[{"data":1,"prerenderedAt":5157},["ShallowReactive",2],{"blog-\u002Fblog\u002Fclothoid-transition-curves":3,"blog-surround-\u002Fblog\u002Fclothoid-transition-curves":5146},{"id":4,"title":5,"author":6,"body":7,"category":5117,"description":5118,"extension":5119,"image":5120,"meta":5121,"navigation":5138,"path":5139,"publishedAt":5140,"readTime":5141,"seo":5142,"sitemap":5143,"stem":5144,"__hash__":5145},"blog\u002Fblog\u002F7.clothoid-transition-curves.md","The Clothoid Spiral: Engineering Seamless Transitions in Road Design","Surveyiiing Expert",{"type":8,"value":9,"toc":5089},"minimark",[10,15,33,37,51,55,240,306,310,344,824,858,1024,1059,1203,1207,2467,2471,2677,4461,4465,4750,4754,4838,4842,5068,5072,5075,5079],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,27,28,32],"p",{},"Moving a vehicle directly from a straight path into a circular curve causes a sudden jolt of centrifugal force. To ensure safety and comfort, engineers use ",[20,21,22],"strong",{},"Transition Curves",". The ",[20,25,26],{},"Clothoid Spiral"," (also known as the Ideal Transition) is the preferred geometry because its radius decreases linearly with its length, providing a constant rate of change of centrifugal acceleration ",[29,30,31],"span",{},"18, 19",".",[11,34,36],{"id":35},"why-this-matters","Why This Matters",[16,38,39,40,43,44,47,48,32],{},"Without a transition curve, a driver would have to turn the steering wheel instantaneously at the tangent point. This causes lateral skidding, structural stress on the vehicle, and driver discomfort ",[29,41,42],{},"18",". Clothoids allow for the gradual introduction of ",[20,45,46],{},"super-elevation"," (banking), keeping vehicles stable at design speeds ",[29,49,50],{},"20",[11,52,54],{"id":53},"theory","Theory",[16,56,57,58,104,105,136,137,235,236,239],{},"The clothoid is defined by the property that its radius (",[29,59,62,84],{"className":60},[61],"katex",[29,63,66],{"className":64},[65],"katex-mathml",[67,68,70],"math",{"xmlns":69},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[71,72,73,80],"semantics",{},[74,75,76],"mrow",{},[77,78,79],"mi",{},"r",[81,82,79],"annotation",{"encoding":83},"application\u002Fx-tex",[29,85,89],{"className":86,"ariaHidden":88},[87],"katex-html","true",[29,90,93,98],{"className":91},[92],"base",[29,94],{"className":95,"style":97},[96],"strut","height:0.4306em;",[29,99,79],{"className":100,"style":103},[101,102],"mord","mathnormal","margin-right:0.0278em;",") at any point is inversely proportional to its distance (",[29,106,108,122],{"className":107},[61],[29,109,111],{"className":110},[65],[67,112,113],{"xmlns":69},[71,114,115,120],{},[74,116,117],{},[77,118,119],{},"l",[81,121,119],{"encoding":83},[29,123,125],{"className":124,"ariaHidden":88},[87],[29,126,128,132],{"className":127},[92],[29,129],{"className":130,"style":131},[96],"height:0.6944em;",[29,133,119],{"className":134,"style":135},[101,102],"margin-right:0.0197em;",") from the start of the curve:\n",[29,138,140,172],{"className":139},[61],[29,141,143],{"className":142},[65],[67,144,145],{"xmlns":69},[71,146,147,169],{},[74,148,149,151,153,157,160,163,165],{},[77,150,79],{},[77,152,119],{},[154,155,156],"mo",{},"=",[77,158,159],{},"R",[77,161,162],{},"L",[154,164,156],{},[166,167,168],"mtext",{},"Constant",[81,170,171],{"encoding":83},"rl = RL = \\text{Constant}",[29,173,175,199,222],{"className":174,"ariaHidden":88},[87],[29,176,178,181,184,187,192,196],{"className":177},[92],[29,179],{"className":180,"style":131},[96],[29,182,79],{"className":183,"style":103},[101,102],[29,185,119],{"className":186,"style":135},[101,102],[29,188],{"className":189,"style":191},[190],"mspace","margin-right:0.2778em;",[29,193,156],{"className":194},[195],"mrel",[29,197],{"className":198,"style":191},[190],[29,200,202,206,210,213,216,219],{"className":201},[92],[29,203],{"className":204,"style":205},[96],"height:0.6833em;",[29,207,159],{"className":208,"style":209},[101,102],"margin-right:0.0077em;",[29,211,162],{"className":212},[101,102],[29,214],{"className":215,"style":191},[190],[29,217,156],{"className":218},[195],[29,220],{"className":221,"style":191},[190],[29,223,225,228],{"className":224},[92],[29,226],{"className":227,"style":205},[96],[29,229,232],{"className":230},[101,231],"text",[29,233,168],{"className":234},[101]," ",[29,237,238],{},"19",".\nWhere:",[241,242,243,275],"ul",{},[244,245,246,274],"li",{},[29,247,249,262],{"className":248},[61],[29,250,252],{"className":251},[65],[67,253,254],{"xmlns":69},[71,255,256,260],{},[74,257,258],{},[77,259,159],{},[81,261,159],{"encoding":83},[29,263,265],{"className":264,"ariaHidden":88},[87],[29,266,268,271],{"className":267},[92],[29,269],{"className":270,"style":205},[96],[29,272,159],{"className":273,"style":209},[101,102]," = Radius of the circular arc to be joined.",[244,276,277,305],{},[29,278,280,293],{"className":279},[61],[29,281,283],{"className":282},[65],[67,284,285],{"xmlns":69},[71,286,287,291],{},[74,288,289],{},[77,290,162],{},[81,292,162],{"encoding":83},[29,294,296],{"className":295,"ariaHidden":88},[87],[29,297,299,302],{"className":298},[92],[29,300],{"className":301,"style":205},[96],[29,303,162],{"className":304},[101,102]," = Total length of the transition curve.",[11,307,309],{"id":308},"mathematical-principles","Mathematical Principles",[311,312,314,315,343],"h3",{"id":313},"_1-the-length-of-transition-lll","1. The Length of Transition (",[29,316,318,331],{"className":317},[61],[29,319,321],{"className":320},[65],[67,322,323],{"xmlns":69},[71,324,325,329],{},[74,326,327],{},[77,328,162],{},[81,330,162],{"encoding":83},[29,332,334],{"className":333,"ariaHidden":88},[87],[29,335,337,340],{"className":336},[92],[29,338],{"className":339,"style":205},[96],[29,341,162],{"className":342},[101,102],")",[16,345,346,347,378,379,407,408,493,494,235,562,565,566,235,789,565,791],{},"The length is determined by the \"Rate of Change of Radial Acceleration\" (",[29,348,350,364],{"className":349},[61],[29,351,353],{"className":352},[65],[67,354,355],{"xmlns":69},[71,356,357,362],{},[74,358,359],{},[77,360,361],{},"q",[81,363,361],{"encoding":83},[29,365,367],{"className":366,"ariaHidden":88},[87],[29,368,370,374],{"className":369},[92],[29,371],{"className":372,"style":373},[96],"height:0.625em;vertical-align:-0.1944em;",[29,375,361],{"className":376,"style":377},[101,102],"margin-right:0.0359em;","). A comfortable value for ",[29,380,382,395],{"className":381},[61],[29,383,385],{"className":384},[65],[67,386,387],{"xmlns":69},[71,388,389,393],{},[74,390,391],{},[77,392,361],{},[81,394,361],{"encoding":83},[29,396,398],{"className":397,"ariaHidden":88},[87],[29,399,401,404],{"className":400},[92],[29,402],{"className":403,"style":373},[96],[29,405,361],{"className":406,"style":377},[101,102]," is typically ",[29,409,411,436],{"className":410},[61],[29,412,414],{"className":413},[65],[67,415,416],{"xmlns":69},[71,417,418,433],{},[74,419,420,424],{},[421,422,423],"mn",{},"0.3",[425,426,427,430],"msup",{},[166,428,429],{}," m\u002Fs",[421,431,432],{},"3",[81,434,435],{"encoding":83},"0.3\\text{ m\u002Fs}^3",[29,437,439],{"className":438,"ariaHidden":88},[87],[29,440,442,446,449],{"className":441},[92],[29,443],{"className":444,"style":445},[96],"height:1.204em;vertical-align:-0.25em;",[29,447,423],{"className":448},[101],[29,450,452,458],{"className":451},[101],[29,453,455],{"className":454},[101,231],[29,456,429],{"className":457},[101],[29,459,462],{"className":460},[461],"msupsub",[29,463,466],{"className":464},[465],"vlist-t",[29,467,470],{"className":468},[469],"vlist-r",[29,471,475],{"className":472,"style":474},[473],"vlist","height:0.954em;",[29,476,478,483],{"style":477},"top:-3.2029em;margin-right:0.05em;",[29,479],{"className":480,"style":482},[481],"pstrut","height:2.7em;",[29,484,490],{"className":485},[486,487,488,489],"sizing","reset-size6","size3","mtight",[29,491,432],{"className":492},[101,489]," to ",[29,495,497,518],{"className":496},[61],[29,498,500],{"className":499},[65],[67,501,502],{"xmlns":69},[71,503,504,515],{},[74,505,506,509],{},[421,507,508],{},"0.6",[425,510,511,513],{},[166,512,429],{},[421,514,432],{},[81,516,517],{"encoding":83},"0.6\\text{ m\u002Fs}^3",[29,519,521],{"className":520,"ariaHidden":88},[87],[29,522,524,527,530],{"className":523},[92],[29,525],{"className":526,"style":445},[96],[29,528,508],{"className":529},[101],[29,531,533,539],{"className":532},[101],[29,534,536],{"className":535},[101,231],[29,537,429],{"className":538},[101],[29,540,542],{"className":541},[461],[29,543,545],{"className":544},[465],[29,546,548],{"className":547},[469],[29,549,551],{"className":550,"style":474},[473],[29,552,553,556],{"style":477},[29,554],{"className":555,"style":482},[481],[29,557,559],{"className":558},[486,487,488,489],[29,560,432],{"className":561},[101,489],[29,563,564],{},"21",".\n",[29,567,569,608],{"className":568},[61],[29,570,572],{"className":571},[65],[67,573,574],{"xmlns":69},[71,575,576,605],{},[74,577,578,580,582],{},[77,579,162],{},[154,581,156],{},[583,584,585,592],"mfrac",{},[425,586,587,590],{},[77,588,589],{},"V",[421,591,432],{},[74,593,594,601,603],{},[425,595,596,599],{},[421,597,598],{},"3.6",[421,600,432],{},[77,602,159],{},[77,604,361],{},[81,606,607],{"encoding":83},"L = \\frac{V^3}{3.6^3 R q}",[29,609,611,629],{"className":610,"ariaHidden":88},[87],[29,612,614,617,620,623,626],{"className":613},[92],[29,615],{"className":616,"style":205},[96],[29,618,162],{"className":619},[101,102],[29,621],{"className":622,"style":191},[190],[29,624,156],{"className":625},[195],[29,627],{"className":628,"style":191},[190],[29,630,632,636],{"className":631},[92],[29,633],{"className":634,"style":635},[96],"height:1.499em;vertical-align:-0.4811em;",[29,637,639,644,785],{"className":638},[101],[29,640],{"className":641},[642,643],"mopen","nulldelimiter",[29,645,647],{"className":646},[583],[29,648,651,776],{"className":649},[465,650],"vlist-t2",[29,652,654,771],{"className":653},[469],[29,655,658,716,727],{"className":656,"style":657},[473],"height:1.0179em;",[29,659,661,665],{"style":660},"top:-2.655em;",[29,662],{"className":663,"style":664},[481],"height:3em;",[29,666,668],{"className":667},[486,487,488,489],[29,669,671,675,710,713],{"className":670},[101,489],[29,672,674],{"className":673},[101,489],"3.",[29,676,678,682],{"className":677},[101,489],[29,679,681],{"className":680},[101,489],"6",[29,683,685],{"className":684},[461],[29,686,688],{"className":687},[465],[29,689,691],{"className":690},[469],[29,692,695],{"className":693,"style":694},[473],"height:0.7463em;",[29,696,698,702],{"style":697},"top:-2.786em;margin-right:0.0714em;",[29,699],{"className":700,"style":701},[481],"height:2.5em;",[29,703,707],{"className":704},[486,705,706,489],"reset-size3","size1",[29,708,432],{"className":709},[101,489],[29,711,159],{"className":712,"style":209},[101,102,489],[29,714,361],{"className":715,"style":377},[101,102,489],[29,717,719,722],{"style":718},"top:-3.23em;",[29,720],{"className":721,"style":664},[481],[29,723],{"className":724,"style":726},[725],"frac-line","border-bottom-width:0.04em;",[29,728,730,733],{"style":729},"top:-3.394em;",[29,731],{"className":732,"style":664},[481],[29,734,736],{"className":735},[486,487,488,489],[29,737,739],{"className":738},[101,489],[29,740,742,746],{"className":741},[101,489],[29,743,589],{"className":744,"style":745},[101,102,489],"margin-right:0.2222em;",[29,747,749],{"className":748},[461],[29,750,752],{"className":751},[465],[29,753,755],{"className":754},[469],[29,756,759],{"className":757,"style":758},[473],"height:0.8913em;",[29,760,762,765],{"style":761},"top:-2.931em;margin-right:0.0714em;",[29,763],{"className":764,"style":701},[481],[29,766,768],{"className":767},[486,705,706,489],[29,769,432],{"className":770},[101,489],[29,772,775],{"className":773},[774],"vlist-s","​",[29,777,779],{"className":778},[469],[29,780,783],{"className":781,"style":782},[473],"height:0.4811em;",[29,784],{},[29,786],{"className":787},[788,643],"mclose",[29,790,564],{},[792,793,794,795,823],"em",{},"(Where ",[29,796,798,811],{"className":797},[61],[29,799,801],{"className":800},[65],[67,802,803],{"xmlns":69},[71,804,805,809],{},[74,806,807],{},[77,808,589],{},[81,810,589],{"encoding":83},[29,812,814],{"className":813,"ariaHidden":88},[87],[29,815,817,820],{"className":816},[92],[29,818],{"className":819,"style":205},[96],[29,821,589],{"className":822,"style":745},[101,102]," is speed in km\u002Fh).",[311,825,827,828,343],{"id":826},"_2-the-shift-sss","2. The Shift (",[29,829,831,845],{"className":830},[61],[29,832,834],{"className":833},[65],[67,835,836],{"xmlns":69},[71,837,838,843],{},[74,839,840],{},[77,841,842],{},"S",[81,844,842],{"encoding":83},[29,846,848],{"className":847,"ariaHidden":88},[87],[29,849,851,854],{"className":850},[92],[29,852],{"className":853,"style":205},[96],[29,855,842],{"className":856,"style":857},[101,102],"margin-right:0.0576em;",[16,859,860,861,235,1022,32],{},"To accommodate the transition, the circular curve must be \"shifted\" inward away from the tangents:\n",[29,862,864,896],{"className":863},[61],[29,865,867],{"className":866},[65],[67,868,869],{"xmlns":69},[71,870,871,893],{},[74,872,873,875,877],{},[77,874,842],{},[154,876,156],{},[583,878,879,886],{},[425,880,881,883],{},[77,882,162],{},[421,884,885],{},"2",[74,887,888,891],{},[421,889,890],{},"24",[77,892,159],{},[81,894,895],{"encoding":83},"S = \\frac{L^2}{24R}",[29,897,899,917],{"className":898,"ariaHidden":88},[87],[29,900,902,905,908,911,914],{"className":901},[92],[29,903],{"className":904,"style":205},[96],[29,906,842],{"className":907,"style":857},[101,102],[29,909],{"className":910,"style":191},[190],[29,912,156],{"className":913},[195],[29,915],{"className":916,"style":191},[190],[29,918,920,924],{"className":919},[92],[29,921],{"className":922,"style":923},[96],"height:1.3629em;vertical-align:-0.345em;",[29,925,927,930,1019],{"className":926},[101],[29,928],{"className":929},[642,643],[29,931,933],{"className":932},[583],[29,934,936,1010],{"className":935},[465,650],[29,937,939,1007],{"className":938},[469],[29,940,942,959,967],{"className":941,"style":657},[473],[29,943,944,947],{"style":660},[29,945],{"className":946,"style":664},[481],[29,948,950],{"className":949},[486,487,488,489],[29,951,953,956],{"className":952},[101,489],[29,954,890],{"className":955},[101,489],[29,957,159],{"className":958,"style":209},[101,102,489],[29,960,961,964],{"style":718},[29,962],{"className":963,"style":664},[481],[29,965],{"className":966,"style":726},[725],[29,968,969,972],{"style":729},[29,970],{"className":971,"style":664},[481],[29,973,975],{"className":974},[486,487,488,489],[29,976,978],{"className":977},[101,489],[29,979,981,984],{"className":980},[101,489],[29,982,162],{"className":983},[101,102,489],[29,985,987],{"className":986},[461],[29,988,990],{"className":989},[465],[29,991,993],{"className":992},[469],[29,994,996],{"className":995,"style":758},[473],[29,997,998,1001],{"style":761},[29,999],{"className":1000,"style":701},[481],[29,1002,1004],{"className":1003},[486,705,706,489],[29,1005,885],{"className":1006},[101,489],[29,1008,775],{"className":1009},[774],[29,1011,1013],{"className":1012},[469],[29,1014,1017],{"className":1015,"style":1016},[473],"height:0.345em;",[29,1018],{},[29,1020],{"className":1021},[788,643],[29,1023,564],{},[311,1025,1027,1028,343],{"id":1026},"_3-spiral-angle-φphiφ","3. Spiral Angle (",[29,1029,1031,1047],{"className":1030},[61],[29,1032,1034],{"className":1033},[65],[67,1035,1036],{"xmlns":69},[71,1037,1038,1044],{},[74,1039,1040],{},[77,1041,1043],{"mathvariant":1042},"normal","Φ",[81,1045,1046],{"encoding":83},"\\Phi",[29,1048,1050],{"className":1049,"ariaHidden":88},[87],[29,1051,1053,1056],{"className":1052},[92],[29,1054],{"className":1055,"style":205},[96],[29,1057,1043],{"className":1058},[101],[16,1060,1061,1062,235,1200,32],{},"The total angle consumed by the transition:\n",[29,1063,1065,1094],{"className":1064},[61],[29,1066,1068],{"className":1067},[65],[67,1069,1070],{"xmlns":69},[71,1071,1072,1091],{},[74,1073,1074,1076,1078,1088],{},[77,1075,1043],{"mathvariant":1042},[154,1077,156],{},[583,1079,1080,1082],{},[77,1081,162],{},[74,1083,1084,1086],{},[421,1085,885],{},[77,1087,159],{},[166,1089,1090],{}," radians",[81,1092,1093],{"encoding":83},"\\Phi = \\frac{L}{2R} \\text{ radians}",[29,1095,1097,1115],{"className":1096,"ariaHidden":88},[87],[29,1098,1100,1103,1106,1109,1112],{"className":1099},[92],[29,1101],{"className":1102,"style":205},[96],[29,1104,1043],{"className":1105},[101],[29,1107],{"className":1108,"style":191},[190],[29,1110,156],{"className":1111},[195],[29,1113],{"className":1114,"style":191},[190],[29,1116,1118,1122,1194],{"className":1117},[92],[29,1119],{"className":1120,"style":1121},[96],"height:1.2173em;vertical-align:-0.345em;",[29,1123,1125,1128,1191],{"className":1124},[101],[29,1126],{"className":1127},[642,643],[29,1129,1131],{"className":1130},[583],[29,1132,1134,1183],{"className":1133},[465,650],[29,1135,1137,1180],{"className":1136},[469],[29,1138,1141,1158,1166],{"className":1139,"style":1140},[473],"height:0.8723em;",[29,1142,1143,1146],{"style":660},[29,1144],{"className":1145,"style":664},[481],[29,1147,1149],{"className":1148},[486,487,488,489],[29,1150,1152,1155],{"className":1151},[101,489],[29,1153,885],{"className":1154},[101,489],[29,1156,159],{"className":1157,"style":209},[101,102,489],[29,1159,1160,1163],{"style":718},[29,1161],{"className":1162,"style":664},[481],[29,1164],{"className":1165,"style":726},[725],[29,1167,1168,1171],{"style":729},[29,1169],{"className":1170,"style":664},[481],[29,1172,1174],{"className":1173},[486,487,488,489],[29,1175,1177],{"className":1176},[101,489],[29,1178,162],{"className":1179},[101,102,489],[29,1181,775],{"className":1182},[774],[29,1184,1186],{"className":1185},[469],[29,1187,1189],{"className":1188,"style":1016},[473],[29,1190],{},[29,1192],{"className":1193},[788,643],[29,1195,1197],{"className":1196},[101,231],[29,1198,1090],{"className":1199},[101],[29,1201,1202],{},"22",[11,1204,1206],{"id":1205},"field-workflow","Field Workflow",[1208,1209,1210,1214,1495,1499,1713,1717,2276,2280],"steps",{},[311,1211,1213],{"id":1212},"calculate-tangent-length","Calculate Tangent Length",[16,1215,1216,1217,1247,1248,1276,1277,235,1493,32],{},"Using the total deflection angle (",[29,1218,1220,1235],{"className":1219},[61],[29,1221,1223],{"className":1222},[65],[67,1224,1225],{"xmlns":69},[71,1226,1227,1232],{},[74,1228,1229],{},[77,1230,1231],{"mathvariant":1042},"Δ",[81,1233,1234],{"encoding":83},"\\Delta",[29,1236,1238],{"className":1237,"ariaHidden":88},[87],[29,1239,1241,1244],{"className":1240},[92],[29,1242],{"className":1243,"style":205},[96],[29,1245,1231],{"className":1246},[101],") and the Shift (",[29,1249,1251,1264],{"className":1250},[61],[29,1252,1254],{"className":1253},[65],[67,1255,1256],{"xmlns":69},[71,1257,1258,1262],{},[74,1259,1260],{},[77,1261,842],{},[81,1263,842],{"encoding":83},[29,1265,1267],{"className":1266,"ariaHidden":88},[87],[29,1268,1270,1273],{"className":1269},[92],[29,1271],{"className":1272,"style":205},[96],[29,1274,842],{"className":1275,"style":857},[101,102],"):\n",[29,1278,1280,1335],{"className":1279},[61],[29,1281,1283],{"className":1282},[65],[67,1284,1285],{"xmlns":69},[71,1286,1287,1332],{},[74,1288,1289,1292,1294,1298,1300,1303,1305,1307,1310,1313,1315,1317,1320,1322,1324,1326],{},[77,1290,1291],{},"T",[154,1293,156],{},[154,1295,1297],{"stretchy":1296},"false","(",[77,1299,159],{},[154,1301,1302],{},"+",[77,1304,842],{},[154,1306,343],{"stretchy":1296},[77,1308,1309],{},"tan",[154,1311,1312],{},"⁡",[154,1314,1297],{"stretchy":1296},[77,1316,1231],{"mathvariant":1042},[77,1318,1319],{"mathvariant":1042},"\u002F",[421,1321,885],{},[154,1323,343],{"stretchy":1296},[154,1325,1302],{},[583,1327,1328,1330],{},[77,1329,162],{},[421,1331,885],{},[81,1333,1334],{"encoding":83},"T = (R+S) \\tan(\\Delta\u002F2) + \\frac{L}{2}",[29,1336,1338,1357,1380,1419],{"className":1337,"ariaHidden":88},[87],[29,1339,1341,1344,1348,1351,1354],{"className":1340},[92],[29,1342],{"className":1343,"style":205},[96],[29,1345,1291],{"className":1346,"style":1347},[101,102],"margin-right:0.1389em;",[29,1349],{"className":1350,"style":191},[190],[29,1352,156],{"className":1353},[195],[29,1355],{"className":1356,"style":191},[190],[29,1358,1360,1364,1367,1370,1373,1377],{"className":1359},[92],[29,1361],{"className":1362,"style":1363},[96],"height:1em;vertical-align:-0.25em;",[29,1365,1297],{"className":1366},[642],[29,1368,159],{"className":1369,"style":209},[101,102],[29,1371],{"className":1372,"style":745},[190],[29,1374,1302],{"className":1375},[1376],"mbin",[29,1378],{"className":1379,"style":745},[190],[29,1381,1383,1386,1389,1392,1396,1400,1403,1407,1410,1413,1416],{"className":1382},[92],[29,1384],{"className":1385,"style":1363},[96],[29,1387,842],{"className":1388,"style":857},[101,102],[29,1390,343],{"className":1391},[788],[29,1393],{"className":1394,"style":1395},[190],"margin-right:0.1667em;",[29,1397,1309],{"className":1398},[1399],"mop",[29,1401,1297],{"className":1402},[642],[29,1404,1406],{"className":1405},[101],"Δ\u002F2",[29,1408,343],{"className":1409},[788],[29,1411],{"className":1412,"style":745},[190],[29,1414,1302],{"className":1415},[1376],[29,1417],{"className":1418,"style":745},[190],[29,1420,1422,1425],{"className":1421},[92],[29,1423],{"className":1424,"style":1121},[96],[29,1426,1428,1431,1490],{"className":1427},[101],[29,1429],{"className":1430},[642,643],[29,1432,1434],{"className":1433},[583],[29,1435,1437,1482],{"className":1436},[465,650],[29,1438,1440,1479],{"className":1439},[469],[29,1441,1443,1457,1465],{"className":1442,"style":1140},[473],[29,1444,1445,1448],{"style":660},[29,1446],{"className":1447,"style":664},[481],[29,1449,1451],{"className":1450},[486,487,488,489],[29,1452,1454],{"className":1453},[101,489],[29,1455,885],{"className":1456},[101,489],[29,1458,1459,1462],{"style":718},[29,1460],{"className":1461,"style":664},[481],[29,1463],{"className":1464,"style":726},[725],[29,1466,1467,1470],{"style":729},[29,1468],{"className":1469,"style":664},[481],[29,1471,1473],{"className":1472},[486,487,488,489],[29,1474,1476],{"className":1475},[101,489],[29,1477,162],{"className":1478},[101,102,489],[29,1480,775],{"className":1481},[774],[29,1483,1485],{"className":1484},[469],[29,1486,1488],{"className":1487,"style":1016},[473],[29,1489],{},[29,1491],{"className":1492},[788,643],[29,1494,564],{},[311,1496,1498],{"id":1497},"locate-tangent-points","Locate Tangent Points",[16,1500,1501,1502,1530,1531,1561,1562,1638,1639,1709,1710,32],{},"Measure ",[29,1503,1505,1518],{"className":1504},[61],[29,1506,1508],{"className":1507},[65],[67,1509,1510],{"xmlns":69},[71,1511,1512,1516],{},[74,1513,1514],{},[77,1515,1291],{},[81,1517,1291],{"encoding":83},[29,1519,1521],{"className":1520,"ariaHidden":88},[87],[29,1522,1524,1527],{"className":1523},[92],[29,1525],{"className":1526,"style":205},[96],[29,1528,1291],{"className":1529,"style":1347},[101,102]," back from the Intersection Point (",[29,1532,1534,1548],{"className":1533},[61],[29,1535,1537],{"className":1536},[65],[67,1538,1539],{"xmlns":69},[71,1540,1541,1546],{},[74,1542,1543],{},[77,1544,1545],{},"I",[81,1547,1545],{"encoding":83},[29,1549,1551],{"className":1550,"ariaHidden":88},[87],[29,1552,1554,1557],{"className":1553},[92],[29,1555],{"className":1556,"style":205},[96],[29,1558,1545],{"className":1559,"style":1560},[101,102],"margin-right:0.0785em;",") to fix ",[29,1563,1565,1585],{"className":1564},[61],[29,1566,1568],{"className":1567},[65],[67,1569,1570],{"xmlns":69},[71,1571,1572,1582],{},[74,1573,1574],{},[1575,1576,1577,1579],"msub",{},[77,1578,1291],{},[421,1580,1581],{},"1",[81,1583,1584],{"encoding":83},"T_1",[29,1586,1588],{"className":1587,"ariaHidden":88},[87],[29,1589,1591,1595],{"className":1590},[92],[29,1592],{"className":1593,"style":1594},[96],"height:0.8333em;vertical-align:-0.15em;",[29,1596,1598,1601],{"className":1597},[101],[29,1599,1291],{"className":1600,"style":1347},[101,102],[29,1602,1604],{"className":1603},[461],[29,1605,1607,1629],{"className":1606},[465,650],[29,1608,1610,1626],{"className":1609},[469],[29,1611,1614],{"className":1612,"style":1613},[473],"height:0.3011em;",[29,1615,1617,1620],{"style":1616},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[29,1618],{"className":1619,"style":482},[481],[29,1621,1623],{"className":1622},[486,487,488,489],[29,1624,1581],{"className":1625},[101,489],[29,1627,775],{"className":1628},[774],[29,1630,1632],{"className":1631},[469],[29,1633,1636],{"className":1634,"style":1635},[473],"height:0.15em;",[29,1637],{}," (start of spiral) and ",[29,1640,1642,1660],{"className":1641},[61],[29,1643,1645],{"className":1644},[65],[67,1646,1647],{"xmlns":69},[71,1648,1649,1657],{},[74,1650,1651],{},[1575,1652,1653,1655],{},[77,1654,1291],{},[421,1656,885],{},[81,1658,1659],{"encoding":83},"T_2",[29,1661,1663],{"className":1662,"ariaHidden":88},[87],[29,1664,1666,1669],{"className":1665},[92],[29,1667],{"className":1668,"style":1594},[96],[29,1670,1672,1675],{"className":1671},[101],[29,1673,1291],{"className":1674,"style":1347},[101,102],[29,1676,1678],{"className":1677},[461],[29,1679,1681,1701],{"className":1680},[465,650],[29,1682,1684,1698],{"className":1683},[469],[29,1685,1687],{"className":1686,"style":1613},[473],[29,1688,1689,1692],{"style":1616},[29,1690],{"className":1691,"style":482},[481],[29,1693,1695],{"className":1694},[486,487,488,489],[29,1696,885],{"className":1697},[101,489],[29,1699,775],{"className":1700},[774],[29,1702,1704],{"className":1703},[469],[29,1705,1707],{"className":1706,"style":1635},[473],[29,1708],{}," (end of spiral) ",[29,1711,1712],{},"23",[311,1714,1716],{"id":1715},"set-out-by-deflection-angles","Set Out by Deflection Angles",[16,1718,1719,1720,1789,1790,1818,1819,1276,1849,235,2273,32],{},"Set up at ",[29,1721,1723,1740],{"className":1722},[61],[29,1724,1726],{"className":1725},[65],[67,1727,1728],{"xmlns":69},[71,1729,1730,1738],{},[74,1731,1732],{},[1575,1733,1734,1736],{},[77,1735,1291],{},[421,1737,1581],{},[81,1739,1584],{"encoding":83},[29,1741,1743],{"className":1742,"ariaHidden":88},[87],[29,1744,1746,1749],{"className":1745},[92],[29,1747],{"className":1748,"style":1594},[96],[29,1750,1752,1755],{"className":1751},[101],[29,1753,1291],{"className":1754,"style":1347},[101,102],[29,1756,1758],{"className":1757},[461],[29,1759,1761,1781],{"className":1760},[465,650],[29,1762,1764,1778],{"className":1763},[469],[29,1765,1767],{"className":1766,"style":1613},[473],[29,1768,1769,1772],{"style":1616},[29,1770],{"className":1771,"style":482},[481],[29,1773,1775],{"className":1774},[486,487,488,489],[29,1776,1581],{"className":1777},[101,489],[29,1779,775],{"className":1780},[774],[29,1782,1784],{"className":1783},[469],[29,1785,1787],{"className":1786,"style":1635},[473],[29,1788],{},". Use cumulative chord lengths (",[29,1791,1793,1806],{"className":1792},[61],[29,1794,1796],{"className":1795},[65],[67,1797,1798],{"xmlns":69},[71,1799,1800,1804],{},[74,1801,1802],{},[77,1803,119],{},[81,1805,119],{"encoding":83},[29,1807,1809],{"className":1808,"ariaHidden":88},[87],[29,1810,1812,1815],{"className":1811},[92],[29,1813],{"className":1814,"style":131},[96],[29,1816,119],{"className":1817,"style":135},[101,102],") to calculate deflection angles (",[29,1820,1822,1837],{"className":1821},[61],[29,1823,1825],{"className":1824},[65],[67,1826,1827],{"xmlns":69},[71,1828,1829,1834],{},[74,1830,1831],{},[77,1832,1833],{},"θ",[81,1835,1836],{"encoding":83},"\\theta",[29,1838,1840],{"className":1839,"ariaHidden":88},[87],[29,1841,1843,1846],{"className":1842},[92],[29,1844],{"className":1845,"style":131},[96],[29,1847,1833],{"className":1848,"style":103},[101,102],[29,1850,1852,1921],{"className":1851},[61],[29,1853,1855],{"className":1854},[65],[67,1856,1857],{"xmlns":69},[71,1858,1859,1918],{},[74,1860,1861,1863,1866,1882,1884,1887,1890,1892,1894,1896,1902],{},[77,1862,1833],{},[154,1864,1865],{},"≈",[583,1867,1868,1874],{},[425,1869,1870,1872],{},[77,1871,119],{},[421,1873,885],{},[74,1875,1876,1878,1880],{},[421,1877,681],{},[77,1879,159],{},[77,1881,162],{},[166,1883,1090],{},[190,1885],{"width":1886},"1em",[166,1888,1889],{},"or",[190,1891],{"width":1886},[77,1893,1833],{},[154,1895,156],{},[583,1897,1898,1900],{},[77,1899,1043],{"mathvariant":1042},[421,1901,432],{},[425,1903,1904,1916],{},[74,1905,1906,1908,1914],{},[154,1907,1297],{"fence":88},[583,1909,1910,1912],{},[77,1911,119],{},[77,1913,162],{},[154,1915,343],{"fence":88},[421,1917,885],{},[81,1919,1920],{"encoding":83},"\\theta \\approx \\frac{l^2}{6RL} \\text{ radians} \\quad \\text{or} \\quad \\theta = \\frac{\\Phi}{3} \\left( \\frac{l}{L} \\right)^2",[29,1922,1924,1942,2079],{"className":1923,"ariaHidden":88},[87],[29,1925,1927,1930,1933,1936,1939],{"className":1926},[92],[29,1928],{"className":1929,"style":131},[96],[29,1931,1833],{"className":1932,"style":103},[101,102],[29,1934],{"className":1935,"style":191},[190],[29,1937,1865],{"className":1938},[195],[29,1940],{"className":1941,"style":191},[190],[29,1943,1945,1948,2048,2054,2058,2064,2067,2070,2073,2076],{"className":1944},[92],[29,1946],{"className":1947,"style":923},[96],[29,1949,1951,1954,2045],{"className":1950},[101],[29,1952],{"className":1953},[642,643],[29,1955,1957],{"className":1956},[583],[29,1958,1960,2037],{"className":1959},[465,650],[29,1961,1963,2034],{"className":1962},[469],[29,1964,1966,1986,1994],{"className":1965,"style":657},[473],[29,1967,1968,1971],{"style":660},[29,1969],{"className":1970,"style":664},[481],[29,1972,1974],{"className":1973},[486,487,488,489],[29,1975,1977,1980,1983],{"className":1976},[101,489],[29,1978,681],{"className":1979},[101,489],[29,1981,159],{"className":1982,"style":209},[101,102,489],[29,1984,162],{"className":1985},[101,102,489],[29,1987,1988,1991],{"style":718},[29,1989],{"className":1990,"style":664},[481],[29,1992],{"className":1993,"style":726},[725],[29,1995,1996,1999],{"style":729},[29,1997],{"className":1998,"style":664},[481],[29,2000,2002],{"className":2001},[486,487,488,489],[29,2003,2005],{"className":2004},[101,489],[29,2006,2008,2011],{"className":2007},[101,489],[29,2009,119],{"className":2010,"style":135},[101,102,489],[29,2012,2014],{"className":2013},[461],[29,2015,2017],{"className":2016},[465],[29,2018,2020],{"className":2019},[469],[29,2021,2023],{"className":2022,"style":758},[473],[29,2024,2025,2028],{"style":761},[29,2026],{"className":2027,"style":701},[481],[29,2029,2031],{"className":2030},[486,705,706,489],[29,2032,885],{"className":2033},[101,489],[29,2035,775],{"className":2036},[774],[29,2038,2040],{"className":2039},[469],[29,2041,2043],{"className":2042,"style":1016},[473],[29,2044],{},[29,2046],{"className":2047},[788,643],[29,2049,2051],{"className":2050},[101,231],[29,2052,1090],{"className":2053},[101],[29,2055],{"className":2056,"style":2057},[190],"margin-right:1em;",[29,2059,2061],{"className":2060},[101,231],[29,2062,1889],{"className":2063},[101],[29,2065],{"className":2066,"style":2057},[190],[29,2068,1833],{"className":2069,"style":103},[101,102],[29,2071],{"className":2072,"style":191},[190],[29,2074,156],{"className":2075},[195],[29,2077],{"className":2078,"style":191},[190],[29,2080,2082,2086,2154,2157],{"className":2081},[92],[29,2083],{"className":2084,"style":2085},[96],"height:1.4341em;vertical-align:-0.35em;",[29,2087,2089,2092,2151],{"className":2088},[101],[29,2090],{"className":2091},[642,643],[29,2093,2095],{"className":2094},[583],[29,2096,2098,2143],{"className":2097},[465,650],[29,2099,2101,2140],{"className":2100},[469],[29,2102,2104,2118,2126],{"className":2103,"style":1140},[473],[29,2105,2106,2109],{"style":660},[29,2107],{"className":2108,"style":664},[481],[29,2110,2112],{"className":2111},[486,487,488,489],[29,2113,2115],{"className":2114},[101,489],[29,2116,432],{"className":2117},[101,489],[29,2119,2120,2123],{"style":718},[29,2121],{"className":2122,"style":664},[481],[29,2124],{"className":2125,"style":726},[725],[29,2127,2128,2131],{"style":729},[29,2129],{"className":2130,"style":664},[481],[29,2132,2134],{"className":2133},[486,487,488,489],[29,2135,2137],{"className":2136},[101,489],[29,2138,1043],{"className":2139},[101,489],[29,2141,775],{"className":2142},[774],[29,2144,2146],{"className":2145},[469],[29,2147,2149],{"className":2148,"style":1016},[473],[29,2150],{},[29,2152],{"className":2153},[788,643],[29,2155],{"className":2156,"style":1395},[190],[29,2158,2161,2248],{"className":2159},[2160],"minner",[29,2162,2164,2173,2242],{"className":2163},[2160],[29,2165,2169],{"className":2166,"style":2168},[642,2167],"delimcenter","top:0em;",[29,2170,1297],{"className":2171},[2172,706],"delimsizing",[29,2174,2176,2179,2239],{"className":2175},[101],[29,2177],{"className":2178},[642,643],[29,2180,2182],{"className":2181},[583],[29,2183,2185,2231],{"className":2184},[465,650],[29,2186,2188,2228],{"className":2187},[469],[29,2189,2192,2206,2214],{"className":2190,"style":2191},[473],"height:0.8801em;",[29,2193,2194,2197],{"style":660},[29,2195],{"className":2196,"style":664},[481],[29,2198,2200],{"className":2199},[486,487,488,489],[29,2201,2203],{"className":2202},[101,489],[29,2204,162],{"className":2205},[101,102,489],[29,2207,2208,2211],{"style":718},[29,2209],{"className":2210,"style":664},[481],[29,2212],{"className":2213,"style":726},[725],[29,2215,2216,2219],{"style":729},[29,2217],{"className":2218,"style":664},[481],[29,2220,2222],{"className":2221},[486,487,488,489],[29,2223,2225],{"className":2224},[101,489],[29,2226,119],{"className":2227,"style":135},[101,102,489],[29,2229,775],{"className":2230},[774],[29,2232,2234],{"className":2233},[469],[29,2235,2237],{"className":2236,"style":1016},[473],[29,2238],{},[29,2240],{"className":2241},[788,643],[29,2243,2245],{"className":2244,"style":2168},[788,2167],[29,2246,343],{"className":2247},[2172,706],[29,2249,2251],{"className":2250},[461],[29,2252,2254],{"className":2253},[465],[29,2255,2257],{"className":2256},[469],[29,2258,2261],{"className":2259,"style":2260},[473],"height:1.0841em;",[29,2262,2264,2267],{"style":2263},"top:-3.333em;margin-right:0.05em;",[29,2265],{"className":2266,"style":482},[481],[29,2268,2270],{"className":2269},[486,487,488,489],[29,2271,885],{"className":2272},[101,489],[29,2274,2275],{},"7, 22",[311,2277,2279],{"id":2278},"transition-into-circle","Transition into Circle",[16,2281,2282,2283,2356,2357,2426,2427,2463,2464,32],{},"Move to the end of the spiral (",[29,2284,2286,2305],{"className":2285},[61],[29,2287,2289],{"className":2288},[65],[67,2290,2291],{"xmlns":69},[71,2292,2293,2302],{},[74,2294,2295],{},[1575,2296,2297,2300],{},[77,2298,2299],{},"t",[421,2301,1581],{},[81,2303,2304],{"encoding":83},"t_1",[29,2306,2308],{"className":2307,"ariaHidden":88},[87],[29,2309,2311,2315],{"className":2310},[92],[29,2312],{"className":2313,"style":2314},[96],"height:0.7651em;vertical-align:-0.15em;",[29,2316,2318,2321],{"className":2317},[101],[29,2319,2299],{"className":2320},[101,102],[29,2322,2324],{"className":2323},[461],[29,2325,2327,2348],{"className":2326},[465,650],[29,2328,2330,2345],{"className":2329},[469],[29,2331,2333],{"className":2332,"style":1613},[473],[29,2334,2336,2339],{"style":2335},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[29,2337],{"className":2338,"style":482},[481],[29,2340,2342],{"className":2341},[486,487,488,489],[29,2343,1581],{"className":2344},[101,489],[29,2346,775],{"className":2347},[774],[29,2349,2351],{"className":2350},[469],[29,2352,2354],{"className":2353,"style":1635},[473],[29,2355],{},"). Backsight to ",[29,2358,2360,2377],{"className":2359},[61],[29,2361,2363],{"className":2362},[65],[67,2364,2365],{"xmlns":69},[71,2366,2367,2375],{},[74,2368,2369],{},[1575,2370,2371,2373],{},[77,2372,1291],{},[421,2374,1581],{},[81,2376,1584],{"encoding":83},[29,2378,2380],{"className":2379,"ariaHidden":88},[87],[29,2381,2383,2386],{"className":2382},[92],[29,2384],{"className":2385,"style":1594},[96],[29,2387,2389,2392],{"className":2388},[101],[29,2390,1291],{"className":2391,"style":1347},[101,102],[29,2393,2395],{"className":2394},[461],[29,2396,2398,2418],{"className":2397},[465,650],[29,2399,2401,2415],{"className":2400},[469],[29,2402,2404],{"className":2403,"style":1613},[473],[29,2405,2406,2409],{"style":1616},[29,2407],{"className":2408,"style":482},[481],[29,2410,2412],{"className":2411},[486,487,488,489],[29,2413,1581],{"className":2414},[101,489],[29,2416,775],{"className":2417},[774],[29,2419,2421],{"className":2420},[469],[29,2422,2424],{"className":2423,"style":1635},[473],[29,2425],{}," with a \"back-angle\" of ",[29,2428,2430,2450],{"className":2429},[61],[29,2431,2433],{"className":2432},[65],[67,2434,2435],{"xmlns":69},[71,2436,2437,2447],{},[74,2438,2439,2441,2443,2445],{},[421,2440,885],{},[77,2442,1043],{"mathvariant":1042},[77,2444,1319],{"mathvariant":1042},[421,2446,432],{},[81,2448,2449],{"encoding":83},"2\\Phi\u002F3",[29,2451,2453],{"className":2452,"ariaHidden":88},[87],[29,2454,2456,2459],{"className":2455},[92],[29,2457],{"className":2458,"style":1363},[96],[29,2460,2462],{"className":2461},[101],"2Φ\u002F3"," to orient the instrument tangentially to the circular arc ",[29,2465,2466],{},"24, 25",[11,2468,2470],{"id":2469},"step-by-step-example","Step-by-Step Example",[16,2472,2473,2476,2477,2517,2518,2546,2547,2586,2587,32],{},[20,2474,2475],{},"Problem:"," Design a transition for a ",[29,2478,2480,2498],{"className":2479},[61],[29,2481,2483],{"className":2482},[65],[67,2484,2485],{"xmlns":69},[71,2486,2487,2495],{},[74,2488,2489,2492],{},[421,2490,2491],{},"400",[166,2493,2494],{}," m",[81,2496,2497],{"encoding":83},"400\\text{ m}",[29,2499,2501],{"className":2500,"ariaHidden":88},[87],[29,2502,2504,2508,2511],{"className":2503},[92],[29,2505],{"className":2506,"style":2507},[96],"height:0.6444em;",[29,2509,2491],{"className":2510},[101],[29,2512,2514],{"className":2513},[101,231],[29,2515,2494],{"className":2516},[101]," radius curve with a design speed (",[29,2519,2521,2534],{"className":2520},[61],[29,2522,2524],{"className":2523},[65],[67,2525,2526],{"xmlns":69},[71,2527,2528,2532],{},[74,2529,2530],{},[77,2531,589],{},[81,2533,589],{"encoding":83},[29,2535,2537],{"className":2536,"ariaHidden":88},[87],[29,2538,2540,2543],{"className":2539},[92],[29,2541],{"className":2542,"style":205},[96],[29,2544,589],{"className":2545,"style":745},[101,102],") of ",[29,2548,2550,2568],{"className":2549},[61],[29,2551,2553],{"className":2552},[65],[67,2554,2555],{"xmlns":69},[71,2556,2557,2565],{},[74,2558,2559,2562],{},[421,2560,2561],{},"100",[166,2563,2564],{}," km\u002Fh",[81,2566,2567],{"encoding":83},"100\\text{ km\u002Fh}",[29,2569,2571],{"className":2570,"ariaHidden":88},[87],[29,2572,2574,2577,2580],{"className":2573},[92],[29,2575],{"className":2576,"style":1363},[96],[29,2578,2561],{"className":2579},[101],[29,2581,2583],{"className":2582},[101,231],[29,2584,2564],{"className":2585},[101]," and ",[29,2588,2590,2615],{"className":2589},[61],[29,2591,2593],{"className":2592},[65],[67,2594,2595],{"xmlns":69},[71,2596,2597,2612],{},[74,2598,2599,2601,2603,2606],{},[77,2600,361],{},[154,2602,156],{},[421,2604,2605],{},"0.45",[425,2607,2608,2610],{},[166,2609,429],{},[421,2611,432],{},[81,2613,2614],{"encoding":83},"q = 0.45\\text{ m\u002Fs}^3",[29,2616,2618,2636],{"className":2617,"ariaHidden":88},[87],[29,2619,2621,2624,2627,2630,2633],{"className":2620},[92],[29,2622],{"className":2623,"style":373},[96],[29,2625,361],{"className":2626,"style":377},[101,102],[29,2628],{"className":2629,"style":191},[190],[29,2631,156],{"className":2632},[195],[29,2634],{"className":2635,"style":191},[190],[29,2637,2639,2642,2645],{"className":2638},[92],[29,2640],{"className":2641,"style":445},[96],[29,2643,2605],{"className":2644},[101],[29,2646,2648,2654],{"className":2647},[101],[29,2649,2651],{"className":2650},[101,231],[29,2652,429],{"className":2653},[101],[29,2655,2657],{"className":2656},[461],[29,2658,2660],{"className":2659},[465],[29,2661,2663],{"className":2662},[469],[29,2664,2666],{"className":2665,"style":474},[473],[29,2667,2668,2671],{"style":477},[29,2669],{"className":2670,"style":482},[481],[29,2672,2674],{"className":2673},[486,487,488,489],[29,2675,432],{"className":2676},[101,489],[2678,2679,2680,2992,3224,3576],"ol",{},[244,2681,2682,2714,235,2989,32],{},[20,2683,2684,2685,2713],{},"Calculate Length (",[29,2686,2688,2701],{"className":2687},[61],[29,2689,2691],{"className":2690},[65],[67,2692,2693],{"xmlns":69},[71,2694,2695,2699],{},[74,2696,2697],{},[77,2698,162],{},[81,2700,162],{"encoding":83},[29,2702,2704],{"className":2703,"ariaHidden":88},[87],[29,2705,2707,2710],{"className":2706},[92],[29,2708],{"className":2709,"style":205},[96],[29,2711,162],{"className":2712},[101,102],"):",[29,2715,2717,2772],{"className":2716},[61],[29,2718,2720],{"className":2719},[65],[67,2721,2722],{"xmlns":69},[71,2723,2724,2769],{},[74,2725,2726,2728,2730,2755,2757,2760,2762,2764,2767],{},[77,2727,162],{},[154,2729,156],{},[583,2731,2732,2738],{},[425,2733,2734,2736],{},[421,2735,2561],{},[421,2737,432],{},[74,2739,2740,2746,2749,2751,2753],{},[425,2741,2742,2744],{},[421,2743,598],{},[421,2745,432],{},[154,2747,2748],{},"×",[421,2750,2491],{},[154,2752,2748],{},[421,2754,2605],{},[154,2756,156],{},[421,2758,2759],{},"119.07",[166,2761,2494],{},[154,2763,1865],{},[421,2765,2766],{},"120",[166,2768,2494],{},[81,2770,2771],{"encoding":83},"L = \\frac{100^3}{3.6^3 \\times 400 \\times 0.45} = 119.07\\text{ m} \\approx 120\\text{ m}",[29,2773,2775,2793,2950,2974],{"className":2774,"ariaHidden":88},[87],[29,2776,2778,2781,2784,2787,2790],{"className":2777},[92],[29,2779],{"className":2780,"style":205},[96],[29,2782,162],{"className":2783},[101,102],[29,2785],{"className":2786,"style":191},[190],[29,2788,156],{"className":2789},[195],[29,2791],{"className":2792,"style":191},[190],[29,2794,2796,2800,2941,2944,2947],{"className":2795},[92],[29,2797],{"className":2798,"style":2799},[96],"height:1.4213em;vertical-align:-0.4033em;",[29,2801,2803,2806,2938],{"className":2802},[101],[29,2804],{"className":2805},[642,643],[29,2807,2809],{"className":2808},[583],[29,2810,2812,2929],{"className":2811},[465,650],[29,2813,2815,2926],{"className":2814},[469],[29,2816,2818,2873,2881],{"className":2817,"style":657},[473],[29,2819,2820,2823],{"style":660},[29,2821],{"className":2822,"style":664},[481],[29,2824,2826],{"className":2825},[486,487,488,489],[29,2827,2829,2832,2861,2864,2867,2870],{"className":2828},[101,489],[29,2830,674],{"className":2831},[101,489],[29,2833,2835,2838],{"className":2834},[101,489],[29,2836,681],{"className":2837},[101,489],[29,2839,2841],{"className":2840},[461],[29,2842,2844],{"className":2843},[465],[29,2845,2847],{"className":2846},[469],[29,2848,2850],{"className":2849,"style":694},[473],[29,2851,2852,2855],{"style":697},[29,2853],{"className":2854,"style":701},[481],[29,2856,2858],{"className":2857},[486,705,706,489],[29,2859,432],{"className":2860},[101,489],[29,2862,2748],{"className":2863},[1376,489],[29,2865,2491],{"className":2866},[101,489],[29,2868,2748],{"className":2869},[1376,489],[29,2871,2605],{"className":2872},[101,489],[29,2874,2875,2878],{"style":718},[29,2876],{"className":2877,"style":664},[481],[29,2879],{"className":2880,"style":726},[725],[29,2882,2883,2886],{"style":729},[29,2884],{"className":2885,"style":664},[481],[29,2887,2889],{"className":2888},[486,487,488,489],[29,2890,2892,2896],{"className":2891},[101,489],[29,2893,2895],{"className":2894},[101,489],"10",[29,2897,2899,2903],{"className":2898},[101,489],[29,2900,2902],{"className":2901},[101,489],"0",[29,2904,2906],{"className":2905},[461],[29,2907,2909],{"className":2908},[465],[29,2910,2912],{"className":2911},[469],[29,2913,2915],{"className":2914,"style":758},[473],[29,2916,2917,2920],{"style":761},[29,2918],{"className":2919,"style":701},[481],[29,2921,2923],{"className":2922},[486,705,706,489],[29,2924,432],{"className":2925},[101,489],[29,2927,775],{"className":2928},[774],[29,2930,2932],{"className":2931},[469],[29,2933,2936],{"className":2934,"style":2935},[473],"height:0.4033em;",[29,2937],{},[29,2939],{"className":2940},[788,643],[29,2942],{"className":2943,"style":191},[190],[29,2945,156],{"className":2946},[195],[29,2948],{"className":2949,"style":191},[190],[29,2951,2953,2956,2959,2965,2968,2971],{"className":2952},[92],[29,2954],{"className":2955,"style":2507},[96],[29,2957,2759],{"className":2958},[101],[29,2960,2962],{"className":2961},[101,231],[29,2963,2494],{"className":2964},[101],[29,2966],{"className":2967,"style":191},[190],[29,2969,1865],{"className":2970},[195],[29,2972],{"className":2973,"style":191},[190],[29,2975,2977,2980,2983],{"className":2976},[92],[29,2978],{"className":2979,"style":2507},[96],[29,2981,2766],{"className":2982},[101],[29,2984,2986],{"className":2985},[101,231],[29,2987,2494],{"className":2988},[101],[29,2990,2991],{},"26",[244,2993,2994,3025,235,3222,32],{},[20,2995,2996,2997,2713],{},"Calculate Shift (",[29,2998,3000,3013],{"className":2999},[61],[29,3001,3003],{"className":3002},[65],[67,3004,3005],{"xmlns":69},[71,3006,3007,3011],{},[74,3008,3009],{},[77,3010,842],{},[81,3012,842],{"encoding":83},[29,3014,3016],{"className":3015,"ariaHidden":88},[87],[29,3017,3019,3022],{"className":3018},[92],[29,3020],{"className":3021,"style":205},[96],[29,3023,842],{"className":3024,"style":857},[101,102],[29,3026,3028,3067],{"className":3027},[61],[29,3029,3031],{"className":3030},[65],[67,3032,3033],{"xmlns":69},[71,3034,3035,3064],{},[74,3036,3037,3039,3041,3057,3059,3062],{},[77,3038,842],{},[154,3040,156],{},[583,3042,3043,3049],{},[425,3044,3045,3047],{},[421,3046,2766],{},[421,3048,885],{},[74,3050,3051,3053,3055],{},[421,3052,890],{},[154,3054,2748],{},[421,3056,2491],{},[154,3058,156],{},[421,3060,3061],{},"1.5",[166,3063,2494],{},[81,3065,3066],{"encoding":83},"S = \\frac{120^2}{24 \\times 400} = 1.5\\text{ m}",[29,3068,3070,3088,3207],{"className":3069,"ariaHidden":88},[87],[29,3071,3073,3076,3079,3082,3085],{"className":3072},[92],[29,3074],{"className":3075,"style":205},[96],[29,3077,842],{"className":3078,"style":857},[101,102],[29,3080],{"className":3081,"style":191},[190],[29,3083,156],{"className":3084},[195],[29,3086],{"className":3087,"style":191},[190],[29,3089,3091,3094,3198,3201,3204],{"className":3090},[92],[29,3092],{"className":3093,"style":2799},[96],[29,3095,3097,3100,3195],{"className":3096},[101],[29,3098],{"className":3099},[642,643],[29,3101,3103],{"className":3102},[583],[29,3104,3106,3187],{"className":3105},[465,650],[29,3107,3109,3184],{"className":3108},[469],[29,3110,3112,3132,3140],{"className":3111,"style":657},[473],[29,3113,3114,3117],{"style":660},[29,3115],{"className":3116,"style":664},[481],[29,3118,3120],{"className":3119},[486,487,488,489],[29,3121,3123,3126,3129],{"className":3122},[101,489],[29,3124,890],{"className":3125},[101,489],[29,3127,2748],{"className":3128},[1376,489],[29,3130,2491],{"className":3131},[101,489],[29,3133,3134,3137],{"style":718},[29,3135],{"className":3136,"style":664},[481],[29,3138],{"className":3139,"style":726},[725],[29,3141,3142,3145],{"style":729},[29,3143],{"className":3144,"style":664},[481],[29,3146,3148],{"className":3147},[486,487,488,489],[29,3149,3151,3155],{"className":3150},[101,489],[29,3152,3154],{"className":3153},[101,489],"12",[29,3156,3158,3161],{"className":3157},[101,489],[29,3159,2902],{"className":3160},[101,489],[29,3162,3164],{"className":3163},[461],[29,3165,3167],{"className":3166},[465],[29,3168,3170],{"className":3169},[469],[29,3171,3173],{"className":3172,"style":758},[473],[29,3174,3175,3178],{"style":761},[29,3176],{"className":3177,"style":701},[481],[29,3179,3181],{"className":3180},[486,705,706,489],[29,3182,885],{"className":3183},[101,489],[29,3185,775],{"className":3186},[774],[29,3188,3190],{"className":3189},[469],[29,3191,3193],{"className":3192,"style":2935},[473],[29,3194],{},[29,3196],{"className":3197},[788,643],[29,3199],{"className":3200,"style":191},[190],[29,3202,156],{"className":3203},[195],[29,3205],{"className":3206,"style":191},[190],[29,3208,3210,3213,3216],{"className":3209},[92],[29,3211],{"className":3212,"style":2507},[96],[29,3214,3061],{"className":3215},[101],[29,3217,3219],{"className":3218},[101,231],[29,3220,2494],{"className":3221},[101],[29,3223,2991],{},[244,3225,3226,3257,235,3573,32],{},[20,3227,3228,3229,2713],{},"Calculate Spiral Angle (",[29,3230,3232,3245],{"className":3231},[61],[29,3233,3235],{"className":3234},[65],[67,3236,3237],{"xmlns":69},[71,3238,3239,3243],{},[74,3240,3241],{},[77,3242,1043],{"mathvariant":1042},[81,3244,1046],{"encoding":83},[29,3246,3248],{"className":3247,"ariaHidden":88},[87],[29,3249,3251,3254],{"className":3250},[92],[29,3252],{"className":3253,"style":205},[96],[29,3255,1043],{"className":3256},[101],[29,3258,3260,3326],{"className":3259},[61],[29,3261,3263],{"className":3262},[65],[67,3264,3265],{"xmlns":69},[71,3266,3267,3323],{},[74,3268,3269,3271,3273,3285,3287,3290,3293,3295,3303,3312],{},[77,3270,1043],{"mathvariant":1042},[154,3272,156],{},[583,3274,3275,3277],{},[421,3276,2766],{},[74,3278,3279,3281,3283],{},[421,3280,885],{},[154,3282,2748],{},[421,3284,2491],{},[154,3286,156],{},[421,3288,3289],{},"0.15",[166,3291,3292],{}," rad",[154,3294,156],{},[425,3296,3297,3300],{},[421,3298,3299],{},"8",[154,3301,3302],{},"∘",[425,3304,3305,3308],{},[421,3306,3307],{},"35",[154,3309,3311],{"mathvariant":1042,"lspace":3310,"rspace":3310},"0em","′",[425,3313,3314,3317],{},[421,3315,3316],{},"40",[74,3318,3319,3321],{},[154,3320,3311],{"mathvariant":1042},[154,3322,3311],{"mathvariant":1042},[81,3324,3325],{"encoding":83},"\\Phi = \\frac{120}{2 \\times 400} = 0.15\\text{ rad} = 8^\\circ 35' 40''",[29,3327,3329,3347,3438,3462],{"className":3328,"ariaHidden":88},[87],[29,3330,3332,3335,3338,3341,3344],{"className":3331},[92],[29,3333],{"className":3334,"style":205},[96],[29,3336,1043],{"className":3337},[101],[29,3339],{"className":3340,"style":191},[190],[29,3342,156],{"className":3343},[195],[29,3345],{"className":3346,"style":191},[190],[29,3348,3350,3354,3429,3432,3435],{"className":3349},[92],[29,3351],{"className":3352,"style":3353},[96],"height:1.2484em;vertical-align:-0.4033em;",[29,3355,3357,3360,3426],{"className":3356},[101],[29,3358],{"className":3359},[642,643],[29,3361,3363],{"className":3362},[583],[29,3364,3366,3418],{"className":3365},[465,650],[29,3367,3369,3415],{"className":3368},[469],[29,3370,3373,3393,3401],{"className":3371,"style":3372},[473],"height:0.8451em;",[29,3374,3375,3378],{"style":660},[29,3376],{"className":3377,"style":664},[481],[29,3379,3381],{"className":3380},[486,487,488,489],[29,3382,3384,3387,3390],{"className":3383},[101,489],[29,3385,885],{"className":3386},[101,489],[29,3388,2748],{"className":3389},[1376,489],[29,3391,2491],{"className":3392},[101,489],[29,3394,3395,3398],{"style":718},[29,3396],{"className":3397,"style":664},[481],[29,3399],{"className":3400,"style":726},[725],[29,3402,3403,3406],{"style":729},[29,3404],{"className":3405,"style":664},[481],[29,3407,3409],{"className":3408},[486,487,488,489],[29,3410,3412],{"className":3411},[101,489],[29,3413,2766],{"className":3414},[101,489],[29,3416,775],{"className":3417},[774],[29,3419,3421],{"className":3420},[469],[29,3422,3424],{"className":3423,"style":2935},[473],[29,3425],{},[29,3427],{"className":3428},[788,643],[29,3430],{"className":3431,"style":191},[190],[29,3433,156],{"className":3434},[195],[29,3436],{"className":3437,"style":191},[190],[29,3439,3441,3444,3447,3453,3456,3459],{"className":3440},[92],[29,3442],{"className":3443,"style":131},[96],[29,3445,3289],{"className":3446},[101],[29,3448,3450],{"className":3449},[101,231],[29,3451,3292],{"className":3452},[101],[29,3454],{"className":3455,"style":191},[190],[29,3457,156],{"className":3458},[195],[29,3460],{"className":3461,"style":191},[190],[29,3463,3465,3469,3500,3503,3536,3540],{"className":3464},[92],[29,3466],{"className":3467,"style":3468},[96],"height:0.7519em;",[29,3470,3472,3475],{"className":3471},[101],[29,3473,3299],{"className":3474},[101],[29,3476,3478],{"className":3477},[461],[29,3479,3481],{"className":3480},[465],[29,3482,3484],{"className":3483},[469],[29,3485,3488],{"className":3486,"style":3487},[473],"height:0.6741em;",[29,3489,3491,3494],{"style":3490},"top:-3.063em;margin-right:0.05em;",[29,3492],{"className":3493,"style":482},[481],[29,3495,3497],{"className":3496},[486,487,488,489],[29,3498,3302],{"className":3499},[1376,489],[29,3501,432],{"className":3502},[101],[29,3504,3506,3510],{"className":3505},[101],[29,3507,3509],{"className":3508},[101],"5",[29,3511,3513],{"className":3512},[461],[29,3514,3516],{"className":3515},[465],[29,3517,3519],{"className":3518},[469],[29,3520,3522],{"className":3521,"style":3468},[473],[29,3523,3524,3527],{"style":3490},[29,3525],{"className":3526,"style":482},[481],[29,3528,3530],{"className":3529},[486,487,488,489],[29,3531,3533],{"className":3532},[101,489],[29,3534,3311],{"className":3535},[101,489],[29,3537,3539],{"className":3538},[101],"4",[29,3541,3543,3546],{"className":3542},[101],[29,3544,2902],{"className":3545},[101],[29,3547,3549],{"className":3548},[461],[29,3550,3552],{"className":3551},[465],[29,3553,3555],{"className":3554},[469],[29,3556,3558],{"className":3557,"style":3468},[473],[29,3559,3560,3563],{"style":3490},[29,3561],{"className":3562,"style":482},[481],[29,3564,3566],{"className":3565},[486,487,488,489],[29,3567,3569],{"className":3568},[101,489],[29,3570,3572],{"className":3571},[101,489],"′′",[29,3574,3575],{},"27, 28",[244,3577,3578,3692,4016,235,4458,32],{},[20,3579,3580,3581,3653,3654,3691],{},"First Deflection Angle (",[29,3582,3584,3602],{"className":3583},[61],[29,3585,3587],{"className":3586},[65],[67,3588,3589],{"xmlns":69},[71,3590,3591,3599],{},[74,3592,3593],{},[1575,3594,3595,3597],{},[77,3596,1833],{},[421,3598,1581],{},[81,3600,3601],{"encoding":83},"\\theta_1",[29,3603,3605],{"className":3604,"ariaHidden":88},[87],[29,3606,3608,3612],{"className":3607},[92],[29,3609],{"className":3610,"style":3611},[96],"height:0.8444em;vertical-align:-0.15em;",[29,3613,3615,3618],{"className":3614},[101],[29,3616,1833],{"className":3617,"style":103},[101,102],[29,3619,3621],{"className":3620},[461],[29,3622,3624,3645],{"className":3623},[465,650],[29,3625,3627,3642],{"className":3626},[469],[29,3628,3630],{"className":3629,"style":1613},[473],[29,3631,3633,3636],{"style":3632},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[29,3634],{"className":3635,"style":482},[481],[29,3637,3639],{"className":3638},[486,487,488,489],[29,3640,1581],{"className":3641},[101,489],[29,3643,775],{"className":3644},[774],[29,3646,3648],{"className":3647},[469],[29,3649,3651],{"className":3650,"style":1635},[473],[29,3652],{},") for a ",[29,3655,3657,3673],{"className":3656},[61],[29,3658,3660],{"className":3659},[65],[67,3661,3662],{"xmlns":69},[71,3663,3664,3670],{},[74,3665,3666,3668],{},[421,3667,2895],{},[166,3669,2494],{},[81,3671,3672],{"encoding":83},"10\\text{ m}",[29,3674,3676],{"className":3675,"ariaHidden":88},[87],[29,3677,3679,3682,3685],{"className":3678},[92],[29,3680],{"className":3681,"style":2507},[96],[29,3683,2895],{"className":3684},[101],[29,3686,3688],{"className":3687},[101,231],[29,3689,2494],{"className":3690},[101]," chord:",[29,3693,3695,3756],{"className":3694},[61],[29,3696,3698],{"className":3697},[65],[67,3699,3700],{"xmlns":69},[71,3701,3702,3753],{},[74,3703,3704,3706,3708,3714,3716,3722,3729,3740,3742],{},[77,3705,1833],{},[154,3707,156],{},[583,3709,3710,3712],{},[77,3711,1043],{"mathvariant":1042},[421,3713,432],{},[154,3715,156],{},[425,3717,3718,3720],{},[421,3719,885],{},[154,3721,3302],{},[425,3723,3724,3727],{},[421,3725,3726],{},"51",[154,3728,3311],{"mathvariant":1042,"lspace":3310,"rspace":3310},[425,3730,3731,3734],{},[421,3732,3733],{},"53",[74,3735,3736,3738],{},[154,3737,3311],{"mathvariant":1042},[154,3739,3311],{"mathvariant":1042},[154,3741,156],{},[425,3743,3744,3747],{},[421,3745,3746],{},"10313",[74,3748,3749,3751],{},[154,3750,3311],{"mathvariant":1042},[154,3752,3311],{"mathvariant":1042},[81,3754,3755],{"encoding":83},"\\theta = \\frac{\\Phi}{3} = 2^\\circ 51' 53'' = 10313''",[29,3757,3759,3777,3860,3974],{"className":3758,"ariaHidden":88},[87],[29,3760,3762,3765,3768,3771,3774],{"className":3761},[92],[29,3763],{"className":3764,"style":131},[96],[29,3766,1833],{"className":3767,"style":103},[101,102],[29,3769],{"className":3770,"style":191},[190],[29,3772,156],{"className":3773},[195],[29,3775],{"className":3776,"style":191},[190],[29,3778,3780,3783,3851,3854,3857],{"className":3779},[92],[29,3781],{"className":3782,"style":1121},[96],[29,3784,3786,3789,3848],{"className":3785},[101],[29,3787],{"className":3788},[642,643],[29,3790,3792],{"className":3791},[583],[29,3793,3795,3840],{"className":3794},[465,650],[29,3796,3798,3837],{"className":3797},[469],[29,3799,3801,3815,3823],{"className":3800,"style":1140},[473],[29,3802,3803,3806],{"style":660},[29,3804],{"className":3805,"style":664},[481],[29,3807,3809],{"className":3808},[486,487,488,489],[29,3810,3812],{"className":3811},[101,489],[29,3813,432],{"className":3814},[101,489],[29,3816,3817,3820],{"style":718},[29,3818],{"className":3819,"style":664},[481],[29,3821],{"className":3822,"style":726},[725],[29,3824,3825,3828],{"style":729},[29,3826],{"className":3827,"style":664},[481],[29,3829,3831],{"className":3830},[486,487,488,489],[29,3832,3834],{"className":3833},[101,489],[29,3835,1043],{"className":3836},[101,489],[29,3838,775],{"className":3839},[774],[29,3841,3843],{"className":3842},[469],[29,3844,3846],{"className":3845,"style":1016},[473],[29,3847],{},[29,3849],{"className":3850},[788,643],[29,3852],{"className":3853,"style":191},[190],[29,3855,156],{"className":3856},[195],[29,3858],{"className":3859,"style":191},[190],[29,3861,3863,3866,3895,3898,3930,3933,3965,3968,3971],{"className":3862},[92],[29,3864],{"className":3865,"style":3468},[96],[29,3867,3869,3872],{"className":3868},[101],[29,3870,885],{"className":3871},[101],[29,3873,3875],{"className":3874},[461],[29,3876,3878],{"className":3877},[465],[29,3879,3881],{"className":3880},[469],[29,3882,3884],{"className":3883,"style":3487},[473],[29,3885,3886,3889],{"style":3490},[29,3887],{"className":3888,"style":482},[481],[29,3890,3892],{"className":3891},[486,487,488,489],[29,3893,3302],{"className":3894},[1376,489],[29,3896,3509],{"className":3897},[101],[29,3899,3901,3904],{"className":3900},[101],[29,3902,1581],{"className":3903},[101],[29,3905,3907],{"className":3906},[461],[29,3908,3910],{"className":3909},[465],[29,3911,3913],{"className":3912},[469],[29,3914,3916],{"className":3915,"style":3468},[473],[29,3917,3918,3921],{"style":3490},[29,3919],{"className":3920,"style":482},[481],[29,3922,3924],{"className":3923},[486,487,488,489],[29,3925,3927],{"className":3926},[101,489],[29,3928,3311],{"className":3929},[101,489],[29,3931,3509],{"className":3932},[101],[29,3934,3936,3939],{"className":3935},[101],[29,3937,432],{"className":3938},[101],[29,3940,3942],{"className":3941},[461],[29,3943,3945],{"className":3944},[465],[29,3946,3948],{"className":3947},[469],[29,3949,3951],{"className":3950,"style":3468},[473],[29,3952,3953,3956],{"style":3490},[29,3954],{"className":3955,"style":482},[481],[29,3957,3959],{"className":3958},[486,487,488,489],[29,3960,3962],{"className":3961},[101,489],[29,3963,3572],{"className":3964},[101,489],[29,3966],{"className":3967,"style":191},[190],[29,3969,156],{"className":3970},[195],[29,3972],{"className":3973,"style":191},[190],[29,3975,3977,3980,3984],{"className":3976},[92],[29,3978],{"className":3979,"style":3468},[96],[29,3981,3983],{"className":3982},[101],"1031",[29,3985,3987,3990],{"className":3986},[101],[29,3988,432],{"className":3989},[101],[29,3991,3993],{"className":3992},[461],[29,3994,3996],{"className":3995},[465],[29,3997,3999],{"className":3998},[469],[29,4000,4002],{"className":4001,"style":3468},[473],[29,4003,4004,4007],{"style":3490},[29,4005],{"className":4006,"style":482},[481],[29,4008,4010],{"className":4009},[486,487,488,489],[29,4011,4013],{"className":4012},[101,489],[29,4014,3572],{"className":4015},[101,489],[29,4017,4019,4098],{"className":4018},[61],[29,4020,4022],{"className":4021},[65],[67,4023,4024],{"xmlns":69},[71,4025,4026,4095],{},[74,4027,4028,4034,4036,4046,4048,4064,4066,4077,4079,4085],{},[1575,4029,4030,4032],{},[77,4031,1833],{},[421,4033,1581],{},[154,4035,156],{},[425,4037,4038,4040],{},[421,4039,3746],{},[74,4041,4042,4044],{},[154,4043,3311],{"mathvariant":1042},[154,4045,3311],{"mathvariant":1042},[154,4047,2748],{},[425,4049,4050,4062],{},[74,4051,4052,4054,4060],{},[154,4053,1297],{"fence":88},[583,4055,4056,4058],{},[421,4057,2895],{},[421,4059,2766],{},[154,4061,343],{"fence":88},[421,4063,885],{},[154,4065,156],{},[425,4067,4068,4071],{},[421,4069,4070],{},"71.6",[74,4072,4073,4075],{},[154,4074,3311],{"mathvariant":1042},[154,4076,3311],{"mathvariant":1042},[154,4078,1865],{},[425,4080,4081,4083],{},[421,4082,1581],{},[154,4084,3311],{"mathvariant":1042,"lspace":3310,"rspace":3310},[425,4086,4087,4089],{},[421,4088,3154],{},[74,4090,4091,4093],{},[154,4092,3311],{"mathvariant":1042},[154,4094,3311],{"mathvariant":1042},[81,4096,4097],{"encoding":83},"\\theta_1 = 10313'' \\times \\left( \\frac{10}{120} \\right)^2 = 71.6'' \\approx 1' 12''",[29,4099,4101,4156,4207,4334,4385],{"className":4100,"ariaHidden":88},[87],[29,4102,4104,4107,4147,4150,4153],{"className":4103},[92],[29,4105],{"className":4106,"style":3611},[96],[29,4108,4110,4113],{"className":4109},[101],[29,4111,1833],{"className":4112,"style":103},[101,102],[29,4114,4116],{"className":4115},[461],[29,4117,4119,4139],{"className":4118},[465,650],[29,4120,4122,4136],{"className":4121},[469],[29,4123,4125],{"className":4124,"style":1613},[473],[29,4126,4127,4130],{"style":3632},[29,4128],{"className":4129,"style":482},[481],[29,4131,4133],{"className":4132},[486,487,488,489],[29,4134,1581],{"className":4135},[101,489],[29,4137,775],{"className":4138},[774],[29,4140,4142],{"className":4141},[469],[29,4143,4145],{"className":4144,"style":1635},[473],[29,4146],{},[29,4148],{"className":4149,"style":191},[190],[29,4151,156],{"className":4152},[195],[29,4154],{"className":4155,"style":191},[190],[29,4157,4159,4163,4166,4198,4201,4204],{"className":4158},[92],[29,4160],{"className":4161,"style":4162},[96],"height:0.8352em;vertical-align:-0.0833em;",[29,4164,3983],{"className":4165},[101],[29,4167,4169,4172],{"className":4168},[101],[29,4170,432],{"className":4171},[101],[29,4173,4175],{"className":4174},[461],[29,4176,4178],{"className":4177},[465],[29,4179,4181],{"className":4180},[469],[29,4182,4184],{"className":4183,"style":3468},[473],[29,4185,4186,4189],{"style":3490},[29,4187],{"className":4188,"style":482},[481],[29,4190,4192],{"className":4191},[486,487,488,489],[29,4193,4195],{"className":4194},[101,489],[29,4196,3572],{"className":4197},[101,489],[29,4199],{"className":4200,"style":745},[190],[29,4202,2748],{"className":4203},[1376],[29,4205],{"className":4206,"style":745},[190],[29,4208,4210,4214,4325,4328,4331],{"className":4209},[92],[29,4211],{"className":4212,"style":4213},[96],"height:1.404em;vertical-align:-0.35em;",[29,4215,4217,4300],{"className":4216},[2160],[29,4218,4220,4226,4294],{"className":4219},[2160],[29,4221,4223],{"className":4222,"style":2168},[642,2167],[29,4224,1297],{"className":4225},[2172,706],[29,4227,4229,4232,4291],{"className":4228},[101],[29,4230],{"className":4231},[642,643],[29,4233,4235],{"className":4234},[583],[29,4236,4238,4283],{"className":4237},[465,650],[29,4239,4241,4280],{"className":4240},[469],[29,4242,4244,4258,4266],{"className":4243,"style":3372},[473],[29,4245,4246,4249],{"style":660},[29,4247],{"className":4248,"style":664},[481],[29,4250,4252],{"className":4251},[486,487,488,489],[29,4253,4255],{"className":4254},[101,489],[29,4256,2766],{"className":4257},[101,489],[29,4259,4260,4263],{"style":718},[29,4261],{"className":4262,"style":664},[481],[29,4264],{"className":4265,"style":726},[725],[29,4267,4268,4271],{"style":729},[29,4269],{"className":4270,"style":664},[481],[29,4272,4274],{"className":4273},[486,487,488,489],[29,4275,4277],{"className":4276},[101,489],[29,4278,2895],{"className":4279},[101,489],[29,4281,775],{"className":4282},[774],[29,4284,4286],{"className":4285},[469],[29,4287,4289],{"className":4288,"style":1016},[473],[29,4290],{},[29,4292],{"className":4293},[788,643],[29,4295,4297],{"className":4296,"style":2168},[788,2167],[29,4298,343],{"className":4299},[2172,706],[29,4301,4303],{"className":4302},[461],[29,4304,4306],{"className":4305},[465],[29,4307,4309],{"className":4308},[469],[29,4310,4313],{"className":4311,"style":4312},[473],"height:1.054em;",[29,4314,4316,4319],{"style":4315},"top:-3.3029em;margin-right:0.05em;",[29,4317],{"className":4318,"style":482},[481],[29,4320,4322],{"className":4321},[486,487,488,489],[29,4323,885],{"className":4324},[101,489],[29,4326],{"className":4327,"style":191},[190],[29,4329,156],{"className":4330},[195],[29,4332],{"className":4333,"style":191},[190],[29,4335,4337,4340,4344,4376,4379,4382],{"className":4336},[92],[29,4338],{"className":4339,"style":3468},[96],[29,4341,4343],{"className":4342},[101],"71.",[29,4345,4347,4350],{"className":4346},[101],[29,4348,681],{"className":4349},[101],[29,4351,4353],{"className":4352},[461],[29,4354,4356],{"className":4355},[465],[29,4357,4359],{"className":4358},[469],[29,4360,4362],{"className":4361,"style":3468},[473],[29,4363,4364,4367],{"style":3490},[29,4365],{"className":4366,"style":482},[481],[29,4368,4370],{"className":4369},[486,487,488,489],[29,4371,4373],{"className":4372},[101,489],[29,4374,3572],{"className":4375},[101,489],[29,4377],{"className":4378,"style":191},[190],[29,4380,1865],{"className":4381},[195],[29,4383],{"className":4384,"style":191},[190],[29,4386,4388,4391,4423,4426],{"className":4387},[92],[29,4389],{"className":4390,"style":3468},[96],[29,4392,4394,4397],{"className":4393},[101],[29,4395,1581],{"className":4396},[101],[29,4398,4400],{"className":4399},[461],[29,4401,4403],{"className":4402},[465],[29,4404,4406],{"className":4405},[469],[29,4407,4409],{"className":4408,"style":3468},[473],[29,4410,4411,4414],{"style":3490},[29,4412],{"className":4413,"style":482},[481],[29,4415,4417],{"className":4416},[486,487,488,489],[29,4418,4420],{"className":4419},[101,489],[29,4421,3311],{"className":4422},[101,489],[29,4424,1581],{"className":4425},[101],[29,4427,4429,4432],{"className":4428},[101],[29,4430,885],{"className":4431},[101],[29,4433,4435],{"className":4434},[461],[29,4436,4438],{"className":4437},[465],[29,4439,4441],{"className":4440},[469],[29,4442,4444],{"className":4443,"style":3468},[473],[29,4445,4446,4449],{"style":3490},[29,4447],{"className":4448,"style":482},[481],[29,4450,4452],{"className":4451},[486,487,488,489],[29,4453,4455],{"className":4454},[101,489],[29,4456,3572],{"className":4457},[101,489],[29,4459,4460],{},"28, 29",[11,4462,4464],{"id":4463},"practical-tips","Practical Tips",[241,4466,4467,4572,4676],{},[244,4468,4469,4472,4473,2586,4538,4568,4569,32],{},[20,4470,4471],{},"Use Highway Tables:"," Formulas for clothoids involve complex series expansions. In practice, use \"Highway Transition Curve Tables\" to find values for ",[29,4474,4476,4503],{"className":4475},[61],[29,4477,4479],{"className":4478},[65],[67,4480,4481],{"xmlns":69},[71,4482,4483,4500],{},[74,4484,4485,4488,4491,4494,4496,4498],{},[77,4486,4487],{},"X",[154,4489,4490],{"separator":88},",",[77,4492,4493],{},"Y",[154,4495,4490],{"separator":88},[77,4497,842],{},[154,4499,4490],{"separator":88},[81,4501,4502],{"encoding":83},"X, Y, S,",[29,4504,4506],{"className":4505,"ariaHidden":88},[87],[29,4507,4509,4513,4516,4520,4523,4526,4529,4532,4535],{"className":4508},[92],[29,4510],{"className":4511,"style":4512},[96],"height:0.8778em;vertical-align:-0.1944em;",[29,4514,4487],{"className":4515,"style":1560},[101,102],[29,4517,4490],{"className":4518},[4519],"mpunct",[29,4521],{"className":4522,"style":1395},[190],[29,4524,4493],{"className":4525,"style":745},[101,102],[29,4527,4490],{"className":4528},[4519],[29,4530],{"className":4531,"style":1395},[190],[29,4533,842],{"className":4534,"style":857},[101,102],[29,4536,4490],{"className":4537},[4519],[29,4539,4541,4555],{"className":4540},[61],[29,4542,4544],{"className":4543},[65],[67,4545,4546],{"xmlns":69},[71,4547,4548,4553],{},[74,4549,4550],{},[77,4551,4552],{},"C",[81,4554,4552],{"encoding":83},[29,4556,4558],{"className":4557,"ariaHidden":88},[87],[29,4559,4561,4564],{"className":4560},[92],[29,4562],{"className":4563,"style":205},[96],[29,4565,4552],{"className":4566,"style":4567},[101,102],"margin-right:0.0715em;"," based on radius and speed ",[29,4570,4571],{},"27, 30",[244,4573,4574,4577,4578,4606,4607,4640,4641,4672,4673,32],{},[20,4575,4576],{},"The 1\u002F3 Rule:"," The deflection angle ",[29,4579,4581,4594],{"className":4580},[61],[29,4582,4584],{"className":4583},[65],[67,4585,4586],{"xmlns":69},[71,4587,4588,4592],{},[74,4589,4590],{},[77,4591,1833],{},[81,4593,1836],{"encoding":83},[29,4595,4597],{"className":4596,"ariaHidden":88},[87],[29,4598,4600,4603],{"className":4599},[92],[29,4601],{"className":4602,"style":131},[96],[29,4604,1833],{"className":4605,"style":103},[101,102]," to any point on the spiral is approximately ",[29,4608,4610,4628],{"className":4609},[61],[29,4611,4613],{"className":4612},[65],[67,4614,4615],{"xmlns":69},[71,4616,4617,4625],{},[74,4618,4619,4621,4623],{},[421,4620,1581],{},[77,4622,1319],{"mathvariant":1042},[421,4624,432],{},[81,4626,4627],{"encoding":83},"1\u002F3",[29,4629,4631],{"className":4630,"ariaHidden":88},[87],[29,4632,4634,4637],{"className":4633},[92],[29,4635],{"className":4636,"style":1363},[96],[29,4638,4627],{"className":4639},[101]," of the angle consumed by the curve (",[29,4642,4644,4659],{"className":4643},[61],[29,4645,4647],{"className":4646},[65],[67,4648,4649],{"xmlns":69},[71,4650,4651,4656],{},[74,4652,4653],{},[77,4654,4655],{},"ϕ",[81,4657,4658],{"encoding":83},"\\phi",[29,4660,4662],{"className":4661,"ariaHidden":88},[87],[29,4663,4665,4669],{"className":4664},[92],[29,4666],{"className":4667,"style":4668},[96],"height:0.8889em;vertical-align:-0.1944em;",[29,4670,4655],{"className":4671},[101,102],") at that point ",[29,4674,4675],{},"22, 31",[244,4677,4678,4681,4682,4716,4717,4747,4748,32],{},[20,4679,4680],{},"Check the Back-Angle:"," When moving from the spiral to the circle, the back-angle to the origin is exactly ",[29,4683,4685,4701],{"className":4684},[61],[29,4686,4688],{"className":4687},[65],[67,4689,4690],{"xmlns":69},[71,4691,4692,4698],{},[74,4693,4694,4696],{},[421,4695,885],{},[77,4697,1833],{},[81,4699,4700],{"encoding":83},"2\\theta",[29,4702,4704],{"className":4703,"ariaHidden":88},[87],[29,4705,4707,4710,4713],{"className":4706},[92],[29,4708],{"className":4709,"style":131},[96],[29,4711,885],{"className":4712},[101],[29,4714,1833],{"className":4715,"style":103},[101,102]," if the spiral is short, but must be corrected by a value ",[29,4718,4720,4734],{"className":4719},[61],[29,4721,4723],{"className":4722},[65],[67,4724,4725],{"xmlns":69},[71,4726,4727,4732],{},[74,4728,4729],{},[77,4730,4731],{},"N",[81,4733,4731],{"encoding":83},[29,4735,4737],{"className":4736,"ariaHidden":88},[87],[29,4738,4740,4743],{"className":4739},[92],[29,4741],{"className":4742,"style":205},[96],[29,4744,4731],{"className":4745,"style":4746},[101,102],"margin-right:0.109em;"," for long spirals ",[29,4749,2466],{},[11,4751,4753],{"id":4752},"common-mistakes","Common Mistakes",[241,4755,4756,4829],{},[244,4757,4758,4761,4762,4765,4766,4825,4826,32],{},[20,4759,4760],{},"Confusing the Spiral types:"," While the Cubic Parabola and Cubic Spiral are similar, the ",[20,4763,4764],{},"Clothoid"," is the only one where ",[29,4767,4769,4789],{"className":4768},[61],[29,4770,4772],{"className":4771},[65],[67,4773,4774],{"xmlns":69},[71,4775,4776,4786],{},[74,4777,4778,4780,4782,4784],{},[77,4779,79],{},[77,4781,119],{},[154,4783,156],{},[166,4785,168],{},[81,4787,4788],{"encoding":83},"rl = \\text{Constant}",[29,4790,4792,4813],{"className":4791,"ariaHidden":88},[87],[29,4793,4795,4798,4801,4804,4807,4810],{"className":4794},[92],[29,4796],{"className":4797,"style":131},[96],[29,4799,79],{"className":4800,"style":103},[101,102],[29,4802,119],{"className":4803,"style":135},[101,102],[29,4805],{"className":4806,"style":191},[190],[29,4808,156],{"className":4809},[195],[29,4811],{"className":4812,"style":191},[190],[29,4814,4816,4819],{"className":4815},[92],[29,4817],{"className":4818,"style":205},[96],[29,4820,4822],{"className":4821},[101,231],[29,4823,168],{"className":4824},[101]," is strictly maintained for large angles ",[29,4827,4828],{},"19, 32",[244,4830,4831,4834,4835,32],{},[20,4832,4833],{},"Inadequate Super-elevation:"," If the transition is too short, the rate of increase of super-elevation will be too steep, creating a \"twist\" in the road surface ",[29,4836,4837],{},"20, 21",[11,4839,4841],{"id":4840},"faq","FAQ",[4843,4844,4845,4930,4993],"accordion",{},[4846,4847,4849],"accordion-item",{"label":4848},"What is the difference between a Cubic Parabola and a Clothoid?",[16,4850,4851,4852,4927,4928,32],{},"The Cubic Parabola is an approximation used for railway work where angles are small (",[29,4853,4855,4876],{"className":4854},[61],[29,4856,4858],{"className":4857},[65],[67,4859,4860],{"xmlns":69},[71,4861,4862,4873],{},[74,4863,4864,4867],{},[154,4865,4866],{},"\u003C",[425,4868,4869,4871],{},[421,4870,3539],{},[154,4872,3302],{},[81,4874,4875],{"encoding":83},"\u003C 4^\\circ",[29,4877,4879,4892],{"className":4878,"ariaHidden":88},[87],[29,4880,4882,4886,4889],{"className":4881},[92],[29,4883],{"className":4884,"style":4885},[96],"height:0.5782em;vertical-align:-0.0391em;",[29,4887,4866],{"className":4888},[195],[29,4890],{"className":4891,"style":191},[190],[29,4893,4895,4898],{"className":4894},[92],[29,4896],{"className":4897,"style":3487},[96],[29,4899,4901,4904],{"className":4900},[101],[29,4902,3539],{"className":4903},[101],[29,4905,4907],{"className":4906},[461],[29,4908,4910],{"className":4909},[465],[29,4911,4913],{"className":4912},[469],[29,4914,4916],{"className":4915,"style":3487},[473],[29,4917,4918,4921],{"style":3490},[29,4919],{"className":4920,"style":482},[481],[29,4922,4924],{"className":4923},[486,487,488,489],[29,4925,3302],{"className":4926},[1376,489],"). The Clothoid is the mathematically exact transition for high-speed road design ",[29,4929,4828],{},[4846,4931,4933],{"label":4932},"When is a transition curve not required?",[16,4934,4935,4936,4989,4990,32],{},"If the radius is very large (e.g., ",[29,4937,4939,4959],{"className":4938},[61],[29,4940,4942],{"className":4941},[65],[67,4943,4944],{"xmlns":69},[71,4945,4946,4956],{},[74,4947,4948,4951,4954],{},[154,4949,4950],{},">",[421,4952,4953],{},"2000",[166,4955,2494],{},[81,4957,4958],{"encoding":83},"> 2000\\text{ m}",[29,4960,4962,4974],{"className":4961,"ariaHidden":88},[87],[29,4963,4965,4968,4971],{"className":4964},[92],[29,4966],{"className":4967,"style":4885},[96],[29,4969,4950],{"className":4970},[195],[29,4972],{"className":4973,"style":191},[190],[29,4975,4977,4980,4983],{"className":4976},[92],[29,4978],{"className":4979,"style":2507},[96],[29,4981,4953],{"className":4982},[101],[29,4984,4986],{"className":4985},[101,231],[29,4987,2494],{"className":4988},[101],"), the lateral acceleration is so small that a transition provides no practical benefit ",[29,4991,4992],{},"33",[4846,4994,4996],{"label":4995},"What is 'Shortt's Factor'?",[16,4997,4998,4999,5065,5066,32],{},"It is an empirical value (",[29,5000,5002,5021],{"className":5001},[61],[29,5003,5005],{"className":5004},[65],[67,5006,5007],{"xmlns":69},[71,5008,5009,5019],{},[74,5010,5011,5013],{},[421,5012,423],{},[425,5014,5015,5017],{},[166,5016,429],{},[421,5018,432],{},[81,5020,435],{"encoding":83},[29,5022,5024],{"className":5023,"ariaHidden":88},[87],[29,5025,5027,5030,5033],{"className":5026},[92],[29,5028],{"className":5029,"style":445},[96],[29,5031,423],{"className":5032},[101],[29,5034,5036,5042],{"className":5035},[101],[29,5037,5039],{"className":5038},[101,231],[29,5040,429],{"className":5041},[101],[29,5043,5045],{"className":5044},[461],[29,5046,5048],{"className":5047},[465],[29,5049,5051],{"className":5050},[469],[29,5052,5054],{"className":5053,"style":474},[473],[29,5055,5056,5059],{"style":477},[29,5057],{"className":5058,"style":482},[481],[29,5060,5062],{"className":5061},[486,487,488,489],[29,5063,432],{"className":5064},[101,489],") for the rate of change of radial acceleration, originally devised for passenger comfort on railways ",[29,5067,564],{},[11,5069,5071],{"id":5070},"conclusion","Conclusion",[16,5073,5074],{},"The Clothoid Spiral is a masterclass in engineering safety. By linking linear straights to circular curves through a variable-radius spiral, surveyors ensure that high-speed transit remains a smooth, controlled experience.",[11,5076,5078],{"id":5077},"references","References",[16,5080,5081,5082,5085,5086,32],{},"Schofield, W. (2001). ",[792,5083,5084],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann. ",[29,5087,5088],{},"16, 17",{"title":5090,"searchDepth":5091,"depth":5091,"links":5092},"",2,[5093,5094,5095,5096,5105,5111,5112,5113,5114,5115,5116],{"id":13,"depth":5091,"text":14},{"id":35,"depth":5091,"text":36},{"id":53,"depth":5091,"text":54},{"id":308,"depth":5091,"text":309,"children":5097},[5098,5101,5103],{"id":313,"depth":5099,"text":5100},3,"1. The Length of Transition (LLL)",{"id":826,"depth":5099,"text":5102},"2. The Shift (SSS)",{"id":1026,"depth":5099,"text":5104},"3. Spiral Angle (Φ\\PhiΦ)",{"id":1205,"depth":5091,"text":1206,"children":5106},[5107,5108,5109,5110],{"id":1212,"depth":5099,"text":1213},{"id":1497,"depth":5099,"text":1498},{"id":1715,"depth":5099,"text":1716},{"id":2278,"depth":5099,"text":2279},{"id":2469,"depth":5091,"text":2470},{"id":4463,"depth":5091,"text":4464},{"id":4752,"depth":5091,"text":4753},{"id":4840,"depth":5091,"text":4841},{"id":5070,"depth":5091,"text":5071},{"id":5077,"depth":5091,"text":5078},"Road Design","A technical guide to the geometry, design, and setting out of clothoid transition curves for high-speed road and rail projects.","md","\u002Fimages\u002Fclothoid-spiral.jpg",{"authorCredentials":5122,"commentCount":5123,"tags":5124,"relatedTools":5127,"relatedArticles":5130,"sources":5133},"P.Eng",0,[5125,22,5117,4764,5126],"Curves","Setting Out",[5128,5129],"Total Station","Transition Tables",[5131,5132],"rankine-method-circular-curves","vertical-parabolic-curves",[5134],{"title":5135,"type":5136,"note":5137},"Engineering Surveying (Fifth Edition)","Book","Primary source for clothoid formulas and radial acceleration design.",true,"\u002Fblog\u002Fclothoid-transition-curves","2024-03-26",20,{"title":5,"description":5118},{"loc":5139},"blog\u002F7.clothoid-transition-curves","t20TUYZdqBFrJrnhSwLKHb9iB_lWYq-f8_zHaQ8XDzE",[5147,5152],{"title":5148,"path":5149,"stem":5150,"description":5151,"children":-1},"The Gyro-Theodolite: Mastering True North Underground","\u002Fblog\u002Fgyro-theodolite-underground-orientation","blog\u002F6.gyro-theodolite-underground-orientation","A comprehensive technical guide to the theory, observation, and calculation of true azimuths using suspended gyroscopes in tunnel and mine surveying.",{"title":5153,"path":5154,"stem":5155,"description":5156,"children":-1},"Reciprocal Levelling: Defeating Earth Curvature and Refraction","\u002Fblog\u002Freciprocal-levelling-and-curvature","blog\u002F8.reciprocal-levelling-and-curvature","A technical deep dive into reciprocal levelling techniques to eliminate instrumental and environmental errors across wide obstacles like rivers and valleys.",1783851951386]