[{"data":1,"prerenderedAt":3231},["ShallowReactive",2],{"blog-\u002Fblog\u002Fwgs84-datum-and-geodetic-projections":3,"blog-surround-\u002Fblog\u002Fwgs84-datum-and-geodetic-projections":3220},{"id":4,"title":5,"author":6,"body":7,"category":3188,"description":3189,"extension":3190,"image":3191,"meta":3192,"navigation":3212,"path":3213,"publishedAt":3214,"readTime":3215,"seo":3216,"sitemap":3217,"stem":3218,"__hash__":3219},"blog\u002Fblog\u002F22.wgs84-datum-and-geodetic-projections.md","WGS84 and Projections: The Mathematics of Global to Local Mapping","Surveyiiing Expert",{"type":8,"value":9,"toc":3161},"minimark",[10,15,37,41,108,112,163,167,298,302,307,310,504,650,1467,1471,1644,1648,1801,2386,2390,2475,2705,2709,2888,2892,2989,2993,3143,3147,3150,3154],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,27,28,31,32,36],"p",{},"Every GPS measurement is initially calculated in a global coordinate system called ",[20,21,22],"strong",{},"WGS84"," (World Geodetic System 1984). However, engineering designs are typically created on flat, rectangular site grids. Mastering the gap between the curved ",[20,25,26],{},"ellipsoidal"," world of WGS84 and the flat ",[20,29,30],{},"projected"," world of the construction site is fundamental to modern geomatics ",[33,34,35],"span",{},"14, 15",".",[11,38,40],{"id":39},"why-this-matters","Why This Matters",[16,42,43,44,100,101,104,105,36],{},"If you treat WGS84 coordinates as if they were on a flat plane, the Earth's curvature will introduce massive distortions over just a few kilometres. Furthermore, WGS84 coordinates refer to a mathematical surface (the ellipsoid) that can be up to ",[33,45,48,75],{"className":46},[47],"katex",[33,49,52],{"className":50},[51],"katex-mathml",[53,54,56],"math",{"xmlns":55},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[57,58,59,70],"semantics",{},[60,61,62,66],"mrow",{},[63,64,65],"mn",{},"100",[67,68,69],"mtext",{}," m",[71,72,74],"annotation",{"encoding":73},"application\u002Fx-tex","100 \\text{ m}",[33,76,80],{"className":77,"ariaHidden":79},[78],"katex-html","true",[33,81,84,89,93],{"className":82},[83],"base",[33,85],{"className":86,"style":88},[87],"strut","height:0.6444em;",[33,90,65],{"className":91},[92],"mord",[33,94,97],{"className":95},[92,96],"text",[33,98,69],{"className":99},[92]," away from the physical \"sea level\" surface (the geoid) ",[33,102,103],{},"16, 17",". Understanding this relationship is critical for accurate heighting and distance measurement ",[33,106,107],{},"13",[11,109,111],{"id":110},"background","Background",[16,113,114,115,117,118,121,122,151,152,155,156,159,160,36],{},"The ",[20,116,22],{}," system is a geocentric datum, meaning its origin is at the Earth's centre of mass ",[33,119,120],{},"15, 18",". It was established using observations from roughly ",[33,123,125,139],{"className":124},[47],[33,126,128],{"className":127},[51],[53,129,130],{"xmlns":55},[57,131,132,137],{},[60,133,134],{},[63,135,136],{},"1600",[71,138,136],{"encoding":73},[33,140,142],{"className":141,"ariaHidden":79},[78],[33,143,145,148],{"className":144},[83],[33,146],{"className":147,"style":88},[87],[33,149,136],{"className":150},[92]," sites globally and is the standard for all satellite positioning ",[33,153,154],{},"18",". For all practical engineering purposes, it is identical to the ",[20,157,158],{},"GRS80"," ellipsoid ",[33,161,162],{},"19",[11,164,166],{"id":165},"theory","Theory",[168,169,170,180,285],"ol",{},[171,172,173,176,177,36],"li",{},[20,174,175],{},"Reference Ellipsoid:"," Since the physical Earth is too irregular to define mathematically, we use a smooth, symmetrical ellipsoid that best fits the Earth's shape ",[33,178,179],{},"14",[171,181,182,185,186,219,220,251,252,281,282,36],{},[20,183,184],{},"Geodetic Coordinates:"," Position is defined by Latitude (",[33,187,189,205],{"className":188},[47],[33,190,192],{"className":191},[51],[53,193,194],{"xmlns":55},[57,195,196,202],{},[60,197,198],{},[199,200,201],"mi",{},"ϕ",[71,203,204],{"encoding":73},"\\phi",[33,206,208],{"className":207,"ariaHidden":79},[78],[33,209,211,215],{"className":210},[83],[33,212],{"className":213,"style":214},[87],"height:0.8889em;vertical-align:-0.1944em;",[33,216,201],{"className":217},[92,218],"mathnormal","), Longitude (",[33,221,223,238],{"className":222},[47],[33,224,226],{"className":225},[51],[53,227,228],{"xmlns":55},[57,229,230,235],{},[60,231,232],{},[199,233,234],{},"λ",[71,236,237],{"encoding":73},"\\lambda",[33,239,241],{"className":240,"ariaHidden":79},[78],[33,242,244,248],{"className":243},[83],[33,245],{"className":246,"style":247},[87],"height:0.6944em;",[33,249,234],{"className":250},[92,218],"), and height above the ellipsoid (",[33,253,255,269],{"className":254},[47],[33,256,258],{"className":257},[51],[53,259,260],{"xmlns":55},[57,261,262,267],{},[60,263,264],{},[199,265,266],{},"h",[71,268,266],{"encoding":73},[33,270,272],{"className":271,"ariaHidden":79},[78],[33,273,275,278],{"className":274},[83],[33,276],{"className":277,"style":247},[87],[33,279,266],{"className":280},[92,218],") ",[33,283,284],{},"20",[171,286,287,290,291,294,295,36],{},[20,288,289],{},"Projections:"," To create a map or site plan, we mathematically project these coordinates onto a flat surface (like a cylinder in the ",[20,292,293],{},"Transverse Mercator"," projection) ",[33,296,297],{},"21",[11,299,301],{"id":300},"mathematical-principles","Mathematical Principles",[303,304,306],"h3",{"id":305},"_1-wgs84-ellipsoid-parameters","1. WGS84 Ellipsoid Parameters",[16,308,309],{},"The size and shape of the WGS84\u002FGRS80 ellipsoid are defined by:",[311,312,313,425],"ul",{},[171,314,315,349,350,349,422],{},[20,316,317,318,348],{},"Semi-major axis (",[33,319,321,335],{"className":320},[47],[33,322,324],{"className":323},[51],[53,325,326],{"xmlns":55},[57,327,328,333],{},[60,329,330],{},[199,331,332],{},"a",[71,334,332],{"encoding":73},[33,336,338],{"className":337,"ariaHidden":79},[78],[33,339,341,345],{"className":340},[83],[33,342],{"className":343,"style":344},[87],"height:0.4306em;",[33,346,332],{"className":347},[92,218],"):"," ",[33,351,353,382],{"className":352},[47],[33,354,356],{"className":355},[51],[53,357,358],{"xmlns":55},[57,359,360,379],{},[60,361,362,365,369,372,374,377],{},[63,363,364],{},"6",[366,367,368],"mo",{"separator":79},",",[63,370,371],{},"378",[366,373,368],{"separator":79},[63,375,376],{},"137.0",[67,378,69],{},[71,380,381],{"encoding":73},"6,378,137.0 \\text{ m}",[33,383,385],{"className":384,"ariaHidden":79},[78],[33,386,388,392,395,399,404,407,410,413,416],{"className":387},[83],[33,389],{"className":390,"style":391},[87],"height:0.8389em;vertical-align:-0.1944em;",[33,393,364],{"className":394},[92],[33,396,368],{"className":397},[398],"mpunct",[33,400],{"className":401,"style":403},[402],"mspace","margin-right:0.1667em;",[33,405,371],{"className":406},[92],[33,408,368],{"className":409},[398],[33,411],{"className":412,"style":403},[402],[33,414,376],{"className":415},[92],[33,417,419],{"className":418},[92,96],[33,420,69],{"className":421},[92],[33,423,424],{},"19, 22",[171,426,427,349,473,349,502],{},[20,428,429,430,348],{},"Inverse Flattening (",[33,431,433,455],{"className":432},[47],[33,434,436],{"className":435},[51],[53,437,438],{"xmlns":55},[57,439,440,452],{},[60,441,442,445,449],{},[63,443,444],{},"1",[199,446,448],{"mathvariant":447},"normal","\u002F",[199,450,451],{},"f",[71,453,454],{"encoding":73},"1\u002Ff",[33,456,458],{"className":457,"ariaHidden":79},[78],[33,459,461,465,469],{"className":460},[83],[33,462],{"className":463,"style":464},[87],"height:1em;vertical-align:-0.25em;",[33,466,468],{"className":467},[92],"1\u002F",[33,470,451],{"className":471,"style":472},[92,218],"margin-right:0.1076em;",[33,474,476,490],{"className":475},[47],[33,477,479],{"className":478},[51],[53,480,481],{"xmlns":55},[57,482,483,488],{},[60,484,485],{},[63,486,487],{},"298.257223563",[71,489,487],{"encoding":73},[33,491,493],{"className":492,"ariaHidden":79},[78],[33,494,496,499],{"className":495},[83],[33,497],{"className":498,"style":88},[87],[33,500,487],{"className":501},[92],[33,503,424],{},[303,505,507,508,584,585],{"id":506},"_2-cartesian-xyzx-y-zxyz-to-geodetic-ϕλhphi-lambda-hϕλh","2. Cartesian ",[33,509,511,543],{"className":510},[47],[33,512,514],{"className":513},[51],[53,515,516],{"xmlns":55},[57,517,518,540],{},[60,519,520,524,527,529,532,534,537],{},[366,521,523],{"stretchy":522},"false","(",[199,525,526],{},"X",[366,528,368],{"separator":79},[199,530,531],{},"Y",[366,533,368],{"separator":79},[199,535,536],{},"Z",[366,538,539],{"stretchy":522},")",[71,541,542],{"encoding":73},"(X, Y, Z)",[33,544,546],{"className":545,"ariaHidden":79},[78],[33,547,549,552,556,560,563,566,570,573,576,580],{"className":548},[83],[33,550],{"className":551,"style":464},[87],[33,553,523],{"className":554},[555],"mopen",[33,557,526],{"className":558,"style":559},[92,218],"margin-right:0.0785em;",[33,561,368],{"className":562},[398],[33,564],{"className":565,"style":403},[402],[33,567,531],{"className":568,"style":569},[92,218],"margin-right:0.2222em;",[33,571,368],{"className":572},[398],[33,574],{"className":575,"style":403},[402],[33,577,536],{"className":578,"style":579},[92,218],"margin-right:0.0715em;",[33,581,539],{"className":582},[583],"mclose"," to Geodetic ",[33,586,588,614],{"className":587},[47],[33,589,591],{"className":590},[51],[53,592,593],{"xmlns":55},[57,594,595,611],{},[60,596,597,599,601,603,605,607,609],{},[366,598,523],{"stretchy":522},[199,600,201],{},[366,602,368],{"separator":79},[199,604,234],{},[366,606,368],{"separator":79},[199,608,266],{},[366,610,539],{"stretchy":522},[71,612,613],{"encoding":73},"(\\phi, \\lambda, h)",[33,615,617],{"className":616,"ariaHidden":79},[78],[33,618,620,623,626,629,632,635,638,641,644,647],{"className":619},[83],[33,621],{"className":622,"style":464},[87],[33,624,523],{"className":625},[555],[33,627,201],{"className":628},[92,218],[33,630,368],{"className":631},[398],[33,633],{"className":634,"style":403},[402],[33,636,234],{"className":637},[92,218],[33,639,368],{"className":640},[398],[33,642],{"className":643,"style":403},[402],[33,645,266],{"className":646},[92,218],[33,648,539],{"className":649},[583],[16,651,652,653,349,811,814,349,1168,1170,349,1366,1368,349,1465,36],{},"To convert satellite Cartesian coordinates to mapping coordinates:\n",[33,654,656,686],{"className":655},[47],[33,657,659],{"className":658},[51],[53,660,661],{"xmlns":55},[57,662,663,683],{},[60,664,665,668,671,673,676],{},[199,666,667],{},"tan",[366,669,670],{},"⁡",[199,672,234],{},[366,674,675],{},"=",[677,678,679,681],"mfrac",{},[199,680,531],{},[199,682,526],{},[71,684,685],{"encoding":73},"\\tan \\lambda = \\frac{Y}{X}",[33,687,689,716],{"className":688,"ariaHidden":79},[78],[33,690,692,695,699,702,705,709,713],{"className":691},[83],[33,693],{"className":694,"style":247},[87],[33,696,667],{"className":697},[698],"mop",[33,700],{"className":701,"style":403},[402],[33,703,234],{"className":704},[92,218],[33,706],{"className":707,"style":708},[402],"margin-right:0.2778em;",[33,710,675],{"className":711},[712],"mrel",[33,714],{"className":715,"style":708},[402],[33,717,719,723],{"className":718},[83],[33,720],{"className":721,"style":722},[87],"height:1.2173em;vertical-align:-0.345em;",[33,724,726,730,808],{"className":725},[92],[33,727],{"className":728},[555,729],"nulldelimiter",[33,731,733],{"className":732},[677],[33,734,738,799],{"className":735},[736,737],"vlist-t","vlist-t2",[33,739,742,794],{"className":740},[741],"vlist-r",[33,743,747,768,779],{"className":744,"style":746},[745],"vlist","height:0.8723em;",[33,748,750,755],{"style":749},"top:-2.655em;",[33,751],{"className":752,"style":754},[753],"pstrut","height:3em;",[33,756,762],{"className":757},[758,759,760,761],"sizing","reset-size6","size3","mtight",[33,763,765],{"className":764},[92,761],[33,766,526],{"className":767,"style":559},[92,218,761],[33,769,771,774],{"style":770},"top:-3.23em;",[33,772],{"className":773,"style":754},[753],[33,775],{"className":776,"style":778},[777],"frac-line","border-bottom-width:0.04em;",[33,780,782,785],{"style":781},"top:-3.394em;",[33,783],{"className":784,"style":754},[753],[33,786,788],{"className":787},[758,759,760,761],[33,789,791],{"className":790},[92,761],[33,792,531],{"className":793,"style":569},[92,218,761],[33,795,798],{"className":796},[797],"vlist-s","​",[33,800,802],{"className":801},[741],[33,803,806],{"className":804,"style":805},[745],"height:0.345em;",[33,807],{},[33,809],{"className":810},[583,729],[33,812,813],{},"23",[33,815,817,895],{"className":816},[47],[33,818,820],{"className":819},[51],[53,821,822],{"xmlns":55},[57,823,824,892],{},[60,825,826,828,830,832,834],{},[199,827,667],{},[366,829,670],{},[199,831,201],{},[366,833,675],{},[677,835,836,862],{},[60,837,838,840,843,852,855,858,860],{},[199,839,536],{},[366,841,842],{},"+",[844,845,846,849],"msup",{},[199,847,848],{},"e",[63,850,851],{},"2",[199,853,854],{},"ν",[199,856,857],{},"sin",[366,859,670],{},[199,861,201],{},[60,863,864,866,872,874,880],{},[366,865,523],{"stretchy":522},[844,867,868,870],{},[199,869,526],{},[63,871,851],{},[366,873,842],{},[844,875,876,878],{},[199,877,531],{},[63,879,851],{},[844,881,882,884],{},[366,883,539],{"stretchy":522},[60,885,886,888,890],{},[63,887,444],{},[199,889,448],{"mathvariant":447},[63,891,851],{},[71,893,894],{"encoding":73},"\\tan \\phi = \\frac{Z + e^2 \\nu \\sin \\phi}{(X^2 + Y^2)^{1\u002F2}}",[33,896,898,922],{"className":897,"ariaHidden":79},[78],[33,899,901,904,907,910,913,916,919],{"className":900},[83],[33,902],{"className":903,"style":214},[87],[33,905,667],{"className":906},[698],[33,908],{"className":909,"style":403},[402],[33,911,201],{"className":912},[92,218],[33,914],{"className":915,"style":708},[402],[33,917,675],{"className":918},[712],[33,920],{"className":921,"style":708},[402],[33,923,925,929],{"className":924},[83],[33,926],{"className":927,"style":928},[87],"height:1.6305em;vertical-align:-0.5604em;",[33,930,932,935,1165],{"className":931},[92],[33,933],{"className":934},[555,729],[33,936,938],{"className":937},[677],[33,939,941,1156],{"className":940},[736,737],[33,942,944,1153],{"className":943},[741],[33,945,948,1067,1075],{"className":946,"style":947},[745],"height:1.07em;",[33,949,951,954],{"style":950},"top:-2.6146em;",[33,952],{"className":953,"style":754},[753],[33,955,957],{"className":956},[758,759,760,761],[33,958,960,963,998,1002,1031],{"className":959},[92,761],[33,961,523],{"className":962},[555,761],[33,964,966,969],{"className":965},[92,761],[33,967,526],{"className":968,"style":559},[92,218,761],[33,970,973],{"className":971},[972],"msupsub",[33,974,976],{"className":975},[736],[33,977,979],{"className":978},[741],[33,980,983],{"className":981,"style":982},[745],"height:0.7463em;",[33,984,986,990],{"style":985},"top:-2.786em;margin-right:0.0714em;",[33,987],{"className":988,"style":989},[753],"height:2.5em;",[33,991,995],{"className":992},[758,993,994,761],"reset-size3","size1",[33,996,851],{"className":997},[92,761],[33,999,842],{"className":1000},[1001,761],"mbin",[33,1003,1005,1008],{"className":1004},[92,761],[33,1006,531],{"className":1007,"style":569},[92,218,761],[33,1009,1011],{"className":1010},[972],[33,1012,1014],{"className":1013},[736],[33,1015,1017],{"className":1016},[741],[33,1018,1020],{"className":1019,"style":982},[745],[33,1021,1022,1025],{"style":985},[33,1023],{"className":1024,"style":989},[753],[33,1026,1028],{"className":1027},[758,993,994,761],[33,1029,851],{"className":1030},[92,761],[33,1032,1034,1037],{"className":1033},[583,761],[33,1035,539],{"className":1036},[583,761],[33,1038,1040],{"className":1039},[972],[33,1041,1043],{"className":1042},[736],[33,1044,1046],{"className":1045},[741],[33,1047,1050],{"className":1048,"style":1049},[745],"height:0.822em;",[33,1051,1053,1057],{"style":1052},"top:-2.822em;margin-right:0.0714em;",[33,1054],{"className":1055,"style":1056},[753],"height:2.5357em;",[33,1058,1060],{"className":1059},[758,993,994,761],[33,1061,1063],{"className":1062},[92,761],[33,1064,1066],{"className":1065},[92,761],"1\u002F2",[33,1068,1069,1072],{"style":770},[33,1070],{"className":1071,"style":754},[753],[33,1073],{"className":1074,"style":778},[777],[33,1076,1078,1081],{"style":1077},"top:-3.4461em;",[33,1079],{"className":1080,"style":754},[753],[33,1082,1084],{"className":1083},[758,759,760,761],[33,1085,1087,1090,1093,1124,1128,1132,1147,1150],{"className":1086},[92,761],[33,1088,536],{"className":1089,"style":579},[92,218,761],[33,1091,842],{"className":1092},[1001,761],[33,1094,1096,1099],{"className":1095},[92,761],[33,1097,848],{"className":1098},[92,218,761],[33,1100,1102],{"className":1101},[972],[33,1103,1105],{"className":1104},[736],[33,1106,1108],{"className":1107},[741],[33,1109,1112],{"className":1110,"style":1111},[745],"height:0.8913em;",[33,1113,1115,1118],{"style":1114},"top:-2.931em;margin-right:0.0714em;",[33,1116],{"className":1117,"style":989},[753],[33,1119,1121],{"className":1120},[758,993,994,761],[33,1122,851],{"className":1123},[92,761],[33,1125,854],{"className":1126,"style":1127},[92,218,761],"margin-right:0.0637em;",[33,1129],{"className":1130,"style":1131},[402,761],"margin-right:0.1952em;",[33,1133,1135,1139,1143],{"className":1134},[698,761],[33,1136,1138],{"className":1137},[761],"s",[33,1140,1142],{"className":1141},[761],"i",[33,1144,1146],{"className":1145},[761],"n",[33,1148],{"className":1149,"style":1131},[402,761],[33,1151,201],{"className":1152},[92,218,761],[33,1154,798],{"className":1155},[797],[33,1157,1159],{"className":1158},[741],[33,1160,1163],{"className":1161,"style":1162},[745],"height:0.5604em;",[33,1164],{},[33,1166],{"className":1167},[583,729],[33,1169,813],{},[33,1171,1173,1213],{"className":1172},[47],[33,1174,1176],{"className":1175},[51],[53,1177,1178],{"xmlns":55},[57,1179,1180,1210],{},[60,1181,1182,1184,1186,1205,1208],{},[199,1183,266],{},[366,1185,675],{},[677,1187,1188,1190],{},[199,1189,526],{},[60,1191,1192,1195,1197,1199,1201,1203],{},[199,1193,1194],{},"cos",[366,1196,670],{},[199,1198,201],{},[199,1200,1194],{},[366,1202,670],{},[199,1204,234],{},[366,1206,1207],{},"−",[199,1209,854],{},[71,1211,1212],{"encoding":73},"h = \\frac{X}{\\cos \\phi \\cos \\lambda} - \\nu",[33,1214,1216,1234,1357],{"className":1215,"ariaHidden":79},[78],[33,1217,1219,1222,1225,1228,1231],{"className":1218},[83],[33,1220],{"className":1221,"style":247},[87],[33,1223,266],{"className":1224},[92,218],[33,1226],{"className":1227,"style":708},[402],[33,1229,675],{"className":1230},[712],[33,1232],{"className":1233,"style":708},[402],[33,1235,1237,1241,1348,1351,1354],{"className":1236},[83],[33,1238],{"className":1239,"style":1240},[87],"height:1.3534em;vertical-align:-0.4811em;",[33,1242,1244,1247,1345],{"className":1243},[92],[33,1245],{"className":1246},[555,729],[33,1248,1250],{"className":1249},[677],[33,1251,1253,1336],{"className":1252},[736,737],[33,1254,1256,1333],{"className":1255},[741],[33,1257,1259,1311,1319],{"className":1258,"style":746},[745],[33,1260,1261,1264],{"style":749},[33,1262],{"className":1263,"style":754},[753],[33,1265,1267],{"className":1266},[758,759,760,761],[33,1268,1270,1284,1287,1290,1293,1305,1308],{"className":1269},[92,761],[33,1271,1273,1277,1281],{"className":1272},[698,761],[33,1274,1276],{"className":1275},[761],"c",[33,1278,1280],{"className":1279},[761],"o",[33,1282,1138],{"className":1283},[761],[33,1285],{"className":1286,"style":1131},[402,761],[33,1288,201],{"className":1289},[92,218,761],[33,1291],{"className":1292,"style":1131},[402,761],[33,1294,1296,1299,1302],{"className":1295},[698,761],[33,1297,1276],{"className":1298},[761],[33,1300,1280],{"className":1301},[761],[33,1303,1138],{"className":1304},[761],[33,1306],{"className":1307,"style":1131},[402,761],[33,1309,234],{"className":1310},[92,218,761],[33,1312,1313,1316],{"style":770},[33,1314],{"className":1315,"style":754},[753],[33,1317],{"className":1318,"style":778},[777],[33,1320,1321,1324],{"style":781},[33,1322],{"className":1323,"style":754},[753],[33,1325,1327],{"className":1326},[758,759,760,761],[33,1328,1330],{"className":1329},[92,761],[33,1331,526],{"className":1332,"style":559},[92,218,761],[33,1334,798],{"className":1335},[797],[33,1337,1339],{"className":1338},[741],[33,1340,1343],{"className":1341,"style":1342},[745],"height:0.4811em;",[33,1344],{},[33,1346],{"className":1347},[583,729],[33,1349],{"className":1350,"style":569},[402],[33,1352,1207],{"className":1353},[1001],[33,1355],{"className":1356,"style":569},[402],[33,1358,1360,1363],{"className":1359},[83],[33,1361],{"className":1362,"style":344},[87],[33,1364,854],{"className":1365,"style":1127},[92,218],[33,1367,813],{},[1369,1370,1371,1372,1401,1402,1464],"em",{},"(Note: ",[33,1373,1375,1389],{"className":1374},[47],[33,1376,1378],{"className":1377},[51],[53,1379,1380],{"xmlns":55},[57,1381,1382,1386],{},[60,1383,1384],{},[199,1385,854],{},[71,1387,1388],{"encoding":73},"\\nu",[33,1390,1392],{"className":1391,"ariaHidden":79},[78],[33,1393,1395,1398],{"className":1394},[83],[33,1396],{"className":1397,"style":344},[87],[33,1399,854],{"className":1400,"style":1127},[92,218]," is the radius of curvature in the prime vertical, and ",[33,1403,1405,1423],{"className":1404},[47],[33,1406,1408],{"className":1407},[51],[53,1409,1410],{"xmlns":55},[57,1411,1412,1420],{},[60,1413,1414],{},[844,1415,1416,1418],{},[199,1417,848],{},[63,1419,851],{},[71,1421,1422],{"encoding":73},"e^2",[33,1424,1426],{"className":1425,"ariaHidden":79},[78],[33,1427,1429,1433],{"className":1428},[83],[33,1430],{"className":1431,"style":1432},[87],"height:0.8141em;",[33,1434,1436,1439],{"className":1435},[92],[33,1437,848],{"className":1438},[92,218],[33,1440,1442],{"className":1441},[972],[33,1443,1445],{"className":1444},[736],[33,1446,1448],{"className":1447},[741],[33,1449,1451],{"className":1450,"style":1432},[745],[33,1452,1454,1458],{"style":1453},"top:-3.063em;margin-right:0.05em;",[33,1455],{"className":1456,"style":1457},[753],"height:2.7em;",[33,1459,1461],{"className":1460},[758,759,760,761],[33,1462,851],{"className":1463},[92,761]," is the eccentricity squared)",[33,1466,813],{},[11,1468,1470],{"id":1469},"field-workflow-wgs84-to-site-grid","Field Workflow: WGS84 to Site Grid",[1472,1473,1474,1478,1484,1488,1498,1502,1599,1603],"steps",{},[303,1475,1477],{"id":1476},"capture-data","Capture Data",[16,1479,1480,1481,36],{},"Observe your control points using high-precision GNSS in the WGS84 system ",[33,1482,1483],{},"24",[303,1485,1487],{"id":1486},"perform-transformation","Perform Transformation",[16,1489,1490,1491,1494,1495,36],{},"Apply a ",[20,1492,1493],{},"Helmert 7-Parameter Transformation"," to shift the origin, rotate the axes, and adjust the scale to match the local datum (e.g., OSGB36) ",[33,1496,1497],{},"25, 26",[303,1499,1501],{"id":1500},"project-to-plane","Project to Plane",[16,1503,1504,1505,1547,1548,1595,1596,36],{},"Convert the transformed ellipsoidal coordinates (",[33,1506,1508,1526],{"className":1507},[47],[33,1509,1511],{"className":1510},[51],[53,1512,1513],{"xmlns":55},[57,1514,1515,1523],{},[60,1516,1517,1519,1521],{},[199,1518,201],{},[366,1520,368],{"separator":79},[199,1522,234],{},[71,1524,1525],{"encoding":73},"\\phi, \\lambda",[33,1527,1529],{"className":1528,"ariaHidden":79},[78],[33,1530,1532,1535,1538,1541,1544],{"className":1531},[83],[33,1533],{"className":1534,"style":214},[87],[33,1536,201],{"className":1537},[92,218],[33,1539,368],{"className":1540},[398],[33,1542],{"className":1543,"style":403},[402],[33,1545,234],{"className":1546},[92,218],") into Northings and Eastings (",[33,1549,1551,1571],{"className":1550},[47],[33,1552,1554],{"className":1553},[51],[53,1555,1556],{"xmlns":55},[57,1557,1558,1568],{},[60,1559,1560,1563,1565],{},[199,1561,1562],{},"E",[366,1564,368],{"separator":79},[199,1566,1567],{},"N",[71,1569,1570],{"encoding":73},"E, N",[33,1572,1574],{"className":1573,"ariaHidden":79},[78],[33,1575,1577,1581,1585,1588,1591],{"className":1576},[83],[33,1578],{"className":1579,"style":1580},[87],"height:0.8778em;vertical-align:-0.1944em;",[33,1582,1562],{"className":1583,"style":1584},[92,218],"margin-right:0.0576em;",[33,1586,368],{"className":1587},[398],[33,1589],{"className":1590,"style":403},[402],[33,1592,1567],{"className":1593,"style":1594},[92,218],"margin-right:0.109em;",") using a projection like the Universal Transverse Mercator (UTM) ",[33,1597,1598],{},"21, 27",[303,1600,1602],{"id":1601},"apply-local-scale-factor","Apply Local Scale Factor",[16,1604,1605,1606,1640,1641,36],{},"Because projections distort distances, multiply all \"grid\" distances by the ",[20,1607,1608,1609,539],{},"Local Scale Factor (",[33,1610,1612,1626],{"className":1611},[47],[33,1613,1615],{"className":1614},[51],[53,1616,1617],{"xmlns":55},[57,1618,1619,1624],{},[60,1620,1621],{},[199,1622,1623],{},"F",[71,1625,1623],{"encoding":73},[33,1627,1629],{"className":1628,"ariaHidden":79},[78],[33,1630,1632,1636],{"className":1631},[83],[33,1633],{"className":1634,"style":1635},[87],"height:0.6833em;",[33,1637,1623],{"className":1638,"style":1639},[92,218],"margin-right:0.1389em;"," to get the actual ground distance for setting out ",[33,1642,1643],{},"28, 29",[11,1645,1647],{"id":1646},"step-by-step-example-ellipsoid-elements","Step-by-Step Example: Ellipsoid Elements",[16,1649,1650,1653,1654,1682,1683,1711,1712,1741,1742,1800],{},[20,1651,1652],{},"Problem:"," Given the semi-major axis (",[33,1655,1657,1670],{"className":1656},[47],[33,1658,1660],{"className":1659},[51],[53,1661,1662],{"xmlns":55},[57,1663,1664,1668],{},[60,1665,1666],{},[199,1667,332],{},[71,1669,332],{"encoding":73},[33,1671,1673],{"className":1672,"ariaHidden":79},[78],[33,1674,1676,1679],{"className":1675},[83],[33,1677],{"className":1678,"style":344},[87],[33,1680,332],{"className":1681},[92,218],") and flattening (",[33,1684,1686,1699],{"className":1685},[47],[33,1687,1689],{"className":1688},[51],[53,1690,1691],{"xmlns":55},[57,1692,1693,1697],{},[60,1694,1695],{},[199,1696,451],{},[71,1698,451],{"encoding":73},[33,1700,1702],{"className":1701,"ariaHidden":79},[78],[33,1703,1705,1708],{"className":1704},[83],[33,1706],{"className":1707,"style":214},[87],[33,1709,451],{"className":1710,"style":472},[92,218],"), find the semi-minor axis (",[33,1713,1715,1729],{"className":1714},[47],[33,1716,1718],{"className":1717},[51],[53,1719,1720],{"xmlns":55},[57,1721,1722,1727],{},[60,1723,1724],{},[199,1725,1726],{},"b",[71,1728,1726],{"encoding":73},[33,1730,1732],{"className":1731,"ariaHidden":79},[78],[33,1733,1735,1738],{"className":1734},[83],[33,1736],{"className":1737,"style":247},[87],[33,1739,1726],{"className":1740},[92,218],") and eccentricity squared (",[33,1743,1745,1762],{"className":1744},[47],[33,1746,1748],{"className":1747},[51],[53,1749,1750],{"xmlns":55},[57,1751,1752,1760],{},[60,1753,1754],{},[844,1755,1756,1758],{},[199,1757,848],{},[63,1759,851],{},[71,1761,1422],{"encoding":73},[33,1763,1765],{"className":1764,"ariaHidden":79},[78],[33,1766,1768,1771],{"className":1767},[83],[33,1769],{"className":1770,"style":1432},[87],[33,1772,1774,1777],{"className":1773},[92],[33,1775,848],{"className":1776},[92,218],[33,1778,1780],{"className":1779},[972],[33,1781,1783],{"className":1782},[736],[33,1784,1786],{"className":1785},[741],[33,1787,1789],{"className":1788,"style":1432},[745],[33,1790,1791,1794],{"style":1453},[33,1792],{"className":1793,"style":1457},[753],[33,1795,1797],{"className":1796},[758,759,760,761],[33,1798,851],{"className":1799},[92,761],").",[168,1802,1803,2066],{},[171,1804,1805,1836,349,2063],{},[20,1806,1807,1808,348],{},"Semi-minor axis (",[33,1809,1811,1824],{"className":1810},[47],[33,1812,1814],{"className":1813},[51],[53,1815,1816],{"xmlns":55},[57,1817,1818,1822],{},[60,1819,1820],{},[199,1821,1726],{},[71,1823,1726],{"encoding":73},[33,1825,1827],{"className":1826,"ariaHidden":79},[78],[33,1828,1830,1833],{"className":1829},[83],[33,1831],{"className":1832,"style":247},[87],[33,1834,1726],{"className":1835},[92,218],[33,1837,1839,1891],{"className":1838},[47],[33,1840,1842],{"className":1841},[51],[53,1843,1844],{"xmlns":55},[57,1845,1846,1888],{},[60,1847,1848,1850,1852,1864,1867,1870,1872,1874,1876,1878,1880,1882,1884,1886],{},[199,1849,451],{},[366,1851,675],{},[677,1853,1854,1862],{},[60,1855,1856,1858,1860],{},[199,1857,332],{},[366,1859,1207],{},[199,1861,1726],{},[199,1863,332],{},[67,1865,1866],{},"  ",[366,1868,1869],{},"⟹",[67,1871,1866],{},[199,1873,1726],{},[366,1875,675],{},[199,1877,332],{},[366,1879,523],{"stretchy":522},[63,1881,444],{},[366,1883,1207],{},[199,1885,451],{},[366,1887,539],{"stretchy":522},[71,1889,1890],{"encoding":73},"f = \\frac{a - b}{a} \\implies b = a(1 - f)",[33,1892,1894,1912,2009,2027,2051],{"className":1893,"ariaHidden":79},[78],[33,1895,1897,1900,1903,1906,1909],{"className":1896},[83],[33,1898],{"className":1899,"style":214},[87],[33,1901,451],{"className":1902,"style":472},[92,218],[33,1904],{"className":1905,"style":708},[402],[33,1907,675],{"className":1908},[712],[33,1910],{"className":1911,"style":708},[402],[33,1913,1915,1919,1994,1997,2000,2003,2006],{"className":1914},[83],[33,1916],{"className":1917,"style":1918},[87],"height:1.2251em;vertical-align:-0.345em;",[33,1920,1922,1925,1991],{"className":1921},[92],[33,1923],{"className":1924},[555,729],[33,1926,1928],{"className":1927},[677],[33,1929,1931,1983],{"className":1930},[736,737],[33,1932,1934,1980],{"className":1933},[741],[33,1935,1938,1952,1960],{"className":1936,"style":1937},[745],"height:0.8801em;",[33,1939,1940,1943],{"style":749},[33,1941],{"className":1942,"style":754},[753],[33,1944,1946],{"className":1945},[758,759,760,761],[33,1947,1949],{"className":1948},[92,761],[33,1950,332],{"className":1951},[92,218,761],[33,1953,1954,1957],{"style":770},[33,1955],{"className":1956,"style":754},[753],[33,1958],{"className":1959,"style":778},[777],[33,1961,1962,1965],{"style":781},[33,1963],{"className":1964,"style":754},[753],[33,1966,1968],{"className":1967},[758,759,760,761],[33,1969,1971,1974,1977],{"className":1970},[92,761],[33,1972,332],{"className":1973},[92,218,761],[33,1975,1207],{"className":1976},[1001,761],[33,1978,1726],{"className":1979},[92,218,761],[33,1981,798],{"className":1982},[797],[33,1984,1986],{"className":1985},[741],[33,1987,1989],{"className":1988,"style":805},[745],[33,1990],{},[33,1992],{"className":1993},[583,729],[33,1995],{"className":1996,"style":708},[402],[33,1998],{"className":1999,"style":708},[402],[33,2001,1869],{"className":2002},[712],[33,2004],{"className":2005,"style":708},[402],[33,2007],{"className":2008,"style":708},[402],[33,2010,2012,2015,2018,2021,2024],{"className":2011},[83],[33,2013],{"className":2014,"style":247},[87],[33,2016,1726],{"className":2017},[92,218],[33,2019],{"className":2020,"style":708},[402],[33,2022,675],{"className":2023},[712],[33,2025],{"className":2026,"style":708},[402],[33,2028,2030,2033,2036,2039,2042,2045,2048],{"className":2029},[83],[33,2031],{"className":2032,"style":464},[87],[33,2034,332],{"className":2035},[92,218],[33,2037,523],{"className":2038},[555],[33,2040,444],{"className":2041},[92],[33,2043],{"className":2044,"style":569},[402],[33,2046,1207],{"className":2047},[1001],[33,2049],{"className":2050,"style":569},[402],[33,2052,2054,2057,2060],{"className":2053},[83],[33,2055],{"className":2056,"style":464},[87],[33,2058,451],{"className":2059,"style":472},[92,218],[33,2061,539],{"className":2062},[583],[33,2064,2065],{},"16",[171,2067,2068,2129,349,2383],{},[20,2069,2070,2071,348],{},"Eccentricity squared (",[33,2072,2074,2091],{"className":2073},[47],[33,2075,2077],{"className":2076},[51],[53,2078,2079],{"xmlns":55},[57,2080,2081,2089],{},[60,2082,2083],{},[844,2084,2085,2087],{},[199,2086,848],{},[63,2088,851],{},[71,2090,1422],{"encoding":73},[33,2092,2094],{"className":2093,"ariaHidden":79},[78],[33,2095,2097,2100],{"className":2096},[83],[33,2098],{"className":2099,"style":1432},[87],[33,2101,2103,2106],{"className":2102},[92],[33,2104,848],{"className":2105},[92,218],[33,2107,2109],{"className":2108},[972],[33,2110,2112],{"className":2111},[736],[33,2113,2115],{"className":2114},[741],[33,2116,2118],{"className":2117,"style":1432},[745],[33,2119,2120,2123],{"style":1453},[33,2121],{"className":2122,"style":1457},[753],[33,2124,2126],{"className":2125},[758,759,760,761],[33,2127,851],{"className":2128},[92,761],[33,2130,2132,2176],{"className":2131},[47],[33,2133,2135],{"className":2134},[51],[53,2136,2137],{"xmlns":55},[57,2138,2139,2173],{},[60,2140,2141,2147,2149],{},[844,2142,2143,2145],{},[199,2144,848],{},[63,2146,851],{},[366,2148,675],{},[677,2150,2151,2167],{},[60,2152,2153,2159,2161],{},[844,2154,2155,2157],{},[199,2156,332],{},[63,2158,851],{},[366,2160,1207],{},[844,2162,2163,2165],{},[199,2164,1726],{},[63,2166,851],{},[844,2168,2169,2171],{},[199,2170,332],{},[63,2172,851],{},[71,2174,2175],{"encoding":73},"e^2 = \\frac{a^2 - b^2}{a^2}",[33,2177,2179,2223],{"className":2178,"ariaHidden":79},[78],[33,2180,2182,2185,2214,2217,2220],{"className":2181},[83],[33,2183],{"className":2184,"style":1432},[87],[33,2186,2188,2191],{"className":2187},[92],[33,2189,848],{"className":2190},[92,218],[33,2192,2194],{"className":2193},[972],[33,2195,2197],{"className":2196},[736],[33,2198,2200],{"className":2199},[741],[33,2201,2203],{"className":2202,"style":1432},[745],[33,2204,2205,2208],{"style":1453},[33,2206],{"className":2207,"style":1457},[753],[33,2209,2211],{"className":2210},[758,759,760,761],[33,2212,851],{"className":2213},[92,761],[33,2215],{"className":2216,"style":708},[402],[33,2218,675],{"className":2219},[712],[33,2221],{"className":2222,"style":708},[402],[33,2224,2226,2230],{"className":2225},[83],[33,2227],{"className":2228,"style":2229},[87],"height:1.3629em;vertical-align:-0.345em;",[33,2231,2233,2236,2380],{"className":2232},[92],[33,2234],{"className":2235},[555,729],[33,2237,2239],{"className":2238},[677],[33,2240,2242,2372],{"className":2241},[736,737],[33,2243,2245,2369],{"className":2244},[741],[33,2246,2249,2289,2297],{"className":2247,"style":2248},[745],"height:1.0179em;",[33,2250,2251,2254],{"style":749},[33,2252],{"className":2253,"style":754},[753],[33,2255,2257],{"className":2256},[758,759,760,761],[33,2258,2260],{"className":2259},[92,761],[33,2261,2263,2266],{"className":2262},[92,761],[33,2264,332],{"className":2265},[92,218,761],[33,2267,2269],{"className":2268},[972],[33,2270,2272],{"className":2271},[736],[33,2273,2275],{"className":2274},[741],[33,2276,2278],{"className":2277,"style":982},[745],[33,2279,2280,2283],{"style":985},[33,2281],{"className":2282,"style":989},[753],[33,2284,2286],{"className":2285},[758,993,994,761],[33,2287,851],{"className":2288},[92,761],[33,2290,2291,2294],{"style":770},[33,2292],{"className":2293,"style":754},[753],[33,2295],{"className":2296,"style":778},[777],[33,2298,2299,2302],{"style":781},[33,2300],{"className":2301,"style":754},[753],[33,2303,2305],{"className":2304},[758,759,760,761],[33,2306,2308,2337,2340],{"className":2307},[92,761],[33,2309,2311,2314],{"className":2310},[92,761],[33,2312,332],{"className":2313},[92,218,761],[33,2315,2317],{"className":2316},[972],[33,2318,2320],{"className":2319},[736],[33,2321,2323],{"className":2322},[741],[33,2324,2326],{"className":2325,"style":1111},[745],[33,2327,2328,2331],{"style":1114},[33,2329],{"className":2330,"style":989},[753],[33,2332,2334],{"className":2333},[758,993,994,761],[33,2335,851],{"className":2336},[92,761],[33,2338,1207],{"className":2339},[1001,761],[33,2341,2343,2346],{"className":2342},[92,761],[33,2344,1726],{"className":2345},[92,218,761],[33,2347,2349],{"className":2348},[972],[33,2350,2352],{"className":2351},[736],[33,2353,2355],{"className":2354},[741],[33,2356,2358],{"className":2357,"style":1111},[745],[33,2359,2360,2363],{"style":1114},[33,2361],{"className":2362,"style":989},[753],[33,2364,2366],{"className":2365},[758,993,994,761],[33,2367,851],{"className":2368},[92,761],[33,2370,798],{"className":2371},[797],[33,2373,2375],{"className":2374},[741],[33,2376,2378],{"className":2377,"style":805},[745],[33,2379],{},[33,2381],{"className":2382},[583,729],[33,2384,2385],{},"17",[11,2387,2389],{"id":2388},"formula-breakdown","Formula Breakdown",[16,2391,2392,2395,349,2471,2474],{},[20,2393,2394],{},"The Height Problem:",[33,2396,2398,2421],{"className":2397},[47],[33,2399,2401],{"className":2400},[51],[53,2402,2403],{"xmlns":55},[57,2404,2405,2418],{},[60,2406,2407,2409,2411,2414,2416],{},[199,2408,266],{},[366,2410,675],{},[199,2412,2413],{},"H",[366,2415,842],{},[199,2417,1567],{},[71,2419,2420],{"encoding":73},"h = H + N",[33,2422,2424,2442,2462],{"className":2423,"ariaHidden":79},[78],[33,2425,2427,2430,2433,2436,2439],{"className":2426},[83],[33,2428],{"className":2429,"style":247},[87],[33,2431,266],{"className":2432},[92,218],[33,2434],{"className":2435,"style":708},[402],[33,2437,675],{"className":2438},[712],[33,2440],{"className":2441,"style":708},[402],[33,2443,2445,2449,2453,2456,2459],{"className":2444},[83],[33,2446],{"className":2447,"style":2448},[87],"height:0.7667em;vertical-align:-0.0833em;",[33,2450,2413],{"className":2451,"style":2452},[92,218],"margin-right:0.0813em;",[33,2454],{"className":2455,"style":569},[402],[33,2457,842],{"className":2458},[1001],[33,2460],{"className":2461,"style":569},[402],[33,2463,2465,2468],{"className":2464},[83],[33,2466],{"className":2467,"style":1635},[87],[33,2469,1567],{"className":2470,"style":1594},[92,218],[33,2472,2473],{},"12","\nWhere:",[311,2476,2477,2515,2551],{},[171,2478,2479,2507,2508,2511,2512,36],{},[33,2480,2482,2495],{"className":2481},[47],[33,2483,2485],{"className":2484},[51],[53,2486,2487],{"xmlns":55},[57,2488,2489,2493],{},[60,2490,2491],{},[199,2492,266],{},[71,2494,266],{"encoding":73},[33,2496,2498],{"className":2497,"ariaHidden":79},[78],[33,2499,2501,2504],{"className":2500},[83],[33,2502],{"className":2503,"style":247},[87],[33,2505,266],{"className":2506},[92,218]," = ",[20,2509,2510],{},"Ellipsoidal Height"," (from GPS) ",[33,2513,2514],{},"30",[171,2516,2517,2507,2545,2548,2549,36],{},[33,2518,2520,2533],{"className":2519},[47],[33,2521,2523],{"className":2522},[51],[53,2524,2525],{"xmlns":55},[57,2526,2527,2531],{},[60,2528,2529],{},[199,2530,2413],{},[71,2532,2413],{"encoding":73},[33,2534,2536],{"className":2535,"ariaHidden":79},[78],[33,2537,2539,2542],{"className":2538},[83],[33,2540],{"className":2541,"style":1635},[87],[33,2543,2413],{"className":2544,"style":2452},[92,218],[20,2546,2547],{},"Orthometric Height"," (Elevation above Mean Sea Level) ",[33,2550,2514],{},[171,2552,2553,2507,2581,349,2584,2587,2588],{},[33,2554,2556,2569],{"className":2555},[47],[33,2557,2559],{"className":2558},[51],[53,2560,2561],{"xmlns":55},[57,2562,2563,2567],{},[60,2564,2565],{},[199,2566,1567],{},[71,2568,1567],{"encoding":73},[33,2570,2572],{"className":2571,"ariaHidden":79},[78],[33,2573,2575,2578],{"className":2574},[83],[33,2576],{"className":2577,"style":1635},[87],[33,2579,1567],{"className":2580,"style":1594},[92,218],[20,2582,2583],{},"Geoid-Ellipsoid Separation",[33,2585,2586],{},"12, 30",".\n",[2589,2590,2592],"callout",{"type":2591},"warning",[16,2593,2594,2595,349,2633,2636,2637,2665,2666,2702,2703,36],{},"In the UK, the WGS84 ellipsoid lies about ",[33,2596,2598,2615],{"className":2597},[47],[33,2599,2601],{"className":2600},[51],[53,2602,2603],{"xmlns":55},[57,2604,2605,2612],{},[60,2606,2607,2610],{},[63,2608,2609],{},"50",[67,2611,69],{},[71,2613,2614],{"encoding":73},"50 \\text{ m}",[33,2616,2618],{"className":2617,"ariaHidden":79},[78],[33,2619,2621,2624,2627],{"className":2620},[83],[33,2622],{"className":2623,"style":88},[87],[33,2625,2609],{"className":2626},[92],[33,2628,2630],{"className":2629},[92,96],[33,2631,69],{"className":2632},[92],[20,2634,2635],{},"below"," the geoid. Failing to account for this ",[33,2638,2640,2653],{"className":2639},[47],[33,2641,2643],{"className":2642},[51],[53,2644,2645],{"xmlns":55},[57,2646,2647,2651],{},[60,2648,2649],{},[199,2650,1567],{},[71,2652,1567],{"encoding":73},[33,2654,2656],{"className":2655,"ariaHidden":79},[78],[33,2657,2659,2662],{"className":2658},[83],[33,2660],{"className":2661,"style":1635},[87],[33,2663,1567],{"className":2664,"style":1594},[92,218]," value will result in elevation errors of ",[33,2667,2669,2684],{"className":2668},[47],[33,2670,2672],{"className":2671},[51],[53,2673,2674],{"xmlns":55},[57,2675,2676,2682],{},[60,2677,2678,2680],{},[63,2679,2609],{},[67,2681,69],{},[71,2683,2614],{"encoding":73},[33,2685,2687],{"className":2686,"ariaHidden":79},[78],[33,2688,2690,2693,2696],{"className":2689},[83],[33,2691],{"className":2692,"style":88},[87],[33,2694,2609],{"className":2695},[92],[33,2697,2699],{"className":2698},[92,96],[33,2700,69],{"className":2701},[92],", even if your horizontal position is perfect ",[33,2704,154],{},[11,2706,2708],{"id":2707},"practical-tips","Practical Tips",[311,2710,2711,2776,2847],{},[171,2712,2713,2716,2717,2772,2773,36],{},[20,2714,2715],{},"Uniform Scale:"," For small engineering sites (",[33,2718,2720,2741],{"className":2719},[47],[33,2721,2723],{"className":2722},[51],[53,2724,2725],{"xmlns":55},[57,2726,2727,2738],{},[60,2728,2729,2732,2735],{},[366,2730,2731],{},"\u003C",[63,2733,2734],{},"10",[67,2736,2737],{}," km",[71,2739,2740],{"encoding":73},"\u003C 10 \\text{ km}",[33,2742,2744,2757],{"className":2743,"ariaHidden":79},[78],[33,2745,2747,2751,2754],{"className":2746},[83],[33,2748],{"className":2749,"style":2750},[87],"height:0.5782em;vertical-align:-0.0391em;",[33,2752,2731],{"className":2753},[712],[33,2755],{"className":2756,"style":708},[402],[33,2758,2760,2763,2766],{"className":2759},[83],[33,2761],{"className":2762,"style":247},[87],[33,2764,2734],{"className":2765},[92],[33,2767,2769],{"className":2768},[92,96],[33,2770,2737],{"className":2771},[92],"), you can often use a single scale factor for the whole site ",[33,2774,2775],{},"31",[171,2777,2778,2781,2782,2810,2811,2839,2840,2843,2844,36],{},[20,2779,2780],{},"Coordinate Order:"," In mapping, the ",[33,2783,2785,2798],{"className":2784},[47],[33,2786,2788],{"className":2787},[51],[53,2789,2790],{"xmlns":55},[57,2791,2792,2796],{},[60,2793,2794],{},[199,2795,526],{},[71,2797,526],{"encoding":73},[33,2799,2801],{"className":2800,"ariaHidden":79},[78],[33,2802,2804,2807],{"className":2803},[83],[33,2805],{"className":2806,"style":1635},[87],[33,2808,526],{"className":2809,"style":559},[92,218],"-axis is Easting and the ",[33,2812,2814,2827],{"className":2813},[47],[33,2815,2817],{"className":2816},[51],[53,2818,2819],{"xmlns":55},[57,2820,2821,2825],{},[60,2822,2823],{},[199,2824,531],{},[71,2826,531],{"encoding":73},[33,2828,2830],{"className":2829,"ariaHidden":79},[78],[33,2831,2833,2836],{"className":2832},[83],[33,2834],{"className":2835,"style":1635},[87],[33,2837,531],{"className":2838,"style":569},[92,218],"-axis is Northing, which is the ",[20,2841,2842],{},"reverse"," of pure mathematical convention ",[33,2845,2846],{},"32",[171,2848,2849,2884,2885,36],{},[20,2850,2851,2852,348],{},"Check the Grid Convergence (",[33,2853,2855,2870],{"className":2854},[47],[33,2856,2858],{"className":2857},[51],[53,2859,2860],{"xmlns":55},[57,2861,2862,2867],{},[60,2863,2864],{},[199,2865,2866],{},"γ",[71,2868,2869],{"encoding":73},"\\gamma",[33,2871,2873],{"className":2872,"ariaHidden":79},[78],[33,2874,2876,2880],{"className":2875},[83],[33,2877],{"className":2878,"style":2879},[87],"height:0.625em;vertical-align:-0.1944em;",[33,2881,2866],{"className":2882,"style":2883},[92,218],"margin-right:0.0556em;"," Grid North and True North are only the same on the Central Meridian. Ensure your Total Station orientation accounts for this ",[33,2886,2887],{},"33, 34",[11,2889,2891],{"id":2890},"common-mistakes","Common Mistakes",[311,2893,2894,2980],{},[171,2895,2896,2899,2900,2976,2977,36],{},[20,2897,2898],{},"Assuming Flatness:"," Neglecting the Earth's radius (",[33,2901,2903,2930],{"className":2902},[47],[33,2904,2906],{"className":2905},[51],[53,2907,2908],{"xmlns":55},[57,2909,2910,2927],{},[60,2911,2912,2915,2918,2920,2922,2925],{},[199,2913,2914],{},"R",[366,2916,2917],{},"≈",[63,2919,364],{},[366,2921,368],{"separator":79},[63,2923,2924],{},"380",[67,2926,2737],{},[71,2928,2929],{"encoding":73},"R \\approx 6,380 \\text{ km}",[33,2931,2933,2952],{"className":2932,"ariaHidden":79},[78],[33,2934,2936,2939,2943,2946,2949],{"className":2935},[83],[33,2937],{"className":2938,"style":1635},[87],[33,2940,2914],{"className":2941,"style":2942},[92,218],"margin-right:0.0077em;",[33,2944],{"className":2945,"style":708},[402],[33,2947,2917],{"className":2948},[712],[33,2950],{"className":2951,"style":708},[402],[33,2953,2955,2958,2961,2964,2967,2970],{"className":2954},[83],[33,2956],{"className":2957,"style":214},[87],[33,2959,364],{"className":2960},[92],[33,2962,368],{"className":2963},[398],[33,2965],{"className":2966,"style":403},[402],[33,2968,2924],{"className":2969},[92],[33,2971,2973],{"className":2972},[92,96],[33,2974,2737],{"className":2975},[92],") in distance reductions ",[33,2978,2979],{},"35",[171,2981,2982,2985,2986,36],{},[20,2983,2984],{},"Sign Errors in Transformations:"," Helmert transformation parameters often have specific sign conventions for rotations. Always test with a known point ",[33,2987,2988],{},"26",[11,2990,2992],{"id":2991},"faq","FAQ",[2994,2995,2996,3005,3067],"accordion",{},[2997,2998,3000],"accordion-item",{"label":2999},"Is WGS84 the same as the GPS datum?",[16,3001,3002,3003,36],{},"Yes. WGS84 is the coordinate system used by the GPS satellites to describe their orbits and your position ",[33,3004,120],{},[2997,3006,3008],{"label":3007},"What is a 'False Origin'?",[16,3009,3010,3011,3063,3064,36],{},"It is a constant value added to coordinates (e.g., ",[33,3012,3014,3036],{"className":3013},[47],[33,3015,3017],{"className":3016},[51],[53,3018,3019],{"xmlns":55},[57,3020,3021,3033],{},[60,3022,3023,3026,3028,3031],{},[63,3024,3025],{},"400",[366,3027,368],{"separator":79},[63,3029,3030],{},"000",[67,3032,69],{},[71,3034,3035],{"encoding":73},"400,000 \\text{ m}",[33,3037,3039],{"className":3038,"ariaHidden":79},[78],[33,3040,3042,3045,3048,3051,3054,3057],{"className":3041},[83],[33,3043],{"className":3044,"style":391},[87],[33,3046,3025],{"className":3047},[92],[33,3049,368],{"className":3050},[398],[33,3052],{"className":3053,"style":403},[402],[33,3055,3030],{"className":3056},[92],[33,3058,3060],{"className":3059},[92,96],[33,3061,69],{"className":3062},[92]," Easting) to ensure that all coordinates in a specific region remain positive ",[33,3065,3066],{},"33",[2997,3068,3070],{"label":3069},"Can I use the same scale factor for $100 \\text{ km}$?",[16,3071,3072,3073,3102,3103,3140,3141,36],{},"No. On long route projects, you must change the scale factor every ",[33,3074,3076,3090],{"className":3075},[47],[33,3077,3079],{"className":3078},[51],[53,3080,3081],{"xmlns":55},[57,3082,3083,3088],{},[60,3084,3085],{},[63,3086,3087],{},"5",[71,3089,3087],{"encoding":73},[33,3091,3093],{"className":3092,"ariaHidden":79},[78],[33,3094,3096,3099],{"className":3095},[83],[33,3097],{"className":3098,"style":88},[87],[33,3100,3087],{"className":3101},[92]," to ",[33,3104,3106,3122],{"className":3105},[47],[33,3107,3109],{"className":3108},[51],[53,3110,3111],{"xmlns":55},[57,3112,3113,3119],{},[60,3114,3115,3117],{},[63,3116,2734],{},[67,3118,2737],{},[71,3120,3121],{"encoding":73},"10 \\text{ km}",[33,3123,3125],{"className":3124,"ariaHidden":79},[78],[33,3126,3128,3131,3134],{"className":3127},[83],[33,3129],{"className":3130,"style":247},[87],[33,3132,2734],{"className":3133},[92],[33,3135,3137],{"className":3136},[92,96],[33,3138,2737],{"className":3139},[92]," to maintain engineering accuracy ",[33,3142,2775],{},[11,3144,3146],{"id":3145},"conclusion","Conclusion",[16,3148,3149],{},"The WGS84 system provides the global foundation for all modern surveying. By rigorously applying geodetic formulas and understanding the transition from ellipsoid to plane, the surveyor ensures that satellite-derived data remains precise and useful for local engineering.",[11,3151,3153],{"id":3152},"references","References",[16,3155,3156,3157,3160],{},"Schofield, W. (2001). ",[1369,3158,3159],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.",{"title":3162,"searchDepth":3163,"depth":3163,"links":3164},"",2,[3165,3166,3167,3168,3169,3175,3181,3182,3183,3184,3185,3186,3187],{"id":13,"depth":3163,"text":14},{"id":39,"depth":3163,"text":40},{"id":110,"depth":3163,"text":111},{"id":165,"depth":3163,"text":166},{"id":300,"depth":3163,"text":301,"children":3170},[3171,3173],{"id":305,"depth":3172,"text":306},3,{"id":506,"depth":3172,"text":3174},"2. Cartesian (X,Y,Z)(X, Y, Z)(X,Y,Z) to Geodetic (ϕ,λ,h)(\\phi, \\lambda, h)(ϕ,λ,h)",{"id":1469,"depth":3163,"text":1470,"children":3176},[3177,3178,3179,3180],{"id":1476,"depth":3172,"text":1477},{"id":1486,"depth":3172,"text":1487},{"id":1500,"depth":3172,"text":1501},{"id":1601,"depth":3172,"text":1602},{"id":1646,"depth":3163,"text":1647},{"id":2388,"depth":3163,"text":2389},{"id":2707,"depth":3163,"text":2708},{"id":2890,"depth":3163,"text":2891},{"id":2991,"depth":3163,"text":2992},{"id":3145,"depth":3163,"text":3146},{"id":3152,"depth":3163,"text":3153},"Position","A deep dive into the WGS84 reference system, geodetic coordinates, and the formulas required to project a curved Earth onto a flat engineering site grid.","md","\u002Fimages\u002Fwgs84-projection.jpg",{"authorCredentials":3193,"commentCount":3194,"tags":3195,"relatedTools":3200,"relatedArticles":3204,"sources":3207},"P.Eng",0,[22,3196,3197,3198,3199],"Datums","Projections","Geodesy","Mapping Math",[3201,3202,3203],"GNSS","GIS","Coordinate Converter",[3205,3206],"rtk-gps-precision","scale-factors-msl-reduction",[3208],{"title":3209,"type":3210,"note":3211},"Engineering Surveying (Fifth Edition)","Book","Primary source for WGS84 parameters and coordinate conversion formulas.",true,"\u002Fblog\u002Fwgs84-datum-and-geodetic-projections","2024-04-10",22,{"title":5,"description":3189},{"loc":3213},"blog\u002F22.wgs84-datum-and-geodetic-projections","-eZAK5wrCXVxa3OhFza9MC-QKXXR6CphREKPVz6zy_s",[3221,3226],{"title":3222,"path":3223,"stem":3224,"description":3225,"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.",{"title":3227,"path":3228,"stem":3229,"description":3230,"children":-1},"Spherical Excess: Adjusting Large-Scale Control Networks","\u002Fblog\u002Fspherical-excess-large-networks","blog\u002F24.spherical-excess-large-networks","A technical guide to calculating spherical excess and adjusting angular observations in primary triangulation networks where the Earth's curvature cannot be ignored.",1783851945667]