[{"data":1,"prerenderedAt":2305},["ShallowReactive",2],{"blog-\u002Fblog\u002Fvertical-parabolic-curves-designing-for-sight-distance-and-safety":3,"blog-surround-\u002Fblog\u002Fvertical-parabolic-curves-designing-for-sight-distance-and-safety":2294},{"id":4,"title":5,"author":6,"body":7,"category":2271,"description":2272,"extension":2273,"image":2274,"meta":2275,"navigation":2286,"path":2287,"publishedAt":2288,"readTime":2289,"seo":2290,"sitemap":2291,"stem":2292,"__hash__":2293},"blog\u002Fblog\u002F4.vertical-parabolic-curves-designing-for-sight-distance-and-safety.md","Vertical Parabolic Curves: Designing for Sight Distance and Safety","Surveyiiing Expert",{"type":8,"value":9,"toc":2249},"minimark",[10,15,33,37,94,98,360,364,550,767,1073,1077,1369,1373,1785,1789,1809,1813,1995,1999,2141,2145,2235,2239],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,28,29,32],"p",{},"Vertical curves connect two different gradients to provide a gradual change in slope, ensuring driver comfort, adequate drainage, and safe sight distances ",[20,21,22],"span",{},"13, 29",". In engineering, the ",[25,26,27],"strong",{},"parabola"," is the preferred shape because it has a constant rate of change of grade ",[20,30,31],{},"30",".",[11,34,36],{"id":35},"why-this-matters","Why This Matters",[16,38,39,40,43,44,90,91,32],{},"A vertical curve that is too short creates a \"hidden dip\" or a crest that blocks the driver’s view of oncoming traffic or obstructions ",[20,41,42],{},"29",". Designing to the correct ",[20,45,48,70],{"className":46},[47],"katex",[20,49,52],{"className":50},[51],"katex-mathml",[53,54,56],"math",{"xmlns":55},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[57,58,59,66],"semantics",{},[60,61,62],"mrow",{},[63,64,65],"mi",{},"K",[67,68,65],"annotation",{"encoding":69},"application\u002Fx-tex",[20,71,75],{"className":72,"ariaHidden":74},[73],"katex-html","true",[20,76,79,84],{"className":77},[78],"base",[20,80],{"className":81,"style":83},[82],"strut","height:0.6833em;",[20,85,65],{"className":86,"style":89},[87,88],"mord","mathnormal","margin-right:0.0715em;","-value ensures safe \"Stopping Sight Distance\" (SSD) ",[20,92,93],{},"29, 31",[11,95,97],{"id":96},"theory","Theory",[16,99,100,101,103,104,216,217,245,246,275,276,357,358,32],{},"Because the gradients in road design are very shallow, the vertical parabola is approximated such that horizontal distances are used instead of distances along the curve ",[20,102,31],{},". The basic equation is:\n",[20,105,107,139],{"className":106},[47],[20,108,110],{"className":109},[51],[53,111,112],{"xmlns":55},[57,113,114,136],{},[60,115,116,119,123,126],{},[63,117,118],{},"y",[120,121,122],"mo",{},"=",[63,124,125],{},"C",[127,128,129,132],"msup",{},[63,130,131],{},"l",[133,134,135],"mn",{},"2",[67,137,138],{"encoding":69},"y = Cl^2",[20,140,142,165],{"className":141,"ariaHidden":74},[73],[20,143,145,149,153,158,162],{"className":144},[78],[20,146],{"className":147,"style":148},[82],"height:0.625em;vertical-align:-0.1944em;",[20,150,118],{"className":151,"style":152},[87,88],"margin-right:0.0359em;",[20,154],{"className":155,"style":157},[156],"mspace","margin-right:0.2778em;",[20,159,122],{"className":160},[161],"mrel",[20,163],{"className":164,"style":157},[156],[20,166,168,172,175],{"className":167},[78],[20,169],{"className":170,"style":171},[82],"height:0.8141em;",[20,173,125],{"className":174,"style":89},[87,88],[20,176,178,182],{"className":177},[87],[20,179,131],{"className":180,"style":181},[87,88],"margin-right:0.0197em;",[20,183,186],{"className":184},[185],"msupsub",[20,187,190],{"className":188},[189],"vlist-t",[20,191,194],{"className":192},[193],"vlist-r",[20,195,198],{"className":196,"style":171},[197],"vlist",[20,199,201,206],{"style":200},"top:-3.063em;margin-right:0.05em;",[20,202],{"className":203,"style":205},[204],"pstrut","height:2.7em;",[20,207,213],{"className":208},[209,210,211,212],"sizing","reset-size6","size3","mtight",[20,214,135],{"className":215},[87,212],"\nWhere ",[20,218,220,233],{"className":219},[47],[20,221,223],{"className":222},[51],[53,224,225],{"xmlns":55},[57,226,227,231],{},[60,228,229],{},[63,230,118],{},[67,232,118],{"encoding":69},[20,234,236],{"className":235,"ariaHidden":74},[73],[20,237,239,242],{"className":238},[78],[20,240],{"className":241,"style":148},[82],[20,243,118],{"className":244,"style":152},[87,88]," is the vertical offset from the tangent at distance ",[20,247,249,262],{"className":248},[47],[20,250,252],{"className":251},[51],[53,253,254],{"xmlns":55},[57,255,256,260],{},[60,257,258],{},[63,259,131],{},[67,261,131],{"encoding":69},[20,263,265],{"className":264,"ariaHidden":74},[73],[20,266,268,272],{"className":267},[78],[20,269],{"className":270,"style":271},[82],"height:0.6944em;",[20,273,131],{"className":274,"style":181},[87,88]," from the start of the curve (",[20,277,279,300],{"className":278},[47],[20,280,282],{"className":281},[51],[53,283,284],{"xmlns":55},[57,285,286,297],{},[60,287,288],{},[289,290,291,294],"msub",{},[63,292,293],{},"T",[133,295,296],{},"1",[67,298,299],{"encoding":69},"T_1",[20,301,303],{"className":302,"ariaHidden":74},[73],[20,304,306,310],{"className":305},[78],[20,307],{"className":308,"style":309},[82],"height:0.8333em;vertical-align:-0.15em;",[20,311,313,317],{"className":312},[87],[20,314,293],{"className":315,"style":316},[87,88],"margin-right:0.1389em;",[20,318,320],{"className":319},[185],[20,321,324,348],{"className":322},[189,323],"vlist-t2",[20,325,327,343],{"className":326},[193],[20,328,331],{"className":329,"style":330},[197],"height:0.3011em;",[20,332,334,337],{"style":333},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[20,335],{"className":336,"style":205},[204],[20,338,340],{"className":339},[209,210,211,212],[20,341,296],{"className":342},[87,212],[20,344,347],{"className":345},[346],"vlist-s","​",[20,349,351],{"className":350},[193],[20,352,355],{"className":353,"style":354},[197],"height:0.15em;",[20,356],{},") ",[20,359,31],{},[11,361,363],{"id":362},"mathematical-principles","Mathematical Principles",[16,365,366,367,401,402,545,546,549],{},"The ",[25,368,369,370,400],{},"Central Offset (",[20,371,373,387],{"className":372},[47],[20,374,376],{"className":375},[51],[53,377,378],{"xmlns":55},[57,379,380,385],{},[60,381,382],{},[63,383,384],{},"Y",[67,386,384],{"encoding":69},[20,388,390],{"className":389,"ariaHidden":74},[73],[20,391,393,396],{"className":392},[78],[20,394],{"className":395,"style":83},[82],[20,397,384],{"className":398,"style":399},[87,88],"margin-right:0.2222em;",")"," is the most important value:\n",[20,403,405,435],{"className":404},[47],[20,406,408],{"className":407},[51],[53,409,410],{"xmlns":55},[57,411,412,432],{},[60,413,414,416,418],{},[63,415,384],{},[120,417,122],{},[419,420,421,429],"mfrac",{},[60,422,423,426],{},[63,424,425],{},"A",[63,427,428],{},"L",[133,430,431],{},"800",[67,433,434],{"encoding":69},"Y = \\frac{AL}{800}",[20,436,438,456],{"className":437,"ariaHidden":74},[73],[20,439,441,444,447,450,453],{"className":440},[78],[20,442],{"className":443,"style":83},[82],[20,445,384],{"className":446,"style":399},[87,88],[20,448],{"className":449,"style":157},[156],[20,451,122],{"className":452},[161],[20,454],{"className":455,"style":157},[156],[20,457,459,463],{"className":458},[78],[20,460],{"className":461,"style":462},[82],"height:1.2173em;vertical-align:-0.345em;",[20,464,466,471,541],{"className":465},[87],[20,467],{"className":468},[469,470],"mopen","nulldelimiter",[20,472,474],{"className":473},[419],[20,475,477,532],{"className":476},[189,323],[20,478,480,529],{"className":479},[193],[20,481,484,500,511],{"className":482,"style":483},[197],"height:0.8723em;",[20,485,487,491],{"style":486},"top:-2.655em;",[20,488],{"className":489,"style":490},[204],"height:3em;",[20,492,494],{"className":493},[209,210,211,212],[20,495,497],{"className":496},[87,212],[20,498,431],{"className":499},[87,212],[20,501,503,506],{"style":502},"top:-3.23em;",[20,504],{"className":505,"style":490},[204],[20,507],{"className":508,"style":510},[509],"frac-line","border-bottom-width:0.04em;",[20,512,514,517],{"style":513},"top:-3.394em;",[20,515],{"className":516,"style":490},[204],[20,518,520],{"className":519},[209,210,211,212],[20,521,523,526],{"className":522},[87,212],[20,524,425],{"className":525},[87,88,212],[20,527,428],{"className":528},[87,88,212],[20,530,347],{"className":531},[346],[20,533,535],{"className":534},[193],[20,536,539],{"className":537,"style":538},[197],"height:0.345em;",[20,540],{},[20,542],{"className":543},[544,470],"mclose"," ",[20,547,548],{},"32","\nWhere:",[551,552,553,736],"ul",{},[554,555,556,584,585,735],"li",{},[20,557,559,572],{"className":558},[47],[20,560,562],{"className":561},[51],[53,563,564],{"xmlns":55},[57,565,566,570],{},[60,567,568],{},[63,569,425],{},[67,571,425],{"encoding":69},[20,573,575],{"className":574,"ariaHidden":74},[73],[20,576,578,581],{"className":577},[78],[20,579],{"className":580,"style":83},[82],[20,582,425],{"className":583},[87,88]," = Algebraic difference in grades (",[20,586,588,622],{"className":587},[47],[20,589,591],{"className":590},[51],[53,592,593],{"xmlns":55},[57,594,595,619],{},[60,596,597,604,608,611,617],{},[289,598,599,602],{},[63,600,601],{},"g",[133,603,296],{},[63,605,607],{"mathvariant":606},"normal","%",[120,609,610],{},"−",[289,612,613,615],{},[63,614,601],{},[133,616,135],{},[63,618,607],{"mathvariant":606},[67,620,621],{"encoding":69},"g_1\\% - g_2\\%",[20,623,625,686],{"className":624,"ariaHidden":74},[73],[20,626,628,632,673,676,679,683],{"className":627},[78],[20,629],{"className":630,"style":631},[82],"height:0.9444em;vertical-align:-0.1944em;",[20,633,635,638],{"className":634},[87],[20,636,601],{"className":637,"style":152},[87,88],[20,639,641],{"className":640},[185],[20,642,644,665],{"className":643},[189,323],[20,645,647,662],{"className":646},[193],[20,648,650],{"className":649,"style":330},[197],[20,651,653,656],{"style":652},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[20,654],{"className":655,"style":205},[204],[20,657,659],{"className":658},[209,210,211,212],[20,660,296],{"className":661},[87,212],[20,663,347],{"className":664},[346],[20,666,668],{"className":667},[193],[20,669,671],{"className":670,"style":354},[197],[20,672],{},[20,674,607],{"className":675},[87],[20,677],{"className":678,"style":399},[156],[20,680,610],{"className":681},[682],"mbin",[20,684],{"className":685,"style":399},[156],[20,687,689,692,732],{"className":688},[78],[20,690],{"className":691,"style":631},[82],[20,693,695,698],{"className":694},[87],[20,696,601],{"className":697,"style":152},[87,88],[20,699,701],{"className":700},[185],[20,702,704,724],{"className":703},[189,323],[20,705,707,721],{"className":706},[193],[20,708,710],{"className":709,"style":330},[197],[20,711,712,715],{"style":652},[20,713],{"className":714,"style":205},[204],[20,716,718],{"className":717},[209,210,211,212],[20,719,135],{"className":720},[87,212],[20,722,347],{"className":723},[346],[20,725,727],{"className":726},[193],[20,728,730],{"className":729,"style":354},[197],[20,731],{},[20,733,607],{"className":734},[87],").",[554,737,738,766],{},[20,739,741,754],{"className":740},[47],[20,742,744],{"className":743},[51],[53,745,746],{"xmlns":55},[57,747,748,752],{},[60,749,750],{},[63,751,428],{},[67,753,428],{"encoding":69},[20,755,757],{"className":756,"ariaHidden":74},[73],[20,758,760,763],{"className":759},[78],[20,761],{"className":762,"style":83},[82],[20,764,428],{"className":765},[87,88]," = Total horizontal length of the curve.",[16,768,769,770,798,799,868,869,545,1071],{},"For any other point at distance ",[20,771,773,786],{"className":772},[47],[20,774,776],{"className":775},[51],[53,777,778],{"xmlns":55},[57,779,780,784],{},[60,781,782],{},[63,783,131],{},[67,785,131],{"encoding":69},[20,787,789],{"className":788,"ariaHidden":74},[73],[20,790,792,795],{"className":791},[78],[20,793],{"className":794,"style":271},[82],[20,796,131],{"className":797,"style":181},[87,88]," from ",[20,800,802,819],{"className":801},[47],[20,803,805],{"className":804},[51],[53,806,807],{"xmlns":55},[57,808,809,817],{},[60,810,811],{},[289,812,813,815],{},[63,814,293],{},[133,816,296],{},[67,818,299],{"encoding":69},[20,820,822],{"className":821,"ariaHidden":74},[73],[20,823,825,828],{"className":824},[78],[20,826],{"className":827,"style":309},[82],[20,829,831,834],{"className":830},[87],[20,832,293],{"className":833,"style":316},[87,88],[20,835,837],{"className":836},[185],[20,838,840,860],{"className":839},[189,323],[20,841,843,857],{"className":842},[193],[20,844,846],{"className":845,"style":330},[197],[20,847,848,851],{"style":333},[20,849],{"className":850,"style":205},[204],[20,852,854],{"className":853},[209,210,211,212],[20,855,296],{"className":856},[87,212],[20,858,347],{"className":859},[346],[20,861,863],{"className":862},[193],[20,864,866],{"className":865,"style":354},[197],[20,867],{},":\n",[20,870,872,914],{"className":871},[47],[20,873,875],{"className":874},[51],[53,876,877],{"xmlns":55},[57,878,879,911],{},[60,880,881,883,885,887],{},[63,882,118],{},[120,884,122],{},[63,886,384],{},[127,888,889,909],{},[60,890,891,894,907],{},[120,892,893],{"fence":74},"(",[419,895,896,898],{},[63,897,131],{},[60,899,900,902,905],{},[63,901,428],{},[63,903,904],{"mathvariant":606},"\u002F",[133,906,135],{},[120,908,400],{"fence":74},[133,910,135],{},[67,912,913],{"encoding":69},"y = Y \\left( \\frac{l}{L\u002F2} \\right)^2",[20,915,917,935],{"className":916,"ariaHidden":74},[73],[20,918,920,923,926,929,932],{"className":919},[78],[20,921],{"className":922,"style":148},[82],[20,924,118],{"className":925,"style":152},[87,88],[20,927],{"className":928,"style":157},[156],[20,930,122],{"className":931},[161],[20,933],{"className":934,"style":157},[156],[20,936,938,942,945,949],{"className":937},[78],[20,939],{"className":940,"style":941},[82],"height:2.004em;vertical-align:-0.65em;",[20,943,384],{"className":944,"style":399},[87,88],[20,946],{"className":947,"style":948},[156],"margin-right:0.1667em;",[20,950,953,1046],{"className":951},[952],"minner",[20,954,956,966,1040],{"className":955},[952],[20,957,961],{"className":958,"style":960},[469,959],"delimcenter","top:0em;",[20,962,893],{"className":963},[964,965],"delimsizing","size2",[20,967,969,972,1037],{"className":968},[87],[20,970],{"className":971},[469,470],[20,973,975],{"className":974},[419],[20,976,978,1028],{"className":977},[189,323],[20,979,981,1025],{"className":980},[193],[20,982,985,1003,1011],{"className":983,"style":984},[197],"height:0.8801em;",[20,986,987,990],{"style":486},[20,988],{"className":989,"style":490},[204],[20,991,993],{"className":992},[209,210,211,212],[20,994,996,999],{"className":995},[87,212],[20,997,428],{"className":998},[87,88,212],[20,1000,1002],{"className":1001},[87,212],"\u002F2",[20,1004,1005,1008],{"style":502},[20,1006],{"className":1007,"style":490},[204],[20,1009],{"className":1010,"style":510},[509],[20,1012,1013,1016],{"style":513},[20,1014],{"className":1015,"style":490},[204],[20,1017,1019],{"className":1018},[209,210,211,212],[20,1020,1022],{"className":1021},[87,212],[20,1023,131],{"className":1024,"style":181},[87,88,212],[20,1026,347],{"className":1027},[346],[20,1029,1031],{"className":1030},[193],[20,1032,1035],{"className":1033,"style":1034},[197],"height:0.52em;",[20,1036],{},[20,1038],{"className":1039},[544,470],[20,1041,1043],{"className":1042,"style":960},[544,959],[20,1044,400],{"className":1045},[964,965],[20,1047,1049],{"className":1048},[185],[20,1050,1052],{"className":1051},[189],[20,1053,1055],{"className":1054},[193],[20,1056,1059],{"className":1057,"style":1058},[197],"height:1.354em;",[20,1060,1062,1065],{"style":1061},"top:-3.6029em;margin-right:0.05em;",[20,1063],{"className":1064,"style":205},[204],[20,1066,1068],{"className":1067},[209,210,211,212],[20,1069,135],{"className":1070},[87,212],[20,1072,548],{},[11,1074,1076],{"id":1075},"field-workflow","Field Workflow",[1078,1079,1080,1085,1199,1203,1237,1241,1308,1312,1359,1363],"steps",{},[1081,1082,1084],"h3",{"id":1083},"determine-grade-angle-a","Determine Grade Angle (A)",[16,1086,1087,1088,357,1197,32],{},"Subtract the second grade from the first (e.g., ",[20,1089,1091,1131],{"className":1090},[47],[20,1092,1094],{"className":1093},[51],[53,1095,1096],{"xmlns":55},[57,1097,1098,1128],{},[60,1099,1100,1103,1105,1107,1109,1112,1114,1117,1119,1121,1123,1126],{},[120,1101,1102],{},"+",[133,1104,135],{},[63,1106,607],{"mathvariant":606},[120,1108,610],{},[120,1110,893],{"stretchy":1111},"false",[120,1113,610],{},[133,1115,1116],{},"4",[63,1118,607],{"mathvariant":606},[120,1120,400],{"stretchy":1111},[120,1122,122],{},[133,1124,1125],{},"6",[63,1127,607],{"mathvariant":606},[67,1129,1130],{"encoding":69},"+2\\% - (-4\\%) = 6\\%",[20,1132,1134,1157,1186],{"className":1133,"ariaHidden":74},[73],[20,1135,1137,1141,1144,1148,1151,1154],{"className":1136},[78],[20,1138],{"className":1139,"style":1140},[82],"height:0.8333em;vertical-align:-0.0833em;",[20,1142,1102],{"className":1143},[87],[20,1145,1147],{"className":1146},[87],"2%",[20,1149],{"className":1150,"style":399},[156],[20,1152,610],{"className":1153},[682],[20,1155],{"className":1156,"style":399},[156],[20,1158,1160,1164,1167,1170,1174,1177,1180,1183],{"className":1159},[78],[20,1161],{"className":1162,"style":1163},[82],"height:1em;vertical-align:-0.25em;",[20,1165,893],{"className":1166},[469],[20,1168,610],{"className":1169},[87],[20,1171,1173],{"className":1172},[87],"4%",[20,1175,400],{"className":1176},[544],[20,1178],{"className":1179,"style":157},[156],[20,1181,122],{"className":1182},[161],[20,1184],{"className":1185,"style":157},[156],[20,1187,1189,1193],{"className":1188},[78],[20,1190],{"className":1191,"style":1192},[82],"height:0.8056em;vertical-align:-0.0556em;",[20,1194,1196],{"className":1195},[87],"6%",[20,1198,548],{},[1081,1200,1202],{"id":1201},"calculate-length-l","Calculate Length (L)",[16,1204,1205,1206,1234,1235,32],{},"Based on design speed and required sight distance (refer to ",[20,1207,1209,1222],{"className":1208},[47],[20,1210,1212],{"className":1211},[51],[53,1213,1214],{"xmlns":55},[57,1215,1216,1220],{},[60,1217,1218],{},[63,1219,65],{},[67,1221,65],{"encoding":69},[20,1223,1225],{"className":1224,"ariaHidden":74},[73],[20,1226,1228,1231],{"className":1227},[78],[20,1229],{"className":1230,"style":83},[82],[20,1232,65],{"className":1233,"style":89},[87,88],"-value tables) ",[20,1236,42],{},[1081,1238,1240],{"id":1239},"compute-central-offset-y","Compute Central Offset (Y)",[16,1242,1243,1244,32],{},"Use ",[20,1245,1247,1271],{"className":1246},[47],[20,1248,1250],{"className":1249},[51],[53,1251,1252],{"xmlns":55},[57,1253,1254,1268],{},[60,1255,1256,1258,1260,1262,1264,1266],{},[63,1257,384],{},[120,1259,122],{},[63,1261,425],{},[63,1263,428],{},[63,1265,904],{"mathvariant":606},[133,1267,431],{},[67,1269,1270],{"encoding":69},"Y = AL \u002F 800",[20,1272,1274,1292],{"className":1273,"ariaHidden":74},[73],[20,1275,1277,1280,1283,1286,1289],{"className":1276},[78],[20,1278],{"className":1279,"style":83},[82],[20,1281,384],{"className":1282,"style":399},[87,88],[20,1284],{"className":1285,"style":157},[156],[20,1287,122],{"className":1288},[161],[20,1290],{"className":1291,"style":157},[156],[20,1293,1295,1298,1301,1304],{"className":1294},[78],[20,1296],{"className":1297,"style":1163},[82],[20,1299,425],{"className":1300},[87,88],[20,1302,428],{"className":1303},[87,88],[20,1305,1307],{"className":1306},[87],"\u002F800",[1081,1309,1311],{"id":1310},"calculate-offsets","Calculate Offsets",[16,1313,1314,1315,357,1357,32],{},"Calculate vertical offsets at regular intervals (e.g., every ",[20,1316,1318,1337],{"className":1317},[47],[20,1319,1321],{"className":1320},[51],[53,1322,1323],{"xmlns":55},[57,1324,1325,1334],{},[60,1326,1327,1330],{},[133,1328,1329],{},"20",[1331,1332,1333],"mtext",{}," m",[67,1335,1336],{"encoding":69},"20 \\text{ m}",[20,1338,1340],{"className":1339,"ariaHidden":74},[73],[20,1341,1343,1347,1350],{"className":1342},[78],[20,1344],{"className":1345,"style":1346},[82],"height:0.6444em;",[20,1348,1329],{"className":1349},[87],[20,1351,1354],{"className":1352},[87,1353],"text",[20,1355,1333],{"className":1356},[87],[20,1358,548],{},[1081,1360,1362],{"id":1361},"determine-elevations","Determine Elevations",[16,1364,1365,1366,32],{},"Calculate levels along the tangent grade and apply (add or subtract) the offsets to get the final curve levels ",[20,1367,1368],{},"32, 33",[11,1370,1372],{"id":1371},"formula-breakdown","Formula Breakdown",[551,1374,1375,1521],{},[554,1376,1377,545,1380,1450,1451,1479,1480,1518,1519,32],{},[25,1378,1379],{},"Radius (R):",[20,1381,1383,1410],{"className":1382},[47],[20,1384,1386],{"className":1385},[51],[53,1387,1388],{"xmlns":55},[57,1389,1390,1407],{},[60,1391,1392,1395,1398,1401,1403,1405],{},[63,1393,1394],{},"R",[120,1396,1397],{},"≈",[133,1399,1400],{},"100",[63,1402,428],{},[63,1404,904],{"mathvariant":606},[63,1406,425],{},[67,1408,1409],{"encoding":69},"R \\approx 100L\u002FA",[20,1411,1413,1432],{"className":1412,"ariaHidden":74},[73],[20,1414,1416,1419,1423,1426,1429],{"className":1415},[78],[20,1417],{"className":1418,"style":83},[82],[20,1420,1394],{"className":1421,"style":1422},[87,88],"margin-right:0.0077em;",[20,1424],{"className":1425,"style":157},[156],[20,1427,1397],{"className":1428},[161],[20,1430],{"className":1431,"style":157},[156],[20,1433,1435,1438,1441,1444,1447],{"className":1434},[78],[20,1436],{"className":1437,"style":1163},[82],[20,1439,1400],{"className":1440},[87],[20,1442,428],{"className":1443},[87,88],[20,1445,904],{"className":1446},[87],[20,1448,425],{"className":1449},[87,88],". Note that ",[20,1452,1454,1467],{"className":1453},[47],[20,1455,1457],{"className":1456},[51],[53,1458,1459],{"xmlns":55},[57,1460,1461,1465],{},[60,1462,1463],{},[63,1464,1394],{},[67,1466,1394],{"encoding":69},[20,1468,1470],{"className":1469,"ariaHidden":74},[73],[20,1471,1473,1476],{"className":1472},[78],[20,1474],{"className":1475,"style":83},[82],[20,1477,1394],{"className":1478,"style":1422},[87,88]," must be massive (e.g., ",[20,1481,1483,1500],{"className":1482},[47],[20,1484,1486],{"className":1485},[51],[53,1487,1488],{"xmlns":55},[57,1489,1490,1497],{},[60,1491,1492,1495],{},[133,1493,1494],{},"3000",[1331,1496,1333],{},[67,1498,1499],{"encoding":69},"3000 \\text{ m}",[20,1501,1503],{"className":1502,"ariaHidden":74},[73],[20,1504,1506,1509,1512],{"className":1505},[78],[20,1507],{"className":1508,"style":1346},[82],[20,1510,1494],{"className":1511},[87],[20,1513,1515],{"className":1514},[87,1353],[20,1516,1333],{"className":1517},[87],") because road grades are shallow ",[20,1520,31],{},[554,1522,1523,1526,1527,1596,1597,545,1782],{},[25,1524,1525],{},"Highest\u002FLowest Point:"," The distance from ",[20,1528,1530,1547],{"className":1529},[47],[20,1531,1533],{"className":1532},[51],[53,1534,1535],{"xmlns":55},[57,1536,1537,1545],{},[60,1538,1539],{},[289,1540,1541,1543],{},[63,1542,293],{},[133,1544,296],{},[67,1546,299],{"encoding":69},[20,1548,1550],{"className":1549,"ariaHidden":74},[73],[20,1551,1553,1556],{"className":1552},[78],[20,1554],{"className":1555,"style":309},[82],[20,1557,1559,1562],{"className":1558},[87],[20,1560,293],{"className":1561,"style":316},[87,88],[20,1563,1565],{"className":1564},[185],[20,1566,1568,1588],{"className":1567},[189,323],[20,1569,1571,1585],{"className":1570},[193],[20,1572,1574],{"className":1573,"style":330},[197],[20,1575,1576,1579],{"style":333},[20,1577],{"className":1578,"style":205},[204],[20,1580,1582],{"className":1581},[209,210,211,212],[20,1583,296],{"className":1584},[87,212],[20,1586,347],{"className":1587},[346],[20,1589,1591],{"className":1590},[193],[20,1592,1594],{"className":1593,"style":354},[197],[20,1595],{}," to the summit or valley is:\n",[20,1598,1600,1634],{"className":1599},[47],[20,1601,1603],{"className":1602},[51],[53,1604,1605],{"xmlns":55},[57,1606,1607,1631],{},[60,1608,1609,1612,1614],{},[63,1610,1611],{},"x",[120,1613,122],{},[419,1615,1616,1629],{},[60,1617,1618,1620,1623],{},[63,1619,428],{},[120,1621,1622],{},"×",[289,1624,1625,1627],{},[63,1626,601],{},[133,1628,296],{},[63,1630,425],{},[67,1632,1633],{"encoding":69},"x = \\frac{L \\times g_1}{A}",[20,1635,1637,1656],{"className":1636,"ariaHidden":74},[73],[20,1638,1640,1644,1647,1650,1653],{"className":1639},[78],[20,1641],{"className":1642,"style":1643},[82],"height:0.4306em;",[20,1645,1611],{"className":1646},[87,88],[20,1648],{"className":1649,"style":157},[156],[20,1651,122],{"className":1652},[161],[20,1654],{"className":1655,"style":157},[156],[20,1657,1659,1663],{"className":1658},[78],[20,1660],{"className":1661,"style":1662},[82],"height:1.2694em;vertical-align:-0.345em;",[20,1664,1666,1669,1779],{"className":1665},[87],[20,1667],{"className":1668},[469,470],[20,1670,1672],{"className":1671},[419],[20,1673,1675,1771],{"className":1674},[189,323],[20,1676,1678,1768],{"className":1677},[193],[20,1679,1682,1696,1704],{"className":1680,"style":1681},[197],"height:0.9244em;",[20,1683,1684,1687],{"style":486},[20,1685],{"className":1686,"style":490},[204],[20,1688,1690],{"className":1689},[209,210,211,212],[20,1691,1693],{"className":1692},[87,212],[20,1694,425],{"className":1695},[87,88,212],[20,1697,1698,1701],{"style":502},[20,1699],{"className":1700,"style":490},[204],[20,1702],{"className":1703,"style":510},[509],[20,1705,1707,1710],{"style":1706},"top:-3.4461em;",[20,1708],{"className":1709,"style":490},[204],[20,1711,1713],{"className":1712},[209,210,211,212],[20,1714,1716,1719,1722],{"className":1715},[87,212],[20,1717,428],{"className":1718},[87,88,212],[20,1720,1622],{"className":1721},[682,212],[20,1723,1725,1728],{"className":1724},[87,212],[20,1726,601],{"className":1727,"style":152},[87,88,212],[20,1729,1731],{"className":1730},[185],[20,1732,1734,1759],{"className":1733},[189,323],[20,1735,1737,1756],{"className":1736},[193],[20,1738,1741],{"className":1739,"style":1740},[197],"height:0.3173em;",[20,1742,1744,1748],{"style":1743},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[20,1745],{"className":1746,"style":1747},[204],"height:2.5em;",[20,1749,1753],{"className":1750},[209,1751,1752,212],"reset-size3","size1",[20,1754,296],{"className":1755},[87,212],[20,1757,347],{"className":1758},[346],[20,1760,1762],{"className":1761},[193],[20,1763,1766],{"className":1764,"style":1765},[197],"height:0.143em;",[20,1767],{},[20,1769,347],{"className":1770},[346],[20,1772,1774],{"className":1773},[193],[20,1775,1777],{"className":1776,"style":538},[197],[20,1778],{},[20,1780],{"className":1781},[544,470],[20,1783,1784],{},"33, 34",[11,1786,1788],{"id":1787},"practical-tips","Practical Tips",[551,1790,1791,1800],{},[554,1792,1793,1796,1797,32],{},[25,1794,1795],{},"The Second Difference Check:"," When calculating curve levels at equal intervals, the \"second difference\" of the offsets should be constant. This is a powerful arithmetical check for your spreadsheet or field book ",[20,1798,1799],{},"30, 32",[554,1801,1802,1805,1806,32],{},[25,1803,1804],{},"Drainage:"," On sag curves, ensure the lowest point isn't exactly at a flat grade to prevent water ponding ",[20,1807,1808],{},"33",[11,1810,1812],{"id":1811},"common-mistakes","Common Mistakes",[551,1814,1815,1983],{},[554,1816,1817,1820,1821,1825,1826,1857,1858,1891,1892,1947,1948,545,1980,32],{},[25,1818,1819],{},"Ignoring the Grade Sign:"," A falling grade is ",[1822,1823,1824],"em",{},"negative",". A ",[20,1827,1829,1845],{"className":1828},[47],[20,1830,1832],{"className":1831},[51],[53,1833,1834],{"xmlns":55},[57,1835,1836,1842],{},[60,1837,1838,1840],{},[133,1839,1116],{},[63,1841,607],{"mathvariant":606},[67,1843,1844],{"encoding":69},"4\\%",[20,1846,1848],{"className":1847,"ariaHidden":74},[73],[20,1849,1851,1854],{"className":1850},[78],[20,1852],{"className":1853,"style":1192},[82],[20,1855,1173],{"className":1856},[87]," falling grade meeting a ",[20,1859,1861,1878],{"className":1860},[47],[20,1862,1864],{"className":1863},[51],[53,1865,1866],{"xmlns":55},[57,1867,1868,1875],{},[60,1869,1870,1873],{},[133,1871,1872],{},"5",[63,1874,607],{"mathvariant":606},[67,1876,1877],{"encoding":69},"5\\%",[20,1879,1881],{"className":1880,"ariaHidden":74},[73],[20,1882,1884,1887],{"className":1883},[78],[20,1885],{"className":1886,"style":1192},[82],[20,1888,1890],{"className":1889},[87],"5%"," rising grade has an ",[20,1893,1895,1916],{"className":1894},[47],[20,1896,1898],{"className":1897},[51],[53,1899,1900],{"xmlns":55},[57,1901,1902,1913],{},[60,1903,1904,1906,1908,1911],{},[63,1905,425],{},[120,1907,122],{},[133,1909,1910],{},"9",[63,1912,607],{"mathvariant":606},[67,1914,1915],{"encoding":69},"A = 9\\%",[20,1917,1919,1937],{"className":1918,"ariaHidden":74},[73],[20,1920,1922,1925,1928,1931,1934],{"className":1921},[78],[20,1923],{"className":1924,"style":83},[82],[20,1926,425],{"className":1927},[87,88],[20,1929],{"className":1930,"style":157},[156],[20,1932,122],{"className":1933},[161],[20,1935],{"className":1936,"style":157},[156],[20,1938,1940,1943],{"className":1939},[78],[20,1941],{"className":1942,"style":1192},[82],[20,1944,1946],{"className":1945},[87],"9%",", not ",[20,1949,1951,1967],{"className":1950},[47],[20,1952,1954],{"className":1953},[51],[53,1955,1956],{"xmlns":55},[57,1957,1958,1964],{},[60,1959,1960,1962],{},[133,1961,296],{},[63,1963,607],{"mathvariant":606},[67,1965,1966],{"encoding":69},"1\\%",[20,1968,1970],{"className":1969,"ariaHidden":74},[73],[20,1971,1973,1976],{"className":1972},[78],[20,1974],{"className":1975,"style":1192},[82],[20,1977,1979],{"className":1978},[87],"1%",[20,1981,1982],{},"35",[554,1984,1985,1988,1989,1992,1993,32],{},[25,1986,1987],{},"Vertical Scale Distortion:"," Scaling levels from a longitudinal section is dangerous because the vertical scale is often exaggerated. Always ",[1822,1990,1991],{},"compute"," the levels ",[20,1994,31],{},[11,1996,1998],{"id":1997},"faq","FAQ",[2000,2001,2002,2132],"accordion",{},[2003,2004,2006],"accordion-item",{"label":2005},"What is the 'K-value'?",[16,2007,366,2008,2036,2037,2067,2068,2129,2130,32],{},[20,2009,2011,2024],{"className":2010},[47],[20,2012,2014],{"className":2013},[51],[53,2015,2016],{"xmlns":55},[57,2017,2018,2022],{},[60,2019,2020],{},[63,2021,65],{},[67,2023,65],{"encoding":69},[20,2025,2027],{"className":2026,"ariaHidden":74},[73],[20,2028,2030,2033],{"className":2029},[78],[20,2031],{"className":2032,"style":83},[82],[20,2034,65],{"className":2035,"style":89},[87,88],"-value is the horizontal distance required to achieve a ",[20,2038,2040,2055],{"className":2039},[47],[20,2041,2043],{"className":2042},[51],[53,2044,2045],{"xmlns":55},[57,2046,2047,2053],{},[60,2048,2049,2051],{},[133,2050,296],{},[63,2052,607],{"mathvariant":606},[67,2054,1966],{"encoding":69},[20,2056,2058],{"className":2057,"ariaHidden":74},[73],[20,2059,2061,2064],{"className":2060},[78],[20,2062],{"className":2063,"style":1192},[82],[20,2065,1979],{"className":2066},[87]," change in gradient (",[20,2069,2071,2093],{"className":2070},[47],[20,2072,2074],{"className":2073},[51],[53,2075,2076],{"xmlns":55},[57,2077,2078,2090],{},[60,2079,2080,2082,2084,2086,2088],{},[63,2081,65],{},[120,2083,122],{},[63,2085,428],{},[63,2087,904],{"mathvariant":606},[63,2089,425],{},[67,2091,2092],{"encoding":69},"K = L\u002FA",[20,2094,2096,2114],{"className":2095,"ariaHidden":74},[73],[20,2097,2099,2102,2105,2108,2111],{"className":2098},[78],[20,2100],{"className":2101,"style":83},[82],[20,2103,65],{"className":2104,"style":89},[87,88],[20,2106],{"className":2107,"style":157},[156],[20,2109,122],{"className":2110},[161],[20,2112],{"className":2113,"style":157},[156],[20,2115,2117,2120,2123,2126],{"className":2116},[78],[20,2118],{"className":2119,"style":1163},[82],[20,2121,428],{"className":2122},[87,88],[20,2124,904],{"className":2125},[87],[20,2127,425],{"className":2128},[87,88],"). It is used to simplify design tables ",[20,2131,42],{},[2003,2133,2135],{"label":2134},"When should I use a sag curve vs a crest curve?",[16,2136,2137,2138,32],{},"A crest curve connects a rising grade to a falling one (convex); a sag curve connects a falling grade to a rising one (concave) ",[20,2139,2140],{},"29, 36",[11,2142,2144],{"id":2143},"conclusion","Conclusion",[16,2146,2147,2148,2234],{},"Vertical curves are the \"invisible\" comfort of road design. By mastering the ",[20,2149,2151,2175],{"className":2150},[47],[20,2152,2154],{"className":2153},[51],[53,2155,2156],{"xmlns":55},[57,2157,2158,2172],{},[60,2159,2160,2162,2164,2166],{},[63,2161,118],{},[120,2163,122],{},[63,2165,125],{},[127,2167,2168,2170],{},[63,2169,131],{},[133,2171,135],{},[67,2173,2174],{"encoding":69},"y=Cl^2",[20,2176,2178,2196],{"className":2177,"ariaHidden":74},[73],[20,2179,2181,2184,2187,2190,2193],{"className":2180},[78],[20,2182],{"className":2183,"style":148},[82],[20,2185,118],{"className":2186,"style":152},[87,88],[20,2188],{"className":2189,"style":157},[156],[20,2191,122],{"className":2192},[161],[20,2194],{"className":2195,"style":157},[156],[20,2197,2199,2202,2205],{"className":2198},[78],[20,2200],{"className":2201,"style":171},[82],[20,2203,125],{"className":2204,"style":89},[87,88],[20,2206,2208,2211],{"className":2207},[87],[20,2209,131],{"className":2210,"style":181},[87,88],[20,2212,2214],{"className":2213},[185],[20,2215,2217],{"className":2216},[189],[20,2218,2220],{"className":2219},[193],[20,2221,2223],{"className":2222,"style":171},[197],[20,2224,2225,2228],{"style":200},[20,2226],{"className":2227,"style":205},[204],[20,2229,2231],{"className":2230},[209,210,211,212],[20,2232,135],{"className":2233},[87,212]," parabola, you ensure that every kilometer of road you set out is safe, drainable, and durable.",[11,2236,2238],{"id":2237},"references","References",[16,2240,2241,2242,2245,2246,32],{},"Schofield, W. (2001). ",[1822,2243,2244],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann. 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