[{"data":1,"prerenderedAt":3574},["ShallowReactive",2],{"blog-\u002Fblog\u002Fscale-factors-and-distance-reduction":3,"blog-surround-\u002Fblog\u002Fscale-factors-and-distance-reduction":3563},{"id":4,"title":5,"author":6,"body":7,"category":3530,"description":3531,"extension":3532,"image":3533,"meta":3534,"navigation":3555,"path":3556,"publishedAt":3557,"readTime":3558,"seo":3559,"sitemap":3560,"stem":3561,"__hash__":3562},"blog\u002Fblog\u002F14.scale-factors-and-distance-reduction.md","Scale Factors & MSL Reduction: Correcting Grid to Ground","Surveyiiing Expert",{"type":8,"value":9,"toc":3506},"minimark",[10,15,33,37,179,183,207,211,216,552,771,775,1075,1298,1302,1721,1725,1770,2024,2975,2979,3244,3248,3372,3376,3487,3491,3494,3498],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,27,28,32],"p",{},"A distance measured on the ground with a Total Station or tape is almost never the distance you should plot on a map. This is due to two factors: the height of the terrain above sea level and the distortion caused by projecting the Earth's curved surface onto a flat map. To correct this, surveyors apply the ",[20,21,22],"strong",{},"Altitude Correction"," (Reduction to MSL) and the ",[20,25,26],{},"Scale Factor (F)"," ",[29,30,31],"span",{},"24, 25",".",[11,34,36],{"id":35},"why-this-matters","Why This Matters",[16,38,39,40,96,97,135,136,174,175,178],{},"In large engineering projects, neglecting these corrections leads to \"scale error.\" For example, at an altitude of ",[29,41,44,71],{"className":42},[43],"katex",[29,45,48],{"className":46},[47],"katex-mathml",[49,50,52],"math",{"xmlns":51},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[53,54,55,66],"semantics",{},[56,57,58,62],"mrow",{},[59,60,61],"mn",{},"500",[63,64,65],"mtext",{}," m",[67,68,70],"annotation",{"encoding":69},"application\u002Fx-tex","500\\text{ m}",[29,72,76],{"className":73,"ariaHidden":75},[74],"katex-html","true",[29,77,80,85,89],{"className":78},[79],"base",[29,81],{"className":82,"style":84},[83],"strut","height:0.6444em;",[29,86,61],{"className":87},[88],"mord",[29,90,93],{"className":91},[88,92],"text",[29,94,65],{"className":95},[88],", a measured ",[29,98,100,117],{"className":99},[43],[29,101,103],{"className":102},[47],[49,104,105],{"xmlns":51},[53,106,107,114],{},[56,108,109,112],{},[59,110,111],{},"1000",[63,113,65],{},[67,115,116],{"encoding":69},"1000\\text{ m}",[29,118,120],{"className":119,"ariaHidden":75},[74],[29,121,123,126,129],{"className":122},[79],[29,124],{"className":125,"style":84},[83],[29,127,111],{"className":128},[88],[29,130,132],{"className":131},[88,92],[29,133,65],{"className":134},[88]," line is actually ",[29,137,139,156],{"className":138},[43],[29,140,142],{"className":141},[47],[49,143,144],{"xmlns":51},[53,145,146,153],{},[56,147,148,151],{},[59,149,150],{},"0.08",[63,152,65],{},[67,154,155],{"encoding":69},"0.08\\text{ m}",[29,157,159],{"className":158,"ariaHidden":75},[74],[29,160,162,165,168],{"className":161},[79],[29,163],{"className":164,"style":84},[83],[29,166,150],{"className":167},[88],[29,169,171],{"className":170},[88,92],[29,172,65],{"className":173},[88]," shorter at sea level ",[29,176,177],{},"24",". Furthermore, if you are at the edge of a map projection zone, the grid distance could differ from the ground distance by several decimetres over a kilometre.",[11,180,182],{"id":181},"theory","Theory",[184,185,186,195],"ol",{},[187,188,189,192,193,32],"li",{},[20,190,191],{},"Reduction to Mean Sea Level (MSL):"," Distances must be reduced to a common datum (the geoid\u002Fellipsoid) so that measurements taken at different altitudes are comparable ",[29,194,177],{},[187,196,197,200,201,203,204,32],{},[20,198,199],{},"Projection to Grid:"," Since maps are flat, the \"ellipsoidal distance\" must be multiplied by a ",[20,202,26],{}," to become a \"grid distance\" ",[29,205,206],{},"25, 26",[11,208,210],{"id":209},"mathematical-principles","Mathematical Principles",[212,213,215],"h3",{"id":214},"_1-altitude-correction-reduction-to-msl","1. Altitude Correction (Reduction to MSL)",[16,217,218,219,313,314,344,345,27,549,551],{},"The correction 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to be subtracted from the horizontal ground distance ",[29,315,317,331],{"className":316},[43],[29,318,320],{"className":319},[47],[49,321,322],{"xmlns":51},[53,323,324,329],{},[56,325,326],{},[235,327,328],{},"L",[67,330,328],{"encoding":69},[29,332,334],{"className":333,"ariaHidden":75},[74],[29,335,337,341],{"className":336},[79],[29,338],{"className":339,"style":340},[83],"height:0.6833em;",[29,342,328],{"className":343},[88,260]," is:\n",[29,346,348,390],{"className":347},[43],[29,349,351],{"className":350},[47],[49,352,353],{"xmlns":51},[53,354,355,387],{},[56,356,357,363,367],{},[232,358,359,361],{},[235,360,237],{},[235,362,240],{},[364,365,366],"mo",{},"=",[368,369,370,377],"mfrac",{},[56,371,372,374],{},[235,373,328],{},[235,375,376],{},"H",[56,378,379,382,385],{},[235,380,381],{},"R",[364,383,384],{},"+",[235,386,376],{},[67,388,389],{"encoding":69},"C_M = \\frac{LH}{R + H}",[29,391,393,451],{"className":392,"ariaHidden":75},[74],[29,394,396,399,439,444,448],{"className":395},[79],[29,397],{"className":398,"style":253},[83],[29,400,402,405],{"className":401},[88],[29,403,237],{"className":404,"style":261},[88,260],[29,406,408],{"className":407},[265],[29,409,411,431],{"className":410},[269,270],[29,412,414,428],{"className":413},[274],[29,415,417],{"className":416,"style":279},[278],[29,418,419,422],{"style":282},[29,420],{"className":421,"style":287},[286],[29,423,425],{"className":424},[291,292,293,294],[29,426,240],{"className":427,"style":298},[88,260,294],[29,429,303],{"className":430},[302],[29,432,434],{"className":433},[274],[29,435,437],{"className":436,"style":310},[278],[29,438],{},[29,440],{"className":441,"style":443},[442],"mspace","margin-right:0.2778em;",[29,445,366],{"className":446},[447],"mrel",[29,449],{"className":450,"style":443},[442],[29,452,454,458],{"className":453},[79],[29,455],{"className":456,"style":457},[83],"height:1.2757em;vertical-align:-0.4033em;",[29,459,461,466,545],{"className":460},[88],[29,462],{"className":463},[464,465],"mopen","nulldelimiter",[29,467,469],{"className":468},[368],[29,470,472,536],{"className":471},[269,270],[29,473,475,533],{"className":474},[274],[29,476,479,504,515],{"className":477,"style":478},[278],"height:0.8723em;",[29,480,482,486],{"style":481},"top:-2.655em;",[29,483],{"className":484,"style":485},[286],"height:3em;",[29,487,489],{"className":488},[291,292,293,294],[29,490,492,496,500],{"className":491},[88,294],[29,493,381],{"className":494,"style":495},[88,260,294],"margin-right:0.0077em;",[29,497,384],{"className":498},[499,294],"mbin",[29,501,376],{"className":502,"style":503},[88,260,294],"margin-right:0.0813em;",[29,505,507,510],{"style":506},"top:-3.23em;",[29,508],{"className":509,"style":485},[286],[29,511],{"className":512,"style":514},[513],"frac-line","border-bottom-width:0.04em;",[29,516,518,521],{"style":517},"top:-3.394em;",[29,519],{"className":520,"style":485},[286],[29,522,524],{"className":523},[291,292,293,294],[29,525,527,530],{"className":526},[88,294],[29,528,328],{"className":529},[88,260,294],[29,531,376],{"className":532,"style":503},[88,260,294],[29,534,303],{"className":535},[302],[29,537,539],{"className":538},[274],[29,540,543],{"className":541,"style":542},[278],"height:0.4033em;",[29,544],{},[29,546],{"className":547},[548,465],"mclose",[29,550,177],{},"\nWhere:",[553,554,555,586,617],"ul",{},[187,556,557,585],{},[29,558,560,573],{"className":559},[43],[29,561,563],{"className":562},[47],[49,564,565],{"xmlns":51},[53,566,567,571],{},[56,568,569],{},[235,570,328],{},[67,572,328],{"encoding":69},[29,574,576],{"className":575,"ariaHidden":75},[74],[29,577,579,582],{"className":578},[79],[29,580],{"className":581,"style":340},[83],[29,583,328],{"className":584},[88,260]," = Mean horizontal distance at ground level.",[187,587,588,616],{},[29,589,591,604],{"className":590},[43],[29,592,594],{"className":593},[47],[49,595,596],{"xmlns":51},[53,597,598,602],{},[56,599,600],{},[235,601,376],{},[67,603,376],{"encoding":69},[29,605,607],{"className":606,"ariaHidden":75},[74],[29,608,610,613],{"className":609},[79],[29,611],{"className":612,"style":340},[83],[29,614,376],{"className":615,"style":503},[88,260]," = Mean height above MSL.",[187,618,619,647,648,767,768,32],{},[29,620,622,635],{"className":621},[43],[29,623,625],{"className":624},[47],[49,626,627],{"xmlns":51},[53,628,629,633],{},[56,630,631],{},[235,632,381],{},[67,634,381],{"encoding":69},[29,636,638],{"className":637,"ariaHidden":75},[74],[29,639,641,644],{"className":640},[79],[29,642],{"className":643,"style":340},[83],[29,645,381],{"className":646,"style":495},[88,260]," = Mean local radius of the Earth (",[29,649,651,683],{"className":650},[43],[29,652,654],{"className":653},[47],[49,655,656],{"xmlns":51},[53,657,658,680],{},[56,659,660,663,666,669,678],{},[364,661,662],{},"≈",[59,664,665],{},"6.4",[364,667,668],{},"×",[670,671,672,675],"msup",{},[59,673,674],{},"10",[59,676,677],{},"6",[63,679,65],{},[67,681,682],{"encoding":69},"\\approx 6.4 \\times 10^6 \\text{ m}",[29,684,686,699,719],{"className":685,"ariaHidden":75},[74],[29,687,689,693,696],{"className":688},[79],[29,690],{"className":691,"style":692},[83],"height:0.4831em;",[29,694,662],{"className":695},[447],[29,697],{"className":698,"style":443},[442],[29,700,702,706,709,713,716],{"className":701},[79],[29,703],{"className":704,"style":705},[83],"height:0.7278em;vertical-align:-0.0833em;",[29,707,665],{"className":708},[88],[29,710],{"className":711,"style":712},[442],"margin-right:0.2222em;",[29,714,668],{"className":715},[499],[29,717],{"className":718,"style":712},[442],[29,720,722,726,730,761],{"className":721},[79],[29,723],{"className":724,"style":725},[83],"height:0.8141em;",[29,727,729],{"className":728},[88],"1",[29,731,733,737],{"className":732},[88],[29,734,736],{"className":735},[88],"0",[29,738,740],{"className":739},[265],[29,741,743],{"className":742},[269],[29,744,746],{"className":745},[274],[29,747,749],{"className":748,"style":725},[278],[29,750,752,755],{"style":751},"top:-3.063em;margin-right:0.05em;",[29,753],{"className":754,"style":287},[286],[29,756,758],{"className":757},[291,292,293,294],[29,759,677],{"className":760},[88,294],[29,762,764],{"className":763},[88,92],[29,765,65],{"className":766},[88],") ",[29,769,770],{},"24, 27",[212,772,774],{"id":773},"_2-point-scale-factor-f","2. Point Scale Factor (F)",[16,776,777,778,808,809,27,1072,551],{},"For the Transverse Mercator projection (used in the UK National Grid), the scale factor ",[29,779,781,795],{"className":780},[43],[29,782,784],{"className":783},[47],[49,785,786],{"xmlns":51},[53,787,788,793],{},[56,789,790],{},[235,791,792],{},"F",[67,794,792],{"encoding":69},[29,796,798],{"className":797,"ariaHidden":75},[74],[29,799,801,804],{"className":800},[79],[29,802],{"className":803,"style":340},[83],[29,805,792],{"className":806,"style":807},[88,260],"margin-right:0.1389em;"," at any point is:\n",[29,810,812,875],{"className":811},[43],[29,813,815],{"className":814},[47],[49,816,817],{"xmlns":51},[53,818,819,872],{},[56,820,821,823,825,832,834,838,841,843,855,857,861,869],{},[235,822,792],{},[364,824,366],{},[232,826,827,829],{},[235,828,792],{},[235,830,831],{},"o",[364,833,384],{},[364,835,837],{"stretchy":836},"false","(",[59,839,840],{},"1.228",[364,842,668],{},[670,844,845,847],{},[59,846,674],{},[56,848,849,852],{},[364,850,851],{},"−",[59,853,854],{},"14",[364,856,668],{},[235,858,860],{"mathvariant":859},"normal","Δ",[670,862,863,866],{},[235,864,865],{},"E",[59,867,868],{},"2",[364,870,871],{"stretchy":836},")",[67,873,874],{"encoding":69},"F = F_o + (1.228 \\times 10^{-14} \\times \\Delta E^2)",[29,876,878,896,953,975,1029],{"className":877,"ariaHidden":75},[74],[29,879,881,884,887,890,893],{"className":880},[79],[29,882],{"className":883,"style":340},[83],[29,885,792],{"className":886,"style":807},[88,260],[29,888],{"className":889,"style":443},[442],[29,891,366],{"className":892},[447],[29,894],{"className":895,"style":443},[442],[29,897,899,902,944,947,950],{"className":898},[79],[29,900],{"className":901,"style":253},[83],[29,903,905,908],{"className":904},[88],[29,906,792],{"className":907,"style":807},[88,260],[29,909,911],{"className":910},[265],[29,912,914,936],{"className":913},[269,270],[29,915,917,933],{"className":916},[274],[29,918,921],{"className":919,"style":920},[278],"height:0.1514em;",[29,922,924,927],{"style":923},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[29,925],{"className":926,"style":287},[286],[29,928,930],{"className":929},[291,292,293,294],[29,931,831],{"className":932},[88,260,294],[29,934,303],{"className":935},[302],[29,937,939],{"className":938},[274],[29,940,942],{"className":941,"style":310},[278],[29,943],{},[29,945],{"className":946,"style":712},[442],[29,948,384],{"className":949},[499],[29,951],{"className":952,"style":712},[442],[29,954,956,960,963,966,969,972],{"className":955},[79],[29,957],{"className":958,"style":959},[83],"height:1em;vertical-align:-0.25em;",[29,961,837],{"className":962},[464],[29,964,840],{"className":965},[88],[29,967],{"className":968,"style":712},[442],[29,970,668],{"className":971},[499],[29,973],{"className":974,"style":712},[442],[29,976,978,982,985,1020,1023,1026],{"className":977},[79],[29,979],{"className":980,"style":981},[83],"height:0.8974em;vertical-align:-0.0833em;",[29,983,729],{"className":984},[88],[29,986,988,991],{"className":987},[88],[29,989,736],{"className":990},[88],[29,992,994],{"className":993},[265],[29,995,997],{"className":996},[269],[29,998,1000],{"className":999},[274],[29,1001,1003],{"className":1002,"style":725},[278],[29,1004,1005,1008],{"style":751},[29,1006],{"className":1007,"style":287},[286],[29,1009,1011],{"className":1010},[291,292,293,294],[29,1012,1014,1017],{"className":1013},[88,294],[29,1015,851],{"className":1016},[88,294],[29,1018,854],{"className":1019},[88,294],[29,1021],{"className":1022,"style":712},[442],[29,1024,668],{"className":1025},[499],[29,1027],{"className":1028,"style":712},[442],[29,1030,1032,1036,1039,1069],{"className":1031},[79],[29,1033],{"className":1034,"style":1035},[83],"height:1.0641em;vertical-align:-0.25em;",[29,1037,860],{"className":1038},[88],[29,1040,1042,1046],{"className":1041},[88],[29,1043,865],{"className":1044,"style":1045},[88,260],"margin-right:0.0576em;",[29,1047,1049],{"className":1048},[265],[29,1050,1052],{"className":1051},[269],[29,1053,1055],{"className":1054},[274],[29,1056,1058],{"className":1057,"style":725},[278],[29,1059,1060,1063],{"style":751},[29,1061],{"className":1062,"style":287},[286],[29,1064,1066],{"className":1065},[291,292,293,294],[29,1067,868],{"className":1068},[88,294],[29,1070,871],{"className":1071},[548],[29,1073,1074],{},"28",[553,1076,1077,1180],{},[187,1078,1079,1149,1150,1179],{},[29,1080,1082,1100],{"className":1081},[43],[29,1083,1085],{"className":1084},[47],[49,1086,1087],{"xmlns":51},[53,1088,1089,1097],{},[56,1090,1091],{},[232,1092,1093,1095],{},[235,1094,792],{},[235,1096,831],{},[67,1098,1099],{"encoding":69},"F_o",[29,1101,1103],{"className":1102,"ariaHidden":75},[74],[29,1104,1106,1109],{"className":1105},[79],[29,1107],{"className":1108,"style":253},[83],[29,1110,1112,1115],{"className":1111},[88],[29,1113,792],{"className":1114,"style":807},[88,260],[29,1116,1118],{"className":1117},[265],[29,1119,1121,1141],{"className":1120},[269,270],[29,1122,1124,1138],{"className":1123},[274],[29,1125,1127],{"className":1126,"style":920},[278],[29,1128,1129,1132],{"style":923},[29,1130],{"className":1131,"style":287},[286],[29,1133,1135],{"className":1134},[291,292,293,294],[29,1136,831],{"className":1137},[88,260,294],[29,1139,303],{"className":1140},[302],[29,1142,1144],{"className":1143},[274],[29,1145,1147],{"className":1146,"style":310},[278],[29,1148],{}," = Scale factor on the central meridian (",[29,1151,1153,1167],{"className":1152},[43],[29,1154,1156],{"className":1155},[47],[49,1157,1158],{"xmlns":51},[53,1159,1160,1165],{},[56,1161,1162],{},[59,1163,1164],{},"0.99960127",[67,1166,1164],{"encoding":69},[29,1168,1170],{"className":1169,"ariaHidden":75},[74],[29,1171,1173,1176],{"className":1172},[79],[29,1174],{"className":1175,"style":84},[83],[29,1177,1164],{"className":1178},[88],").",[187,1181,1182,1216,1217,767,1296,32],{},[29,1183,1185,1201],{"className":1184},[43],[29,1186,1188],{"className":1187},[47],[49,1189,1190],{"xmlns":51},[53,1191,1192,1198],{},[56,1193,1194,1196],{},[235,1195,860],{"mathvariant":859},[235,1197,865],{},[67,1199,1200],{"encoding":69},"\\Delta E",[29,1202,1204],{"className":1203,"ariaHidden":75},[74],[29,1205,1207,1210,1213],{"className":1206},[79],[29,1208],{"className":1209,"style":340},[83],[29,1211,860],{"className":1212},[88],[29,1214,865],{"className":1215,"style":1045},[88,260]," = Easting distance from the central meridian (",[29,1218,1220,1247],{"className":1219},[43],[29,1221,1223],{"className":1222},[47],[49,1224,1225],{"xmlns":51},[53,1226,1227,1244],{},[56,1228,1229,1231,1233,1236,1239,1242],{},[235,1230,865],{},[364,1232,851],{},[59,1234,1235],{},"400",[364,1237,1238],{"separator":75},",",[59,1240,1241],{},"000",[63,1243,65],{},[67,1245,1246],{"encoding":69},"E - 400,000 \\text{ m}",[29,1248,1250,1269],{"className":1249,"ariaHidden":75},[74],[29,1251,1253,1257,1260,1263,1266],{"className":1252},[79],[29,1254],{"className":1255,"style":1256},[83],"height:0.7667em;vertical-align:-0.0833em;",[29,1258,865],{"className":1259,"style":1045},[88,260],[29,1261],{"className":1262,"style":712},[442],[29,1264,851],{"className":1265},[499],[29,1267],{"className":1268,"style":712},[442],[29,1270,1272,1276,1279,1283,1287,1290],{"className":1271},[79],[29,1273],{"className":1274,"style":1275},[83],"height:0.8389em;vertical-align:-0.1944em;",[29,1277,1235],{"className":1278},[88],[29,1280,1238],{"className":1281},[1282],"mpunct",[29,1284],{"className":1285,"style":1286},[442],"margin-right:0.1667em;",[29,1288,1241],{"className":1289},[88],[29,1291,1293],{"className":1292},[88,92],[29,1294,65],{"className":1295},[88],[29,1297,1074],{},[11,1299,1301],{"id":1300},"field-workflow","Field Workflow",[1303,1304,1305,1309,1346,1350,1570,1574,1609,1613],"steps",{},[212,1306,1308],{"id":1307},"measure-horizontal-distance","Measure Horizontal Distance",[16,1310,1311,1312,1342,1343,32],{},"Obtain the mean horizontal ground distance (",[29,1313,1315,1329],{"className":1314},[43],[29,1316,1318],{"className":1317},[47],[49,1319,1320],{"xmlns":51},[53,1321,1322,1327],{},[56,1323,1324],{},[235,1325,1326],{},"D",[67,1328,1326],{"encoding":69},[29,1330,1332],{"className":1331,"ariaHidden":75},[74],[29,1333,1335,1338],{"className":1334},[79],[29,1336],{"className":1337,"style":340},[83],[29,1339,1326],{"className":1340,"style":1341},[88,260],"margin-right:0.0278em;",") through standard survey techniques ",[29,1344,1345],{},"29",[212,1347,1349],{"id":1348},"reduce-to-msl","Reduce to MSL",[16,1351,1352,1353,1382,1383,1497,1498,1567,1568,32],{},"Apply the altitude correction to find the ellipsoidal distance (",[29,1354,1356,1370],{"className":1355},[43],[29,1357,1359],{"className":1358},[47],[49,1360,1361],{"xmlns":51},[53,1362,1363,1368],{},[56,1364,1365],{},[235,1366,1367],{},"S",[67,1369,1367],{"encoding":69},[29,1371,1373],{"className":1372,"ariaHidden":75},[74],[29,1374,1376,1379],{"className":1375},[79],[29,1377],{"className":1378,"style":340},[83],[29,1380,1367],{"className":1381,"style":1045},[88,260],"):\n",[29,1384,1386,1412],{"className":1385},[43],[29,1387,1389],{"className":1388},[47],[49,1390,1391],{"xmlns":51},[53,1392,1393,1409],{},[56,1394,1395,1397,1399,1401,1403],{},[235,1396,1367],{},[364,1398,366],{},[235,1400,1326],{},[364,1402,384],{},[232,1404,1405,1407],{},[235,1406,237],{},[235,1408,240],{},[67,1410,1411],{"encoding":69},"S = D + C_M",[29,1413,1415,1433,1451],{"className":1414,"ariaHidden":75},[74],[29,1416,1418,1421,1424,1427,1430],{"className":1417},[79],[29,1419],{"className":1420,"style":340},[83],[29,1422,1367],{"className":1423,"style":1045},[88,260],[29,1425],{"className":1426,"style":443},[442],[29,1428,366],{"className":1429},[447],[29,1431],{"className":1432,"style":443},[442],[29,1434,1436,1439,1442,1445,1448],{"className":1435},[79],[29,1437],{"className":1438,"style":1256},[83],[29,1440,1326],{"className":1441,"style":1341},[88,260],[29,1443],{"className":1444,"style":712},[442],[29,1446,384],{"className":1447},[499],[29,1449],{"className":1450,"style":712},[442],[29,1452,1454,1457],{"className":1453},[79],[29,1455],{"className":1456,"style":253},[83],[29,1458,1460,1463],{"className":1459},[88],[29,1461,237],{"className":1462,"style":261},[88,260],[29,1464,1466],{"className":1465},[265],[29,1467,1469,1489],{"className":1468},[269,270],[29,1470,1472,1486],{"className":1471},[274],[29,1473,1475],{"className":1474,"style":279},[278],[29,1476,1477,1480],{"style":282},[29,1478],{"className":1479,"style":287},[286],[29,1481,1483],{"className":1482},[291,292,293,294],[29,1484,240],{"className":1485,"style":298},[88,260,294],[29,1487,303],{"className":1488},[302],[29,1490,1492],{"className":1491},[274],[29,1493,1495],{"className":1494,"style":310},[278],[29,1496],{}," (Note: ",[29,1499,1501,1518],{"className":1500},[43],[29,1502,1504],{"className":1503},[47],[49,1505,1506],{"xmlns":51},[53,1507,1508,1516],{},[56,1509,1510],{},[232,1511,1512,1514],{},[235,1513,237],{},[235,1515,240],{},[67,1517,243],{"encoding":69},[29,1519,1521],{"className":1520,"ariaHidden":75},[74],[29,1522,1524,1527],{"className":1523},[79],[29,1525],{"className":1526,"style":253},[83],[29,1528,1530,1533],{"className":1529},[88],[29,1531,237],{"className":1532,"style":261},[88,260],[29,1534,1536],{"className":1535},[265],[29,1537,1539,1559],{"className":1538},[269,270],[29,1540,1542,1556],{"className":1541},[274],[29,1543,1545],{"className":1544,"style":279},[278],[29,1546,1547,1550],{"style":282},[29,1548],{"className":1549,"style":287},[286],[29,1551,1553],{"className":1552},[291,292,293,294],[29,1554,240],{"className":1555,"style":298},[88,260,294],[29,1557,303],{"className":1558},[302],[29,1560,1562],{"className":1561},[274],[29,1563,1565],{"className":1564,"style":310},[278],[29,1566],{}," is negative) ",[29,1569,1345],{},[212,1571,1573],{"id":1572},"compute-scale-factor","Compute Scale Factor",[16,1575,1576,1577,1605,1606,32],{},"Calculate the point scale factor (",[29,1578,1580,1593],{"className":1579},[43],[29,1581,1583],{"className":1582},[47],[49,1584,1585],{"xmlns":51},[53,1586,1587,1591],{},[56,1588,1589],{},[235,1590,792],{},[67,1592,792],{"encoding":69},[29,1594,1596],{"className":1595,"ariaHidden":75},[74],[29,1597,1599,1602],{"className":1598},[79],[29,1600],{"className":1601,"style":340},[83],[29,1603,792],{"className":1604,"style":807},[88,260],") based on the Easting coordinate of the line's midpoint ",[29,1607,1608],{},"30",[212,1610,1612],{"id":1611},"convert-to-grid","Convert to Grid",[16,1614,1615,1616,1382,1645,27,1718,32],{},"Multiply the ellipsoidal distance by the scale factor to get the Grid Distance (",[29,1617,1619,1633],{"className":1618},[43],[29,1620,1622],{"className":1621},[47],[49,1623,1624],{"xmlns":51},[53,1625,1626,1631],{},[56,1627,1628],{},[235,1629,1630],{},"G",[67,1632,1630],{"encoding":69},[29,1634,1636],{"className":1635,"ariaHidden":75},[74],[29,1637,1639,1642],{"className":1638},[79],[29,1640],{"className":1641,"style":340},[83],[29,1643,1630],{"className":1644},[88,260],[29,1646,1648,1670],{"className":1647},[43],[29,1649,1651],{"className":1650},[47],[49,1652,1653],{"xmlns":51},[53,1654,1655,1667],{},[56,1656,1657,1659,1661,1663,1665],{},[235,1658,1630],{},[364,1660,366],{},[235,1662,1367],{},[364,1664,668],{},[235,1666,792],{},[67,1668,1669],{"encoding":69},"G = S \\times F",[29,1671,1673,1691,1709],{"className":1672,"ariaHidden":75},[74],[29,1674,1676,1679,1682,1685,1688],{"className":1675},[79],[29,1677],{"className":1678,"style":340},[83],[29,1680,1630],{"className":1681},[88,260],[29,1683],{"className":1684,"style":443},[442],[29,1686,366],{"className":1687},[447],[29,1689],{"className":1690,"style":443},[442],[29,1692,1694,1697,1700,1703,1706],{"className":1693},[79],[29,1695],{"className":1696,"style":1256},[83],[29,1698,1367],{"className":1699,"style":1045},[88,260],[29,1701],{"className":1702,"style":712},[442],[29,1704,668],{"className":1705},[499],[29,1707],{"className":1708,"style":712},[442],[29,1710,1712,1715],{"className":1711},[79],[29,1713],{"className":1714,"style":340},[83],[29,1716,792],{"className":1717,"style":807},[88,260],[29,1719,1720],{},"31",[11,1722,1724],{"id":1723},"step-by-step-example","Step-by-Step Example",[16,1726,1727,1730,1731,1769],{},[20,1728,1729],{},"Problem:"," Convert a ground length of ",[29,1732,1734,1751],{"className":1733},[43],[29,1735,1737],{"className":1736},[47],[49,1738,1739],{"xmlns":51},[53,1740,1741,1748],{},[56,1742,1743,1746],{},[59,1744,1745],{},"7696.328",[63,1747,65],{},[67,1749,1750],{"encoding":69},"7696.328\\text{ m}",[29,1752,1754],{"className":1753,"ariaHidden":75},[74],[29,1755,1757,1760,1763],{"className":1756},[79],[29,1758],{"className":1759,"style":84},[83],[29,1761,1745],{"className":1762},[88],[29,1764,1766],{"className":1765},[88,92],[29,1767,65],{"className":1768},[88]," to a Grid Distance.",[553,1771,1772,1842,1925],{},[187,1773,1774,1775,1803,1804,32],{},"Mean Altitude (",[29,1776,1778,1791],{"className":1777},[43],[29,1779,1781],{"className":1780},[47],[49,1782,1783],{"xmlns":51},[53,1784,1785,1789],{},[56,1786,1787],{},[235,1788,376],{},[67,1790,376],{"encoding":69},[29,1792,1794],{"className":1793,"ariaHidden":75},[74],[29,1795,1797,1800],{"className":1796},[79],[29,1798],{"className":1799,"style":340},[83],[29,1801,376],{"className":1802,"style":503},[88,260],") = ",[29,1805,1807,1824],{"className":1806},[43],[29,1808,1810],{"className":1809},[47],[49,1811,1812],{"xmlns":51},[53,1813,1814,1821],{},[56,1815,1816,1819],{},[59,1817,1818],{},"250",[63,1820,65],{},[67,1822,1823],{"encoding":69},"250\\text{ m}",[29,1825,1827],{"className":1826,"ariaHidden":75},[74],[29,1828,1830,1833,1836],{"className":1829},[79],[29,1831],{"className":1832,"style":84},[83],[29,1834,1818],{"className":1835},[88],[29,1837,1839],{"className":1838},[88,92],[29,1840,65],{"className":1841},[88],[187,1843,1844,1845,1803,1873,32],{},"Mid-Easting (",[29,1846,1848,1861],{"className":1847},[43],[29,1849,1851],{"className":1850},[47],[49,1852,1853],{"xmlns":51},[53,1854,1855,1859],{},[56,1856,1857],{},[235,1858,865],{},[67,1860,865],{"encoding":69},[29,1862,1864],{"className":1863,"ariaHidden":75},[74],[29,1865,1867,1870],{"className":1866},[79],[29,1868],{"className":1869,"style":340},[83],[29,1871,865],{"className":1872,"style":1045},[88,260],[29,1874,1876,1898],{"className":1875},[43],[29,1877,1879],{"className":1878},[47],[49,1880,1881],{"xmlns":51},[53,1882,1883,1895],{},[56,1884,1885,1888,1890,1893],{},[59,1886,1887],{},"638",[364,1889,1238],{"separator":75},[59,1891,1892],{},"824",[63,1894,65],{},[67,1896,1897],{"encoding":69},"638,824\\text{ m}",[29,1899,1901],{"className":1900,"ariaHidden":75},[74],[29,1902,1904,1907,1910,1913,1916,1919],{"className":1903},[79],[29,1905],{"className":1906,"style":1275},[83],[29,1908,1887],{"className":1909},[88],[29,1911,1238],{"className":1912},[1282],[29,1914],{"className":1915,"style":1286},[442],[29,1917,1892],{"className":1918},[88],[29,1920,1922],{"className":1921},[88,92],[29,1923,65],{"className":1924},[88],[187,1926,1927,1928,1803,1956,27,2021,32],{},"Earth Radius (",[29,1929,1931,1944],{"className":1930},[43],[29,1932,1934],{"className":1933},[47],[49,1935,1936],{"xmlns":51},[53,1937,1938,1942],{},[56,1939,1940],{},[235,1941,381],{},[67,1943,381],{"encoding":69},[29,1945,1947],{"className":1946,"ariaHidden":75},[74],[29,1948,1950,1953],{"className":1949},[79],[29,1951],{"className":1952,"style":340},[83],[29,1954,381],{"className":1955,"style":495},[88,260],[29,1957,1959,1985],{"className":1958},[43],[29,1960,1962],{"className":1961},[47],[49,1963,1964],{"xmlns":51},[53,1965,1966,1982],{},[56,1967,1968,1970,1972,1975,1977,1980],{},[59,1969,677],{},[364,1971,1238],{"separator":75},[59,1973,1974],{},"384",[364,1976,1238],{"separator":75},[59,1978,1979],{},"100",[63,1981,65],{},[67,1983,1984],{"encoding":69},"6,384,100\\text{ m}",[29,1986,1988],{"className":1987,"ariaHidden":75},[74],[29,1989,1991,1994,1997,2000,2003,2006,2009,2012,2015],{"className":1990},[79],[29,1992],{"className":1993,"style":1275},[83],[29,1995,677],{"className":1996},[88],[29,1998,1238],{"className":1999},[1282],[29,2001],{"className":2002,"style":1286},[442],[29,2004,1974],{"className":2005},[88],[29,2007,1238],{"className":2008},[1282],[29,2010],{"className":2011,"style":1286},[442],[29,2013,1979],{"className":2014},[88],[29,2016,2018],{"className":2017},[88,92],[29,2019,65],{"className":2020},[88],[29,2022,2023],{},"32",[184,2025,2026,2273,2413,2838],{},[187,2027,2028,2031,27,2270,32],{},[20,2029,2030],{},"Calculate Altitude Correction:",[29,2032,2034,2087],{"className":2033},[43],[29,2035,2037],{"className":2036},[47],[49,2038,2039],{"xmlns":51},[53,2040,2041,2084],{},[56,2042,2043,2049,2051,2053,2075,2077,2079,2082],{},[232,2044,2045,2047],{},[235,2046,237],{},[235,2048,240],{},[364,2050,366],{},[364,2052,851],{},[368,2054,2055,2063],{},[56,2056,2057,2059,2061],{},[59,2058,1745],{},[364,2060,668],{},[59,2062,1818],{},[56,2064,2065,2067,2069,2071,2073],{},[59,2066,677],{},[364,2068,1238],{"separator":75},[59,2070,1974],{},[364,2072,1238],{"separator":75},[59,2074,1979],{},[364,2076,366],{},[364,2078,851],{},[59,2080,2081],{},"0.301",[63,2083,65],{},[67,2085,2086],{"encoding":69},"C_M = -\\frac{7696.328 \\times 250}{6,384,100} = -0.301\\text{ m}",[29,2088,2090,2145,2252],{"className":2089,"ariaHidden":75},[74],[29,2091,2093,2096,2136,2139,2142],{"className":2092},[79],[29,2094],{"className":2095,"style":253},[83],[29,2097,2099,2102],{"className":2098},[88],[29,2100,237],{"className":2101,"style":261},[88,260],[29,2103,2105],{"className":2104},[265],[29,2106,2108,2128],{"className":2107},[269,270],[29,2109,2111,2125],{"className":2110},[274],[29,2112,2114],{"className":2113,"style":279},[278],[29,2115,2116,2119],{"style":282},[29,2117],{"className":2118,"style":287},[286],[29,2120,2122],{"className":2121},[291,292,293,294],[29,2123,240],{"className":2124,"style":298},[88,260,294],[29,2126,303],{"className":2127},[302],[29,2129,2131],{"className":2130},[274],[29,2132,2134],{"className":2133,"style":310},[278],[29,2135],{},[29,2137],{"className":2138,"style":443},[442],[29,2140,366],{"className":2141},[447],[29,2143],{"className":2144,"style":443},[442],[29,2146,2148,2152,2155,2243,2246,2249],{"className":2147},[79],[29,2149],{"className":2150,"style":2151},[83],"height:1.3262em;vertical-align:-0.4811em;",[29,2153,851],{"className":2154},[88],[29,2156,2158,2161,2240],{"className":2157},[88],[29,2159],{"className":2160},[464,465],[29,2162,2164],{"className":2163},[368],[29,2165,2167,2231],{"className":2166},[269,270],[29,2168,2170,2228],{"className":2169},[274],[29,2171,2174,2200,2208],{"className":2172,"style":2173},[278],"height:0.8451em;",[29,2175,2176,2179],{"style":481},[29,2177],{"className":2178,"style":485},[286],[29,2180,2182],{"className":2181},[291,292,293,294],[29,2183,2185,2188,2191,2194,2197],{"className":2184},[88,294],[29,2186,677],{"className":2187},[88,294],[29,2189,1238],{"className":2190},[1282,294],[29,2192,1974],{"className":2193},[88,294],[29,2195,1238],{"className":2196},[1282,294],[29,2198,1979],{"className":2199},[88,294],[29,2201,2202,2205],{"style":506},[29,2203],{"className":2204,"style":485},[286],[29,2206],{"className":2207,"style":514},[513],[29,2209,2210,2213],{"style":517},[29,2211],{"className":2212,"style":485},[286],[29,2214,2216],{"className":2215},[291,292,293,294],[29,2217,2219,2222,2225],{"className":2218},[88,294],[29,2220,1745],{"className":2221},[88,294],[29,2223,668],{"className":2224},[499,294],[29,2226,1818],{"className":2227},[88,294],[29,2229,303],{"className":2230},[302],[29,2232,2234],{"className":2233},[274],[29,2235,2238],{"className":2236,"style":2237},[278],"height:0.4811em;",[29,2239],{},[29,2241],{"className":2242},[548,465],[29,2244],{"className":2245,"style":443},[442],[29,2247,366],{"className":2248},[447],[29,2250],{"className":2251,"style":443},[442],[29,2253,2255,2258,2261,2264],{"className":2254},[79],[29,2256],{"className":2257,"style":705},[83],[29,2259,851],{"className":2260},[88],[29,2262,2081],{"className":2263},[88],[29,2265,2267],{"className":2266},[88,92],[29,2268,65],{"className":2269},[88],[29,2271,2272],{},"31, 33",[187,2274,2275,2307,27,2411,32],{},[20,2276,2277,2278,2306],{},"Ellipsoidal Distance (",[29,2279,2281,2294],{"className":2280},[43],[29,2282,2284],{"className":2283},[47],[49,2285,2286],{"xmlns":51},[53,2287,2288,2292],{},[56,2289,2290],{},[235,2291,1367],{},[67,2293,1367],{"encoding":69},[29,2295,2297],{"className":2296,"ariaHidden":75},[74],[29,2298,2300,2303],{"className":2299},[79],[29,2301],{"className":2302,"style":340},[83],[29,2304,1367],{"className":2305,"style":1045},[88,260],"):",[29,2308,2310,2339],{"className":2309},[43],[29,2311,2313],{"className":2312},[47],[49,2314,2315],{"xmlns":51},[53,2316,2317,2336],{},[56,2318,2319,2321,2323,2325,2327,2329,2331,2334],{},[235,2320,1367],{},[364,2322,366],{},[59,2324,1745],{},[364,2326,851],{},[59,2328,2081],{},[364,2330,366],{},[59,2332,2333],{},"7696.027",[63,2335,65],{},[67,2337,2338],{"encoding":69},"S = 7696.328 - 0.301 = 7696.027\\text{ m}",[29,2340,2342,2360,2378,2396],{"className":2341,"ariaHidden":75},[74],[29,2343,2345,2348,2351,2354,2357],{"className":2344},[79],[29,2346],{"className":2347,"style":340},[83],[29,2349,1367],{"className":2350,"style":1045},[88,260],[29,2352],{"className":2353,"style":443},[442],[29,2355,366],{"className":2356},[447],[29,2358],{"className":2359,"style":443},[442],[29,2361,2363,2366,2369,2372,2375],{"className":2362},[79],[29,2364],{"className":2365,"style":705},[83],[29,2367,1745],{"className":2368},[88],[29,2370],{"className":2371,"style":712},[442],[29,2373,851],{"className":2374},[499],[29,2376],{"className":2377,"style":712},[442],[29,2379,2381,2384,2387,2390,2393],{"className":2380},[79],[29,2382],{"className":2383,"style":84},[83],[29,2385,2081],{"className":2386},[88],[29,2388],{"className":2389,"style":443},[442],[29,2391,366],{"className":2392},[447],[29,2394],{"className":2395,"style":443},[442],[29,2397,2399,2402,2405],{"className":2398},[79],[29,2400],{"className":2401,"style":84},[83],[29,2403,2333],{"className":2404},[88],[29,2406,2408],{"className":2407},[88,92],[29,2409,65],{"className":2410},[88],[29,2412,1720],{},[187,2414,2415,2446,2594,2595,27,2836,32],{},[20,2416,2417,2418,2306],{},"Calculate Scale Factor (",[29,2419,2421,2434],{"className":2420},[43],[29,2422,2424],{"className":2423},[47],[49,2425,2426],{"xmlns":51},[53,2427,2428,2432],{},[56,2429,2430],{},[235,2431,792],{},[67,2433,792],{"encoding":69},[29,2435,2437],{"className":2436,"ariaHidden":75},[74],[29,2438,2440,2443],{"className":2439},[79],[29,2441],{"className":2442,"style":340},[83],[29,2444,792],{"className":2445,"style":807},[88,260],[29,2447,2449,2492],{"className":2448},[43],[29,2450,2452],{"className":2451},[47],[49,2453,2454],{"xmlns":51},[53,2455,2456,2489],{},[56,2457,2458,2460,2462,2464,2466,2468,2470,2472,2474,2476,2478,2480,2483,2485,2487],{},[235,2459,860],{"mathvariant":859},[235,2461,865],{},[364,2463,366],{},[59,2465,1887],{},[364,2467,1238],{"separator":75},[59,2469,1892],{},[364,2471,851],{},[59,2473,1235],{},[364,2475,1238],{"separator":75},[59,2477,1241],{},[364,2479,366],{},[59,2481,2482],{},"238",[364,2484,1238],{"separator":75},[59,2486,1892],{},[63,2488,65],{},[67,2490,2491],{"encoding":69},"\\Delta E = 638,824 - 400,000 = 238,824\\text{ m}",[29,2493,2495,2516,2543,2570],{"className":2494,"ariaHidden":75},[74],[29,2496,2498,2501,2504,2507,2510,2513],{"className":2497},[79],[29,2499],{"className":2500,"style":340},[83],[29,2502,860],{"className":2503},[88],[29,2505,865],{"className":2506,"style":1045},[88,260],[29,2508],{"className":2509,"style":443},[442],[29,2511,366],{"className":2512},[447],[29,2514],{"className":2515,"style":443},[442],[29,2517,2519,2522,2525,2528,2531,2534,2537,2540],{"className":2518},[79],[29,2520],{"className":2521,"style":1275},[83],[29,2523,1887],{"className":2524},[88],[29,2526,1238],{"className":2527},[1282],[29,2529],{"className":2530,"style":1286},[442],[29,2532,1892],{"className":2533},[88],[29,2535],{"className":2536,"style":712},[442],[29,2538,851],{"className":2539},[499],[29,2541],{"className":2542,"style":712},[442],[29,2544,2546,2549,2552,2555,2558,2561,2564,2567],{"className":2545},[79],[29,2547],{"className":2548,"style":1275},[83],[29,2550,1235],{"className":2551},[88],[29,2553,1238],{"className":2554},[1282],[29,2556],{"className":2557,"style":1286},[442],[29,2559,1241],{"className":2560},[88],[29,2562],{"className":2563,"style":443},[442],[29,2565,366],{"className":2566},[447],[29,2568],{"className":2569,"style":443},[442],[29,2571,2573,2576,2579,2582,2585,2588],{"className":2572},[79],[29,2574],{"className":2575,"style":1275},[83],[29,2577,2482],{"className":2578},[88],[29,2580,1238],{"className":2581},[1282],[29,2583],{"className":2584,"style":1286},[442],[29,2586,1892],{"className":2587},[88],[29,2589,2591],{"className":2590},[88,92],[29,2592,65],{"className":2593},[88],".\n",[29,2596,2598,2653],{"className":2597},[43],[29,2599,2601],{"className":2600},[47],[49,2602,2603],{"xmlns":51},[53,2604,2605,2650],{},[56,2606,2607,2609,2611,2613,2615,2617,2619,2621,2631,2633,2635,2637,2643,2645,2647],{},[235,2608,792],{},[364,2610,366],{},[59,2612,1164],{},[364,2614,384],{},[364,2616,837],{"stretchy":836},[59,2618,840],{},[364,2620,668],{},[670,2622,2623,2625],{},[59,2624,674],{},[56,2626,2627,2629],{},[364,2628,851],{},[59,2630,854],{},[364,2632,668],{},[59,2634,2482],{},[364,2636,1238],{"separator":75},[670,2638,2639,2641],{},[59,2640,1892],{},[59,2642,868],{},[364,2644,871],{"stretchy":836},[364,2646,366],{},[59,2648,2649],{},"1.0003016",[67,2651,2652],{"encoding":69},"F = 0.99960127 + (1.228 \\times 10^{-14} \\times 238,824^2) = 1.0003016",[29,2654,2656,2674,2692,2713,2766,2827],{"className":2655,"ariaHidden":75},[74],[29,2657,2659,2662,2665,2668,2671],{"className":2658},[79],[29,2660],{"className":2661,"style":340},[83],[29,2663,792],{"className":2664,"style":807},[88,260],[29,2666],{"className":2667,"style":443},[442],[29,2669,366],{"className":2670},[447],[29,2672],{"className":2673,"style":443},[442],[29,2675,2677,2680,2683,2686,2689],{"className":2676},[79],[29,2678],{"className":2679,"style":705},[83],[29,2681,1164],{"className":2682},[88],[29,2684],{"className":2685,"style":712},[442],[29,2687,384],{"className":2688},[499],[29,2690],{"className":2691,"style":712},[442],[29,2693,2695,2698,2701,2704,2707,2710],{"className":2694},[79],[29,2696],{"className":2697,"style":959},[83],[29,2699,837],{"className":2700},[464],[29,2702,840],{"className":2703},[88],[29,2705],{"className":2706,"style":712},[442],[29,2708,668],{"className":2709},[499],[29,2711],{"className":2712,"style":712},[442],[29,2714,2716,2719,2722,2757,2760,2763],{"className":2715},[79],[29,2717],{"className":2718,"style":981},[83],[29,2720,729],{"className":2721},[88],[29,2723,2725,2728],{"className":2724},[88],[29,2726,736],{"className":2727},[88],[29,2729,2731],{"className":2730},[265],[29,2732,2734],{"className":2733},[269],[29,2735,2737],{"className":2736},[274],[29,2738,2740],{"className":2739,"style":725},[278],[29,2741,2742,2745],{"style":751},[29,2743],{"className":2744,"style":287},[286],[29,2746,2748],{"className":2747},[291,292,293,294],[29,2749,2751,2754],{"className":2750},[88,294],[29,2752,851],{"className":2753},[88,294],[29,2755,854],{"className":2756},[88,294],[29,2758],{"className":2759,"style":712},[442],[29,2761,668],{"className":2762},[499],[29,2764],{"className":2765,"style":712},[442],[29,2767,2769,2772,2775,2778,2781,2785,2815,2818,2821,2824],{"className":2768},[79],[29,2770],{"className":2771,"style":1035},[83],[29,2773,2482],{"className":2774},[88],[29,2776,1238],{"className":2777},[1282],[29,2779],{"className":2780,"style":1286},[442],[29,2782,2784],{"className":2783},[88],"82",[29,2786,2788,2792],{"className":2787},[88],[29,2789,2791],{"className":2790},[88],"4",[29,2793,2795],{"className":2794},[265],[29,2796,2798],{"className":2797},[269],[29,2799,2801],{"className":2800},[274],[29,2802,2804],{"className":2803,"style":725},[278],[29,2805,2806,2809],{"style":751},[29,2807],{"className":2808,"style":287},[286],[29,2810,2812],{"className":2811},[291,292,293,294],[29,2813,868],{"className":2814},[88,294],[29,2816,871],{"className":2817},[548],[29,2819],{"className":2820,"style":443},[442],[29,2822,366],{"className":2823},[447],[29,2825],{"className":2826,"style":443},[442],[29,2828,2830,2833],{"className":2829},[79],[29,2831],{"className":2832,"style":84},[83],[29,2834,2649],{"className":2835},[88],[29,2837,1074],{},[187,2839,2840,2871,32],{},[20,2841,2842,2843,2306],{},"Grid Distance (",[29,2844,2846,2859],{"className":2845},[43],[29,2847,2849],{"className":2848},[47],[49,2850,2851],{"xmlns":51},[53,2852,2853,2857],{},[56,2854,2855],{},[235,2856,1630],{},[67,2858,1630],{"encoding":69},[29,2860,2862],{"className":2861,"ariaHidden":75},[74],[29,2863,2865,2868],{"className":2864},[79],[29,2866],{"className":2867,"style":340},[83],[29,2869,1630],{"className":2870},[88,260],[29,2872,2874,2903],{"className":2873},[43],[29,2875,2877],{"className":2876},[47],[49,2878,2879],{"xmlns":51},[53,2880,2881,2900],{},[56,2882,2883,2885,2887,2889,2891,2893,2895,2898],{},[235,2884,1630],{},[364,2886,366],{},[59,2888,2333],{},[364,2890,668],{},[59,2892,2649],{},[364,2894,366],{},[59,2896,2897],{},"7698.348",[63,2899,65],{},[67,2901,2902],{"encoding":69},"G = 7696.027 \\times 1.0003016 = 7698.348\\text{ m}",[29,2904,2906,2924,2942,2960],{"className":2905,"ariaHidden":75},[74],[29,2907,2909,2912,2915,2918,2921],{"className":2908},[79],[29,2910],{"className":2911,"style":340},[83],[29,2913,1630],{"className":2914},[88,260],[29,2916],{"className":2917,"style":443},[442],[29,2919,366],{"className":2920},[447],[29,2922],{"className":2923,"style":443},[442],[29,2925,2927,2930,2933,2936,2939],{"className":2926},[79],[29,2928],{"className":2929,"style":705},[83],[29,2931,2333],{"className":2932},[88],[29,2934],{"className":2935,"style":712},[442],[29,2937,668],{"className":2938},[499],[29,2940],{"className":2941,"style":712},[442],[29,2943,2945,2948,2951,2954,2957],{"className":2944},[79],[29,2946],{"className":2947,"style":84},[83],[29,2949,2649],{"className":2950},[88],[29,2952],{"className":2953,"style":443},[442],[29,2955,366],{"className":2956},[447],[29,2958],{"className":2959,"style":443},[442],[29,2961,2963,2966,2969],{"className":2962},[79],[29,2964],{"className":2965,"style":84},[83],[29,2967,2897],{"className":2968},[88],[29,2970,2972],{"className":2971},[88,92],[29,2973,65],{"className":2974},[88],[11,2976,2978],{"id":2977},"practical-tips","Practical Tips",[553,2980,2981,3060,3097],{},[187,2982,2983,2986,2987,3016,3017,3057,3058,32],{},[20,2984,2985],{},"The 180km Rule:"," In the UK, the scale factor is exactly ",[29,2988,2990,3004],{"className":2989},[43],[29,2991,2993],{"className":2992},[47],[49,2994,2995],{"xmlns":51},[53,2996,2997,3002],{},[56,2998,2999],{},[59,3000,3001],{},"1.000",[67,3003,3001],{"encoding":69},[29,3005,3007],{"className":3006,"ariaHidden":75},[74],[29,3008,3010,3013],{"className":3009},[79],[29,3011],{"className":3012,"style":84},[83],[29,3014,3001],{"className":3015},[88]," at approximately ",[29,3018,3020,3038],{"className":3019},[43],[29,3021,3023],{"className":3022},[47],[49,3024,3025],{"xmlns":51},[53,3026,3027,3035],{},[56,3028,3029,3032],{},[59,3030,3031],{},"180",[63,3033,3034],{}," km",[67,3036,3037],{"encoding":69},"180\\text{ km}",[29,3039,3041],{"className":3040,"ariaHidden":75},[74],[29,3042,3044,3048,3051],{"className":3043},[79],[29,3045],{"className":3046,"style":3047},[83],"height:0.6944em;",[29,3049,3031],{"className":3050},[88],[29,3052,3054],{"className":3053},[88,92],[29,3055,3034],{"className":3056},[88]," East or West of the central meridian ",[29,3059,1074],{},[187,3061,3062,3065,3066,3094,3095,32],{},[20,3063,3064],{},"Setting Out:"," When taking a distance from a map to set out on the ground, the rules are reversed: divide by ",[29,3067,3069,3082],{"className":3068},[43],[29,3070,3072],{"className":3071},[47],[49,3073,3074],{"xmlns":51},[53,3075,3076,3080],{},[56,3077,3078],{},[235,3079,792],{},[67,3081,792],{"encoding":69},[29,3083,3085],{"className":3084,"ariaHidden":75},[74],[29,3086,3088,3091],{"className":3087},[79],[29,3089],{"className":3090,"style":340},[83],[29,3092,792],{"className":3093,"style":807},[88,260]," and add the altitude correction ",[29,3096,1345],{},[187,3098,3099,3102,3103,3132,3133,3161,3162,3200,3201,3240,3241,32],{},[20,3100,3101],{},"N-values:"," Theoretically, you should also reduce from MSL to the Ellipsoid using the geoid-ellipsoid separation (",[29,3104,3106,3120],{"className":3105},[43],[29,3107,3109],{"className":3108},[47],[49,3110,3111],{"xmlns":51},[53,3112,3113,3118],{},[56,3114,3115],{},[235,3116,3117],{},"N",[67,3119,3117],{"encoding":69},[29,3121,3123],{"className":3122,"ariaHidden":75},[74],[29,3124,3126,3129],{"className":3125},[79],[29,3127],{"className":3128,"style":340},[83],[29,3130,3117],{"className":3131,"style":298},[88,260],"). However, in the UK, ",[29,3134,3136,3149],{"className":3135},[43],[29,3137,3139],{"className":3138},[47],[49,3140,3141],{"xmlns":51},[53,3142,3143,3147],{},[56,3144,3145],{},[235,3146,3117],{},[67,3148,3117],{"encoding":69},[29,3150,3152],{"className":3151,"ariaHidden":75},[74],[29,3153,3155,3158],{"className":3154},[79],[29,3156],{"className":3157,"style":340},[83],[29,3159,3117],{"className":3160,"style":298},[88,260]," is maximum ",[29,3163,3165,3182],{"className":3164},[43],[29,3166,3168],{"className":3167},[47],[49,3169,3170],{"xmlns":51},[53,3171,3172,3179],{},[56,3173,3174,3177],{},[59,3175,3176],{},"4.5",[63,3178,65],{},[67,3180,3181],{"encoding":69},"4.5\\text{ m}",[29,3183,3185],{"className":3184,"ariaHidden":75},[74],[29,3186,3188,3191,3194],{"className":3187},[79],[29,3189],{"className":3190,"style":84},[83],[29,3192,3176],{"className":3193},[88],[29,3195,3197],{"className":3196},[88,92],[29,3198,65],{"className":3199},[88],", resulting in only ",[29,3202,3204,3222],{"className":3203},[43],[29,3205,3207],{"className":3206},[47],[49,3208,3209],{"xmlns":51},[53,3210,3211,3219],{},[56,3212,3213,3216],{},[59,3214,3215],{},"0.7",[63,3217,3218],{}," ppm",[67,3220,3221],{"encoding":69},"0.7\\text{ ppm}",[29,3223,3225],{"className":3224,"ariaHidden":75},[74],[29,3226,3228,3231,3234],{"className":3227},[79],[29,3229],{"className":3230,"style":1275},[83],[29,3232,3215],{"className":3233},[88],[29,3235,3237],{"className":3236},[88,92],[29,3238,3218],{"className":3239},[88]," error, which is often ignored for scale ",[29,3242,3243],{},"34, 35",[11,3245,3247],{"id":3246},"common-mistakes","Common Mistakes",[553,3249,3250,3363],{},[187,3251,3252,3255,3256,3293,3294,3323,3324,3360,3361,32],{},[20,3253,3254],{},"Using a Single Scale Factor:"," On long route projects (over ",[29,3257,3259,3275],{"className":3258},[43],[29,3260,3262],{"className":3261},[47],[49,3263,3264],{"xmlns":51},[53,3265,3266,3272],{},[56,3267,3268,3270],{},[59,3269,674],{},[63,3271,3034],{},[67,3273,3274],{"encoding":69},"10\\text{ km}",[29,3276,3278],{"className":3277,"ariaHidden":75},[74],[29,3279,3281,3284,3287],{"className":3280},[79],[29,3282],{"className":3283,"style":3047},[83],[29,3285,674],{"className":3286},[88],[29,3288,3290],{"className":3289},[88,92],[29,3291,3034],{"className":3292},[88],"), the scale factor changes significantly. You must use a different scale factor for every ",[29,3295,3297,3311],{"className":3296},[43],[29,3298,3300],{"className":3299},[47],[49,3301,3302],{"xmlns":51},[53,3303,3304,3309],{},[56,3305,3306],{},[59,3307,3308],{},"5",[67,3310,3308],{"encoding":69},[29,3312,3314],{"className":3313,"ariaHidden":75},[74],[29,3315,3317,3320],{"className":3316},[79],[29,3318],{"className":3319,"style":84},[83],[29,3321,3308],{"className":3322},[88]," to ",[29,3325,3327,3342],{"className":3326},[43],[29,3328,3330],{"className":3329},[47],[49,3331,3332],{"xmlns":51},[53,3333,3334,3340],{},[56,3335,3336,3338],{},[59,3337,674],{},[63,3339,3034],{},[67,3341,3274],{"encoding":69},[29,3343,3345],{"className":3344,"ariaHidden":75},[74],[29,3346,3348,3351,3354],{"className":3347},[79],[29,3349],{"className":3350,"style":3047},[83],[29,3352,674],{"className":3353},[88],[29,3355,3357],{"className":3356},[88,92],[29,3358,3034],{"className":3359},[88]," section ",[29,3362,1608],{},[187,3364,3365,3368,3369,32],{},[20,3366,3367],{},"Confusing MSL with Ellipsoid:"," For high-precision GPS work, ensuring you are reducing to the correct surface (geoid vs ellipsoid) is critical for heighting ",[29,3370,3371],{},"36",[11,3373,3375],{"id":3374},"faq","FAQ",[3377,3378,3379,3441,3478],"accordion",{},[3380,3381,3383],"accordion-item",{"label":3382},"What is 'unity' in scale factors?",[16,3384,3385,3386,3437,3438,32],{},"A scale factor of unity (",[29,3387,3389,3407],{"className":3388},[43],[29,3390,3392],{"className":3391},[47],[49,3393,3394],{"xmlns":51},[53,3395,3396,3404],{},[56,3397,3398,3400,3402],{},[235,3399,792],{},[364,3401,366],{},[59,3403,729],{},[67,3405,3406],{"encoding":69},"F=1",[29,3408,3410,3428],{"className":3409,"ariaHidden":75},[74],[29,3411,3413,3416,3419,3422,3425],{"className":3412},[79],[29,3414],{"className":3415,"style":340},[83],[29,3417,792],{"className":3418,"style":807},[88,260],[29,3420],{"className":3421,"style":443},[442],[29,3423,366],{"className":3424},[447],[29,3426],{"className":3427,"style":443},[442],[29,3429,3431,3434],{"className":3430},[79],[29,3432],{"className":3433,"style":84},[83],[29,3435,729],{"className":3436},[88],") means the distance on the ground (at MSL) is identical to the distance on the map. This occurs where the map projection cylinder intersects the Earth's surface ",[29,3439,3440],{},"37, 38",[3380,3442,3444],{"label":3443},"Is the scale factor constant across a site?",[16,3445,3446,3447,3475,3476,32],{},"For small engineering sites, a single value for ",[29,3448,3450,3463],{"className":3449},[43],[29,3451,3453],{"className":3452},[47],[49,3454,3455],{"xmlns":51},[53,3456,3457,3461],{},[56,3458,3459],{},[235,3460,792],{},[67,3462,792],{"encoding":69},[29,3464,3466],{"className":3465,"ariaHidden":75},[74],[29,3467,3469,3472],{"className":3468},[79],[29,3470],{"className":3471,"style":340},[83],[29,3473,792],{"className":3474,"style":807},[88,260]," at the centre of the site can be regarded as constant throughout ",[29,3477,1608],{},[3380,3479,3481],{"label":3480},"Does the altitude correction apply to GPS?",[16,3482,3483,3484,32],{},"GPS measures on the ellipsoid directly. However, the altitude correction is still needed to convert those results into heights above Mean Sea Level (MSL) for drainage and engineering ",[29,3485,3486],{},"39",[11,3488,3490],{"id":3489},"conclusion","Conclusion",[16,3492,3493],{},"Managing the gap between \"Grid\" and \"Ground\" is a core competency of the engineering surveyor. By rigorously applying altitude reductions and scale factors, you ensure that the digital design and the physical construction exist in the same spatial reality.",[11,3495,3497],{"id":3496},"references","References",[16,3499,3500,3501,3505],{},"Schofield, W. (2001). ",[3502,3503,3504],"em",{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.",{"title":3507,"searchDepth":3508,"depth":3508,"links":3509},"",2,[3510,3511,3512,3513,3518,3524,3525,3526,3527,3528,3529],{"id":13,"depth":3508,"text":14},{"id":35,"depth":3508,"text":36},{"id":181,"depth":3508,"text":182},{"id":209,"depth":3508,"text":210,"children":3514},[3515,3517],{"id":214,"depth":3516,"text":215},3,{"id":773,"depth":3516,"text":774},{"id":1300,"depth":3508,"text":1301,"children":3519},[3520,3521,3522,3523],{"id":1307,"depth":3516,"text":1308},{"id":1348,"depth":3516,"text":1349},{"id":1572,"depth":3516,"text":1573},{"id":1611,"depth":3516,"text":1612},{"id":1723,"depth":3508,"text":1724},{"id":2977,"depth":3508,"text":2978},{"id":3246,"depth":3508,"text":3247},{"id":3374,"depth":3508,"text":3375},{"id":3489,"depth":3508,"text":3490},{"id":3496,"depth":3508,"text":3497},"Position","A technical guide to reducing measured distances to the ellipsoid and applying scale factors for National Grid consistency.","md","\u002Fimages\u002Fscale-factors.jpg",{"authorCredentials":3535,"commentCount":3536,"tags":3537,"relatedTools":3543,"relatedArticles":3547,"sources":3550},"P.Eng",0,[3538,3539,3540,3541,3542],"Scale Factor","MSL Reduction","National Grid","Distance Measurement","Math",[3544,3545,3546],"Total Station","EDM","GPS",[3548,3549],"precise-taping-corrections","helmert-transformations-math",[3551],{"title":3552,"type":3553,"note":3554},"Engineering Surveying (Fifth Edition)","Book","Primary source for distance reduction formulas and OSGB scale factors.",true,"\u002Fblog\u002Fscale-factors-and-distance-reduction","2024-04-02",19,{"title":5,"description":3531},{"loc":3556},"blog\u002F14.scale-factors-and-distance-reduction","YbzjEamjQhF_cqpRxTR3MqRcvT5lHikt68xSzaxzlEo",[3564,3569],{"title":3565,"path":3566,"stem":3567,"description":3568,"children":-1},"Coordinate Transformations: Mastering the Helmert Transformation","\u002Fblog\u002Fhelmert-transformations-datum-conversion","blog\u002F13.helmert-transformations-datum-conversion","A technical exploration of 7-parameter coordinate transformations used to convert data between different geodetic datums.",{"title":3570,"path":3571,"stem":3572,"description":3573,"children":-1},"Inverted Sights: Levelling Techniques for Overheads and Soffits","\u002Fblog\u002Finverted-sights-underground-and-overhead-levelling","blog\u002F15.inverted-sights-underground-and-overhead-levelling","A technical guide to performing precise levelling on the underside of structures, tunnels, and bridges using inverted staff techniques.",1783851950588]