80:
1905:
43:
54:
31:
161:
for further details) and of the object velocity which flowing therethrough (river, avalanche, car, bicycle, bobsleigh, skier, high speed train, etc.): the sinuosity of the same curved line could be considered very strong for a high speed train but low for a river. Nevertheless, it is possible to see
1383:
It is also possible to distinguish the case where the stream flowing on the line could not physically travel the distance between the ends: in some hydraulic studies, this leads to assign a sinuosity value of 1 for a torrent flowing over rocky bedrock along a horizontal rectilinear projection, even
1375:
The difference from the general form happens because the downvalley path is not perfectly straight. The sinuosity index can be explained, then, as the deviations from a path defined by the direction of maximum downslope. For this reason, bedrock streams that flow directly downslope have a sinuosity
138:
The curve must be continuous (no jump) between the two ends. The sinuosity value is really significant when the line is continuously differentiable (no angular point). The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the
129:
can also be rephrased as the "actual path length" divided by the "shortest path length" of a curve. The value ranges from 1 (case of straight line) to infinity (case of a closed loop, where the shortest path length is zero for an infinitely-long actual path).
336:
342:
1370:
1319:
956:
794:
463:
217:
1134:
1045:
689:
600:
1258:
1214:
1178:
1089:
1000:
895:
733:
644:
237:
851:
563:
527:
499:
410:
823:
1890:
1340:
1566:
1528:
1264:
901:
739:
150:
of the curve in the selected plan; "classic" sinuosity on the horizontal plane, longitudinal profile sinuosity on the vertical plane).
1445:
Leopold, Luna B., Wolman, M.G., and Miller, J.P., 1964, Fluvial
Processes in Geomorphology, San Francisco, W.H. Freeman and Co., 522p.
1944:
416:
1599:
1559:
181:
1095:
1006:
650:
1334:
In studies of rivers, the sinuosity index is similar but not identical to the general form given above, being given by:
1408:
that causes their average sinuosity (measured in terms of the source-to-mouth distance, not channel length) to be
569:
1908:
1552:
1228:
1184:
1148:
1059:
970:
865:
703:
614:
1722:
331:{\displaystyle S=\textstyle {\tfrac {1}{n\pi }}\int _{0}^{n\pi }{\sqrt {1+(\cos x)^{2}}}dx\approx 1.216...}
1652:
126:
99:
1834:
1829:
147:
829:
541:
505:
477:
388:
1855:
1493:
142:
The calculation of the sinuosity is valid in a 3-dimensional space (e.g. for the central axis of the
162:
a very strong sinuosity in the succession of few river bends, or of laces on some mountain roads.
1885:
1839:
1509:
154:
118:
114:
1412:, but this has not been borne out by later studies, which found an average value less than 2.
1880:
1717:
1484:
1455:
Mueller, Jerry (1968). "An
Introduction to the Hydraulic and Topographic Sinuosity Indexes1".
1405:
158:
79:
64:
1647:
1575:
1501:
1464:
106:
1939:
1934:
1929:
1875:
1662:
1624:
1609:
808:
143:
27:
Ratio of arc length and straight-line distance between two points on a wave-like function
1497:
1865:
1732:
1682:
1634:
1594:
1589:
1468:
365:
1923:
1870:
1752:
1712:
1657:
1365:{\displaystyle {\text{SI}}={\frac {\text{channel length}}{\text{downvalley length}}}}
355:
122:
1513:
1803:
1614:
1533:
1505:
349:
With similar opposite arcs joints in the same plane, continuously differentiable:
1819:
1770:
1737:
1727:
1619:
59:
47:
1482:
Stølum, Hans-Henrik (1996), "River
Meandering as a Self-Organization Process",
17:
1824:
1765:
1692:
1672:
1426:
42:
1697:
1687:
1642:
1421:
231:
153:
The classification of a sinuosity (e.g. strong / weak) often depends on the
53:
341:
1793:
1667:
230:
of half-periods), which can be calculated by computing the sine curve's
1860:
1780:
1742:
1677:
1604:
1377:
30:
1760:
1707:
1702:
370:
83:
Two ski tracks with different degrees of sinuosity on the same slope
1544:
146:), although it is often performed in a plane (with then a possible
1314:{\displaystyle {\frac {11\cdot \pi }{3({\sqrt {6}}-{\sqrt {2}})}}}
340:
110:
102:
78:
52:
41:
951:{\displaystyle {\frac {7\cdot \pi }{3({\sqrt {6}}+{\sqrt {2}})}}}
789:{\displaystyle {\frac {5\cdot \pi }{3({\sqrt {6}}+{\sqrt {2}})}}}
1798:
223:
68:
1548:
178:
2 inverted continuous semicircles located in the same plane is
1788:
1387:
For rivers, the conventional classes of sinuosity, SI, are:
71:, is not taking the shortest path downslope. Therefore, its
351:
458:{\displaystyle {\frac {\pi }{3({\sqrt {6}}-{\sqrt {2}})}}}
1404:
It has been claimed that river shapes are governed by a
1380:
streams have a sinuosity index that is greater than 1.
249:
247:
192:
1343:
1267:
1231:
1187:
1151:
1098:
1062:
1009:
973:
904:
868:
832:
811:
742:
706:
653:
617:
572:
544:
508:
480:
419:
391:
240:
212:{\displaystyle S={\tfrac {\pi }{2}}\approx 1.5708...}
184:
139:
successive inflection points (sinuosity of order 2).
46:
Switchbacks on mountain road with high sinuosity at
1848:
1812:
1779:
1751:
1633:
1582:
1129:{\displaystyle {\frac {3\cdot \pi }{2{\sqrt {2}}}}}
1040:{\displaystyle {\frac {4\cdot \pi }{3{\sqrt {3}}}}}
684:{\displaystyle {\frac {2\cdot \pi }{3{\sqrt {3}}}}}
1529:"A meandering tale: the truth about pi and rivers"
1364:
1313:
1252:
1208:
1172:
1128:
1083:
1039:
994:
950:
889:
845:
817:
788:
727:
683:
638:
594:
557:
521:
493:
457:
404:
330:
211:
1457:Annals of the Association of American Geographers
1560:
8:
595:{\displaystyle {\frac {\pi }{2{\sqrt {2}}}}}
125:) between the end points of the curve. This
1891:List of rivers that have reversed direction
1567:
1553:
1545:
1531:, Alex Bellos's Adventures in Numberland,
1352:
1344:
1342:
1298:
1288:
1268:
1266:
1253:{\displaystyle {\frac {11\cdot \pi }{6}}}
1232:
1230:
1188:
1186:
1152:
1150:
1116:
1099:
1097:
1063:
1061:
1027:
1010:
1008:
974:
972:
935:
925:
905:
903:
869:
867:
833:
831:
810:
773:
763:
743:
741:
707:
705:
671:
654:
652:
618:
616:
582:
573:
571:
545:
543:
509:
507:
481:
479:
442:
432:
420:
418:
392:
390:
307:
283:
274:
269:
248:
239:
219:. It is independent of the circle radius;
191:
183:
1209:{\displaystyle {\frac {5\cdot \pi }{3}}}
1173:{\displaystyle {\frac {5\cdot \pi }{3}}}
1084:{\displaystyle {\frac {3\cdot \pi }{2}}}
995:{\displaystyle {\frac {4\cdot \pi }{3}}}
890:{\displaystyle {\frac {7\cdot \pi }{6}}}
728:{\displaystyle {\frac {5\cdot \pi }{6}}}
639:{\displaystyle {\frac {2\cdot \pi }{3}}}
29:
1438:
7:
1469:10.1111/j.1467-8306.1968.tb00650.x
25:
846:{\displaystyle {\frac {\pi }{2}}}
558:{\displaystyle {\frac {\pi }{2}}}
522:{\displaystyle {\frac {\pi }{3}}}
494:{\displaystyle {\frac {\pi }{3}}}
405:{\displaystyle {\frac {\pi }{6}}}
1904:
1903:
1600:Drainage system (geomorphology)
1527:Grime, James (March 14, 2015),
1610:Strahler number (stream order)
1305:
1285:
942:
922:
780:
760:
449:
429:
304:
291:
226:function (over a whole number
1:
1506:10.1126/science.271.5256.1710
1391:SI <1.05: almost straight
1322:
1261:
1225:
1222:
1217:
1181:
1145:
1142:
1137:
1092:
1056:
1053:
1048:
1003:
967:
964:
959:
898:
862:
859:
854:
826:
805:
802:
797:
736:
700:
697:
692:
647:
611:
608:
603:
566:
538:
535:
530:
502:
474:
471:
466:
413:
385:
382:
377:
374:
369:
364:
1394:1.05 ≤ SI <1.25: winding
1384:if the slope angle varies.
100:continuously differentiable
1961:
1397:1.25 ≤ SI <1.50: twisty
117:(along the curve) and the
1899:
359:
354:
38:for an oscillating curve.
1945:Curvature (mathematics)
1723:River channel migration
345:Example with 270° angle
1653:Bar (river morphology)
1406:self-organizing system
1366:
1315:
1254:
1210:
1174:
1130:
1085:
1041:
996:
952:
891:
847:
819:
790:
729:
685:
640:
596:
559:
523:
495:
459:
406:
346:
332:
213:
157:of the curve (see the
127:dimensionless quantity
84:
76:
50:
39:
1835:Erosion and tectonics
1830:Degradation (geology)
1400:1.50 ≤ SI: meandering
1367:
1316:
1255:
1211:
1175:
1131:
1086:
1042:
997:
953:
892:
848:
820:
791:
730:
686:
641:
597:
560:
524:
496:
460:
407:
344:
333:
234:on those periods, is
214:
148:orthogonal projection
96:sinuosity coefficient
82:
56:
45:
33:
1856:Deposition (geology)
1583:Large-scale features
1341:
1265:
1229:
1185:
1149:
1096:
1060:
1007:
971:
902:
866:
830:
818:{\displaystyle \pi }
809:
740:
704:
651:
615:
570:
542:
506:
478:
417:
389:
238:
182:
105:having at least one
75:is > 1.
1498:1996Sci...271.1710S
1492:(5256): 1710–1713,
282:
1886:Sediment transport
1840:River rejuvenation
1813:Regional processes
1362:
1311:
1250:
1206:
1170:
1126:
1081:
1037:
992:
948:
887:
843:
815:
786:
725:
681:
636:
592:
555:
519:
491:
455:
402:
347:
328:
327:
265:
263:
209:
201:
155:cartographic scale
119:Euclidean distance
115:curvilinear length
85:
77:
51:
40:
1917:
1916:
1718:River bifurcation
1360:
1359:
1358:downvalley length
1356:
1347:
1327:
1326:
1309:
1303:
1293:
1248:
1204:
1168:
1124:
1121:
1079:
1035:
1032:
990:
946:
940:
930:
885:
841:
784:
778:
768:
723:
679:
676:
634:
590:
587:
553:
517:
489:
453:
447:
437:
400:
313:
262:
200:
159:coastline paradox
65:Guamo Embarcadero
16:(Redirected from
1952:
1907:
1906:
1648:Avulsion (river)
1576:River morphology
1569:
1562:
1555:
1546:
1539:
1537:
1524:
1518:
1516:
1479:
1473:
1472:
1452:
1446:
1443:
1411:
1376:index of 1, and
1371:
1369:
1368:
1363:
1361:
1357:
1354:
1353:
1348:
1345:
1320:
1318:
1317:
1312:
1310:
1308:
1304:
1299:
1294:
1289:
1280:
1269:
1259:
1257:
1256:
1251:
1249:
1244:
1233:
1215:
1213:
1212:
1207:
1205:
1200:
1189:
1179:
1177:
1176:
1171:
1169:
1164:
1153:
1135:
1133:
1132:
1127:
1125:
1123:
1122:
1117:
1111:
1100:
1090:
1088:
1087:
1082:
1080:
1075:
1064:
1046:
1044:
1043:
1038:
1036:
1034:
1033:
1028:
1022:
1011:
1001:
999:
998:
993:
991:
986:
975:
957:
955:
954:
949:
947:
945:
941:
936:
931:
926:
917:
906:
896:
894:
893:
888:
886:
881:
870:
852:
850:
849:
844:
842:
834:
824:
822:
821:
816:
795:
793:
792:
787:
785:
783:
779:
774:
769:
764:
755:
744:
734:
732:
731:
726:
724:
719:
708:
690:
688:
687:
682:
680:
678:
677:
672:
666:
655:
645:
643:
642:
637:
635:
630:
619:
601:
599:
598:
593:
591:
589:
588:
583:
574:
564:
562:
561:
556:
554:
546:
528:
526:
525:
520:
518:
510:
500:
498:
497:
492:
490:
482:
464:
462:
461:
456:
454:
452:
448:
443:
438:
433:
421:
411:
409:
408:
403:
401:
393:
352:
337:
335:
334:
329:
314:
312:
311:
284:
281:
273:
264:
261:
250:
218:
216:
215:
210:
202:
193:
107:inflection point
21:
1960:
1959:
1955:
1954:
1953:
1951:
1950:
1949:
1920:
1919:
1918:
1913:
1895:
1876:Helicoidal flow
1844:
1808:
1775:
1747:
1663:Channel pattern
1635:Alluvial rivers
1629:
1625:River sinuosity
1578:
1573:
1543:
1542:
1526:
1525:
1521:
1481:
1480:
1476:
1454:
1453:
1449:
1444:
1440:
1435:
1418:
1409:
1339:
1338:
1332:
1281:
1270:
1263:
1262:
1234:
1227:
1226:
1190:
1183:
1182:
1154:
1147:
1146:
1112:
1101:
1094:
1093:
1065:
1058:
1057:
1023:
1012:
1005:
1004:
976:
969:
968:
918:
907:
900:
899:
871:
864:
863:
828:
827:
807:
806:
756:
745:
738:
737:
709:
702:
701:
667:
656:
649:
648:
620:
613:
612:
578:
568:
567:
540:
539:
504:
503:
476:
475:
425:
415:
414:
387:
386:
303:
254:
236:
235:
180:
179:
168:
144:small intestine
136:
92:sinuosity index
73:sinuosity index
57:The meandering
34:Calculation of
28:
23:
22:
18:River sinuosity
15:
12:
11:
5:
1958:
1956:
1948:
1947:
1942:
1937:
1932:
1922:
1921:
1915:
1914:
1912:
1911:
1900:
1897:
1896:
1894:
1893:
1888:
1883:
1881:Playfair's law
1878:
1873:
1868:
1866:Exner equation
1863:
1858:
1852:
1850:
1846:
1845:
1843:
1842:
1837:
1832:
1827:
1822:
1816:
1814:
1810:
1809:
1807:
1806:
1804:Current ripple
1801:
1796:
1791:
1785:
1783:
1777:
1776:
1774:
1773:
1768:
1763:
1757:
1755:
1749:
1748:
1746:
1745:
1740:
1735:
1733:Slip-off slope
1730:
1725:
1720:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1683:Meander cutoff
1680:
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1639:
1637:
1631:
1630:
1628:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1595:Drainage basin
1592:
1590:Alluvial plain
1586:
1584:
1580:
1579:
1574:
1572:
1571:
1564:
1557:
1549:
1541:
1540:
1519:
1474:
1463:(2): 371–385.
1447:
1437:
1436:
1434:
1431:
1430:
1429:
1424:
1417:
1414:
1402:
1401:
1398:
1395:
1392:
1373:
1372:
1355:channel length
1351:
1331:
1328:
1325:
1324:
1321:
1307:
1302:
1297:
1292:
1287:
1284:
1279:
1276:
1273:
1260:
1247:
1243:
1240:
1237:
1224:
1220:
1219:
1216:
1203:
1199:
1196:
1193:
1180:
1167:
1163:
1160:
1157:
1144:
1140:
1139:
1136:
1120:
1115:
1110:
1107:
1104:
1091:
1078:
1074:
1071:
1068:
1055:
1051:
1050:
1047:
1031:
1026:
1021:
1018:
1015:
1002:
989:
985:
982:
979:
966:
962:
961:
958:
944:
939:
934:
929:
924:
921:
916:
913:
910:
897:
884:
880:
877:
874:
861:
857:
856:
853:
840:
837:
825:
814:
804:
800:
799:
796:
782:
777:
772:
767:
762:
759:
754:
751:
748:
735:
722:
718:
715:
712:
699:
695:
694:
691:
675:
670:
665:
662:
659:
646:
633:
629:
626:
623:
610:
606:
605:
602:
586:
581:
577:
565:
552:
549:
537:
533:
532:
529:
516:
513:
501:
488:
485:
473:
469:
468:
465:
451:
446:
441:
436:
431:
428:
424:
412:
399:
396:
384:
380:
379:
376:
373:
368:
362:
361:
358:
339:
338:
326:
323:
320:
317:
310:
306:
302:
299:
296:
293:
290:
287:
280:
277:
272:
268:
260:
257:
253:
246:
243:
220:
208:
205:
199:
196:
190:
187:
170:The sinuosity
167:
166:Notable values
164:
135:
134:Interpretation
132:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1957:
1946:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1927:
1925:
1910:
1902:
1901:
1898:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1861:Water erosion
1859:
1857:
1854:
1853:
1851:
1847:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1817:
1815:
1811:
1805:
1802:
1800:
1797:
1795:
1792:
1790:
1787:
1786:
1784:
1782:
1778:
1772:
1769:
1767:
1764:
1762:
1759:
1758:
1756:
1754:
1753:Bedrock river
1750:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1713:Riparian zone
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1658:Braided river
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1640:
1638:
1636:
1632:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1587:
1585:
1581:
1577:
1570:
1565:
1563:
1558:
1556:
1551:
1550:
1547:
1536:
1535:
1530:
1523:
1520:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1487:
1486:
1478:
1475:
1470:
1466:
1462:
1458:
1451:
1448:
1442:
1439:
1432:
1428:
1425:
1423:
1420:
1419:
1415:
1413:
1407:
1399:
1396:
1393:
1390:
1389:
1388:
1385:
1381:
1379:
1349:
1337:
1336:
1335:
1329:
1300:
1295:
1290:
1282:
1277:
1274:
1271:
1245:
1241:
1238:
1235:
1221:
1201:
1197:
1194:
1191:
1165:
1161:
1158:
1155:
1141:
1118:
1113:
1108:
1105:
1102:
1076:
1072:
1069:
1066:
1052:
1029:
1024:
1019:
1016:
1013:
987:
983:
980:
977:
963:
937:
932:
927:
919:
914:
911:
908:
882:
878:
875:
872:
858:
838:
835:
812:
801:
775:
770:
765:
757:
752:
749:
746:
720:
716:
713:
710:
696:
673:
668:
663:
660:
657:
631:
627:
624:
621:
607:
584:
579:
575:
550:
547:
534:
514:
511:
486:
483:
470:
444:
439:
434:
426:
422:
397:
394:
381:
372:
367:
363:
357:
356:Central angle
353:
350:
343:
324:
321:
318:
315:
308:
300:
297:
294:
288:
285:
278:
275:
270:
266:
258:
255:
251:
244:
241:
233:
229:
225:
221:
206:
203:
197:
194:
188:
185:
177:
176:
175:
173:
165:
163:
160:
156:
151:
149:
145:
140:
133:
131:
128:
124:
123:straight line
120:
116:
112:
108:
104:
101:
97:
93:
89:
81:
74:
70:
66:
62:
61:
55:
49:
44:
37:
32:
19:
1615:River valley
1534:The Guardian
1532:
1522:
1489:
1483:
1477:
1460:
1456:
1450:
1441:
1403:
1386:
1382:
1374:
1333:
348:
227:
171:
169:
152:
141:
137:
95:
91:
87:
86:
72:
58:
35:
1820:Aggradation
1771:Plunge pool
1738:Stream pool
1728:River mouth
1620:River delta
48:Luz Ardiden
1924:Categories
1871:Hack's law
1825:Base level
1766:Knickpoint
1693:Oxbow lake
1673:Floodplain
1433:References
1427:Oxbow lake
1378:meandering
360:Sinuosity
1849:Mechanics
1698:Point bar
1688:Mouth bar
1643:Anabranch
1422:Curvature
1296:−
1278:π
1275:⋅
1242:π
1239:⋅
1198:π
1195:⋅
1162:π
1159:⋅
1109:π
1106:⋅
1073:π
1070:⋅
1020:π
1017:⋅
984:π
981:⋅
915:π
912:⋅
879:π
876:⋅
836:π
813:π
753:π
750:⋅
717:π
714:⋅
664:π
661:⋅
628:π
625:⋅
576:π
548:π
512:π
484:π
440:−
423:π
395:π
322:≈
298:
279:π
267:∫
259:π
232:arclength
207:1.5708...
204:≈
195:π
88:Sinuosity
60:Rio Cauto
36:sinuosity
1909:Category
1794:Antidune
1781:Bedforms
1668:Cut bank
1514:19219185
1416:See also
1323:11.1267
378:Decimal
325:1.216...
1743:Thalweg
1678:Meander
1605:Estuary
1494:Bibcode
1485:Science
1218:5.2360
1138:3.3322
1049:2.4184
960:1.8972
855:1.5708
798:1.3552
693:1.2092
604:1.1107
531:1.0472
467:1.0115
371:Radians
366:Degrees
113:of the
109:is the
1940:Ratios
1935:Curves
1930:Rivers
1761:Canyon
1708:Rapids
1703:Riffle
1512:
1330:Rivers
375:Exact
1510:S2CID
111:ratio
103:curve
98:of a
94:, or
1799:Dune
1223:330°
1143:300°
1054:270°
965:240°
860:210°
803:180°
698:150°
609:120°
224:sine
174:of:
69:Cuba
1789:Ait
1502:doi
1490:271
1465:doi
536:90°
472:60°
383:30°
295:cos
63:at
1926::
1508:,
1500:,
1488:,
1461:58
1459:.
1346:SI
1272:11
1236:11
222:a
90:,
67:,
1568:e
1561:t
1554:v
1538:.
1517:.
1504::
1496::
1471:.
1467::
1410:π
1350:=
1306:)
1301:2
1291:6
1286:(
1283:3
1246:6
1202:3
1192:5
1166:3
1156:5
1119:2
1114:2
1103:3
1077:2
1067:3
1030:3
1025:3
1014:4
988:3
978:4
943:)
938:2
933:+
928:6
923:(
920:3
909:7
883:6
873:7
839:2
781:)
776:2
771:+
766:6
761:(
758:3
747:5
721:6
711:5
674:3
669:3
658:2
632:3
622:2
585:2
580:2
551:2
515:3
487:3
450:)
445:2
435:6
430:(
427:3
398:6
319:x
316:d
309:2
305:)
301:x
292:(
289:+
286:1
276:n
271:0
256:n
252:1
245:=
242:S
228:n
198:2
189:=
186:S
172:S
121:(
20:)
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