3173,3192],{"className":3172},[53],[25,3174,3176],{"className":3175},[57],[59,3177,3178],{"xmlns":61},[63,3179,3180,3189],{},[66,3181,3182],{},[629,3183,3184,3187],{},[141,3185,3186],{},"90",[131,3188,1344],{},[73,3190,3191],{"encoding":75},"90^\\circ",[25,3193,3195],{"className":3194,"ariaHidden":80},[79],[25,3196,3198,3201,3205],{"className":3197},[84],[25,3199],{"className":3200,"style":1416},[88],[25,3202,3204],{"className":3203},[93],"9",[25,3206,3208,3211],{"className":3207},[93],[25,3209,527],{"className":3210},[93],[25,3212,3214],{"className":3213},[201],[25,3215,3217],{"className":3216},[205],[25,3218,3220],{"className":3219},[210],[25,3221,3223],{"className":3222,"style":1416},[214],[25,3224,3225,3228],{"style":705},[25,3226],{"className":3227,"style":223},[222],[25,3229,3231],{"className":3230},[227,228,229,230],[25,3232,1344],{"className":3233},[258,230]," before using the tacheometry formulas ",[25,3236,3237],{},"24",[11,3239,3241],{"id":3240},"faq","FAQ",[3243,3244,3245,3330,3338],"accordion",{},[3246,3247,3249],"accordion-item",{"label":3248},"What is the maximum range of tacheometry?",[16,3250,3251,3252,3288,3289,3327,3328,28],{},"For accurate results, sights should be limited to ",[25,3253,3255,3270],{"className":3254},[53],[25,3256,3258],{"className":3257},[57],[59,3259,3260],{"xmlns":61},[63,3261,3262,3268],{},[66,3263,3264,3266],{},[141,3265,450],{},[1163,3267,1165],{},[73,3269,2942],{"encoding":75},[25,3271,3273],{"className":3272,"ariaHidden":80},[79],[25,3274,3276,3279,3282],{"className":3275},[84],[25,3277],{"className":3278,"style":1200},[88],[25,3280,450],{"className":3281},[93],[25,3283,3285],{"className":3284},[93,1207],[25,3286,1165],{"className":3287},[93]," or ",[25,3290,3292,3309],{"className":3291},[53],[25,3293,3295],{"className":3294},[57],[59,3296,3297],{"xmlns":61},[63,3298,3299,3306],{},[66,3300,3301,3304],{},[141,3302,3303],{},"150",[1163,3305,1165],{},[73,3307,3308],{"encoding":75},"150\\text{ m}",[25,3310,3312],{"className":3311,"ariaHidden":80},[79],[25,3313,3315,3318,3321],{"className":3314},[84],[25,3316],{"className":3317,"style":1200},[88],[25,3319,3303],{"className":3320},[93],[25,3322,3324],{"className":3323},[93,1207],[25,3325,1165],{"className":3326},[93],". Beyond this, the staff graduations become difficult to read precisely ",[25,3329,38],{},[3246,3331,3333],{"label":3332},"Why are there three hair readings?",[16,3334,3335,3336,28],{},"The top and bottom hairs give the distance. The middle hair gives the height. Taking all three serves as a vital arithmetical check in the field book ",[25,3337,99],{},[3246,3339,3341],{"label":3340},"What is 'Stadia'?",[16,3342,3343,3344,28],{},"The term \"Stadia\" refers to the fixed horizontal cross-hairs etched onto the reticule of the telescope ",[25,3345,99],{},[11,3347,3349],{"id":3348},"conclusion","Conclusion",[16,3351,3352,3353,3423],{},"Stadia tacheometry is a brilliant application of optical geometry. By mastering the ",[25,3354,3356,3379],{"className":3355},[53],[25,3357,3359],{"className":3358},[57],[59,3360,3361],{"xmlns":61},[63,3362,3363,3377],{},[66,3364,3365,3375],{},[629,3366,3367,3373],{},[66,3368,3369,3371],{},[69,3370,635],{},[131,3372,638],{},[141,3374,155],{},[69,3376,550],{},[73,3378,1035],{"encoding":75},[25,3380,3382],{"className":3381,"ariaHidden":80},[79],[25,3383,3385,3388,3417,3420],{"className":3384},[84],[25,3386],{"className":3387,"style":673},[88],[25,3389,3391,3394],{"className":3390},[687],[25,3392,635],{"className":3393},[687],[25,3395,3397],{"className":3396},[201],[25,3398,3400],{"className":3399},[205],[25,3401,3403],{"className":3402},[210],[25,3404,3406],{"className":3405,"style":673},[214],[25,3407,3408,3411],{"style":705},[25,3409],{"className":3410,"style":223},[222],[25,3412,3414],{"className":3413},[227,228,229,230],[25,3415,155],{"className":3416},[93,230],[25,3418],{"className":3419,"style":683},[175],[25,3421,550],{"className":3422,"style":171},[93,94]," reduction, you gain a deep understanding of the relationship between angular measurement and spatial distance—the core of all surveying science.",[11,3425,3427],{"id":3426},"references","References",[16,3429,3430,3431,3434],{},"Schofield, W. (2001). ",[957,3432,3433],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.",{"title":3436,"searchDepth":3437,"depth":3437,"links":3438},"",2,[3439,3440,3441,3442,3447,3454,3455,3456,3457,3458,3459],{"id":13,"depth":3437,"text":14},{"id":31,"depth":3437,"text":32},{"id":41,"depth":3437,"text":42},{"id":102,"depth":3437,"text":103,"children":3443},[3444,3446],{"id":107,"depth":3445,"text":108},3,{"id":531,"depth":3445,"text":532},{"id":852,"depth":3437,"text":853,"children":3448},[3449,3450,3451,3452,3453],{"id":859,"depth":3445,"text":860},{"id":944,"depth":3445,"text":945},{"id":951,"depth":3445,"text":952},{"id":966,"depth":3445,"text":967},{"id":1004,"depth":3445,"text":1005},{"id":1131,"depth":3437,"text":1132},{"id":2796,"depth":3437,"text":2797},{"id":3033,"depth":3437,"text":3034},{"id":3240,"depth":3437,"text":3241},{"id":3348,"depth":3437,"text":3349},{"id":3426,"depth":3437,"text":3427},"Vertical Control","A technical exploration of tacheometry theory and formulas for determining distances and elevations using a theodolite and stadia hair readings.","md","\u002Fimages\u002Ftacheometry.jpg",{"authorCredentials":3465,"commentCount":3466,"tags":3467,"relatedTools":3473,"relatedArticles":3475,"sources":3478},"LS",0,[3468,3469,3470,3471,3472],"Tacheometry","Stadia","Optical Distance","Theodolite","Math",[3471,3474],"Levelling Staff",[3476,3477],"combination-of-errors-propagation","precise-levelling-techniques",[3479],{"title":3480,"type":3481,"note":3482},"Engineering Surveying (Fifth Edition)","Book","Primary source for the mathematical derivation of stadia formulas.",true,"\u002Fblog\u002Fstadia-tacheometry-optical-measurement","2024-04-08",18,{"title":5,"description":3461},{"loc":3484},"blog\u002F20.stadia-tacheometry-optical-measurement","EgjddQZKVvHFUA0Q-Dm7bKCNOU3bhHQudD9bCusvEjc",[3492,3497],{"title":3493,"path":3494,"stem":3495,"description":3496,"children":-1},"Sight Rails and Boning Rods: Practical Dimensional Control for Grading","\u002Fblog\u002Fsight-rails-boning-rods-grading","blog\u002F19.sight-rails-boning-rods-grading","A technical guide to using sight rails, boning rods, and travellers for controlling gradients in trench excavation and drainage projects.",{"title":3498,"path":3499,"stem":3500,"description":3501,"children":-1},"Real-Time Kinematic (RTK): Instantaneous Precision in GPS Surveying","\u002Fblog\u002Frtk-gps-real-time-kinematic","blog\u002F21.rtk-gps-real-time-kinematic","A technical guide to RTK GPS procedures, mobile data communication requirements, and real-time ambiguity resolution for engineering site work.",1783851946035]