365:
325:
292:
125:
85:
52:
1319:
1335:
759:
1378:
To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, and decides to go in the direction where the slope is steepest. The other one is more cautious and does not want to either climb or descend, choosing a path which stays at the same
1342:
whose graph looks like a hill. The blue curves are the level sets; the red curves follow the direction of the gradient. The cautious hiker follows the blue paths; the bold hiker follows the red paths. Note that blue and red paths always cross at right
1852:
1567:
638:
789:
The name isocontour is also used, which means a contour of equal height. In various application areas, isocontours have received specific names, which indicate often the nature of the values of the considered function, such as
1991:
1710:
885:
1214:
999:
1283:
1040:
921:
1721:
1436:
1313:
1157:
1248:
1111:
1072:
506:
1131:
941:
1887:
1606:
1223:
shown in the figure to the right. Each curve shown is a level curve of the function, and they are spaced logarithmically: if a curve represents
2001:
825:
1411:
778:
is a level curve, which is considered independently of its neighbor curves, emphasizing that such a curve is defined by an
1379:
height. In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to each other.
485:
2059:
Simionescu, P.A. (2011). "Some
Advancements to Visualizing Constrained Functions and Inequalities of Two Variables".
2143:
2086:
Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization".
1162:
2012:
sublevel set and the lower-semicontinuity of the function implies that a function attains its minimum. The
1395:
17:
2039:
950:
1997:
1323:
1220:
1359:
795:
1847:{\displaystyle L_{c}^{+}(f)=\left\{(x_{1},\dots ,x_{n})\mid f(x_{1},\dots ,x_{n})\geq c\right\}}
1562:{\displaystyle L_{c}^{-}(f)=\left\{(x_{1},\dots ,x_{n})\mid f(x_{1},\dots ,x_{n})\leq c\right\}}
2029:
493:
1253:
1004:
890:
1288:
688:
38:
1136:
698:); so a level surface is the set of all real-valued roots of an equation in three variables
633:{\displaystyle L_{c}(f)=\left\{(x_{1},\ldots ,x_{n})\mid f(x_{1},\ldots ,x_{n})=c\right\}~.}
2115:
2017:
1226:
1084:
1045:
791:
660:; so a level curve is the set of all real-valued solutions of an equation in two variables
475:
8:
2005:
1415:
747:
2119:
1419:
1116:
926:
811:
774:
Level sets show up in many applications, often under different names. For example, an
2103:
779:
489:
2123:
364:
324:
291:
2095:
2068:
2034:
31:
1986:{\displaystyle \left\{(x_{1},\dots ,x_{n})\mid f(x_{1},\dots ,x_{n})>c\right\}}
1705:{\displaystyle \left\{(x_{1},\dots ,x_{n})\mid f(x_{1},\dots ,x_{n})<c\right\}}
2111:
799:
124:
84:
51:
1403:
1402:. At a critical point, a level set may be reduced to a point (for example at a
775:
2137:
2107:
803:
782:. Analogously, a level surface is sometimes called an implicit surface or an
758:
1387:
1079:
767:
732:
652:
770:. Red curves are closest to the viewer, while yellow curves are farthest.
1074:
lies on the circle of radius 5 centered at the origin. More generally, a
763:
643:
When the number of independent variables is two, a level set is called a
467:
37:"Level surface" redirects here. For the application to force fields, see
2099:
2013:
783:
694:
2072:
2009:
1318:
923:
of this function consists of those points that lie at a distance of
1391:
1363:
807:
1370:
at a point is either zero, or perpendicular to the level set of
1349:
1075:
944:
1334:
646:
2061:
Journal of
Computing and Information Science in Engineering
1382:
A consequence of this theorem (and its proof) is that if
27:
Subset of a function's domain on which its value is equal
1890:
1724:
1609:
1439:
1291:
1256:
1229:
1165:
1139:
1119:
1087:
1048:
1007:
953:
929:
893:
828:
736:, the set of all real-valued roots of an equation in
509:
1985:
1846:
1704:
1561:
1307:
1277:
1242:
1208:
1151:
1125:
1105:
1066:
1034:
993:
935:
915:
879:
632:
178:-dimensional level sets for functions of the form
1329:
2135:
418:-dimensional level sets of non-linear functions
1425:
822:Consider the 2-dimensional Euclidean distance:
1285:, and the curve directly "outside" represents
880:{\displaystyle d(x,y)={\sqrt {x^{2}+y^{2}}}}
1042:. Geometrically, this means that the point
2058:
2094:(1). Berlin, Heidelberg: Springer: 1–25.
1250:, the curve directly "within" represents
1333:
1317:
757:
2016:of all the sublevel sets characterizes
14:
2136:
2085:
1209:{\displaystyle L_{r}(y\mapsto m(x,y))}
370:Curved surfaces at constant slices of
330:Contour curves at constant slices of
30:For the computational technique, see
1386:is differentiable, a level set is a
753:
492:where the function takes on a given
746:A level set is a special case of a
24:
2088:Mathematical Programming, Series A
453:-dimensional Euclidean space, for
273:-dimensional Euclidean space, for
25:
2155:
994:{\displaystyle (3,4)\in L_{5}(d)}
766:function's level surfaces with a
1219:A second example is the plot of
1159:can be defined as the level set
363:
323:
290:
123:
83:
50:
1996:Sublevel sets are important in
1322:Log-spaced level curve plot of
2079:
2052:
1969:
1937:
1928:
1896:
1830:
1798:
1789:
1757:
1746:
1740:
1688:
1656:
1647:
1615:
1545:
1513:
1504:
1472:
1461:
1455:
1330:Level sets versus the gradient
1203:
1200:
1188:
1182:
1176:
1100:
1088:
1061:
1049:
1023:
1011:
988:
982:
966:
954:
910:
904:
844:
832:
610:
578:
569:
537:
526:
520:
13:
1:
2045:
943:from the origin, that make a
297:Points at constant slices of
130:Planes at constant slices of
57:Points at constant slices of
1426:Sublevel and superlevel sets
90:Lines at constant slices of
7:
2040:Level set (data structures)
2023:
817:
10:
2160:
685:, a level set is called a
36:
29:
1278:{\displaystyle L_{x/10}}
1035:{\displaystyle d(3,4)=5}
916:{\displaystyle L_{r}(d)}
1865:(or, alternatively, an
1416:self-intersection point
1308:{\displaystyle L_{10x}}
725:. For higher values of
2144:Multivariable calculus
1987:
1848:
1706:
1580:(or, alternatively, a
1563:
1344:
1326:
1309:
1279:
1244:
1210:
1153:
1152:{\displaystyle x\in M}
1127:
1107:
1068:
1036:
995:
937:
917:
881:
771:
634:
2018:quasiconvex functions
2002:Weierstrass's theorem
1988:
1875:strict superlevel set
1849:
1707:
1564:
1337:
1324:Himmelblau's function
1321:
1310:
1280:
1245:
1243:{\displaystyle L_{x}}
1221:Himmelblau's function
1211:
1154:
1128:
1108:
1106:{\displaystyle (M,m)}
1069:
1067:{\displaystyle (3,4)}
1037:
996:
938:
918:
882:
761:
729:, the level set is a
635:
39:Equipotential surface
1888:
1722:
1607:
1437:
1338:Consider a function
1289:
1254:
1227:
1163:
1137:
1117:
1085:
1046:
1005:
951:
927:
891:
826:
507:
476:real-valued function
1998:minimization theory
1739:
1454:
762:Intersections of a
2100:10.1007/PL00011414
1983:
1844:
1725:
1702:
1559:
1440:
1430:A set of the form
1345:
1327:
1305:
1275:
1240:
1206:
1149:
1123:
1103:
1064:
1032:
991:
933:
913:
877:
812:indifference curve
772:
630:
265:are constants, in
2073:10.1115/1.3570770
1410:) or may have a
1126:{\displaystyle r}
936:{\displaystyle r}
875:
780:implicit equation
754:Alternative names
626:
16:(Redirected from
2151:
2128:
2127:
2083:
2077:
2076:
2056:
2035:Level-set method
1992:
1990:
1989:
1984:
1982:
1978:
1968:
1967:
1949:
1948:
1927:
1926:
1908:
1907:
1853:
1851:
1850:
1845:
1843:
1839:
1829:
1828:
1810:
1809:
1788:
1787:
1769:
1768:
1738:
1733:
1711:
1709:
1708:
1703:
1701:
1697:
1687:
1686:
1668:
1667:
1646:
1645:
1627:
1626:
1568:
1566:
1565:
1560:
1558:
1554:
1544:
1543:
1525:
1524:
1503:
1502:
1484:
1483:
1453:
1448:
1409:
1401:
1385:
1373:
1369:
1357:
1354:If the function
1314:
1312:
1311:
1306:
1304:
1303:
1284:
1282:
1281:
1276:
1274:
1273:
1269:
1249:
1247:
1246:
1241:
1239:
1238:
1215:
1213:
1212:
1207:
1175:
1174:
1158:
1156:
1155:
1150:
1132:
1130:
1129:
1124:
1112:
1110:
1109:
1104:
1073:
1071:
1070:
1065:
1041:
1039:
1038:
1033:
1000:
998:
997:
992:
981:
980:
942:
940:
939:
934:
922:
920:
919:
914:
903:
902:
886:
884:
883:
878:
876:
874:
873:
861:
860:
851:
742:
728:
724:
715:
706:
684:
677:
668:
650:, also known as
639:
637:
636:
631:
624:
623:
619:
609:
608:
590:
589:
568:
567:
549:
548:
519:
518:
499:
484:
480:
459:
452:
444:
417:
404:
367:
357:
327:
317:
294:
279:
272:
264:
241:
177:
164:
127:
117:
87:
77:
54:
32:Level-set method
21:
2159:
2158:
2154:
2153:
2152:
2150:
2149:
2148:
2134:
2133:
2132:
2131:
2084:
2080:
2057:
2053:
2048:
2026:
1963:
1959:
1944:
1940:
1922:
1918:
1903:
1899:
1895:
1891:
1889:
1886:
1885:
1867:upper level set
1824:
1820:
1805:
1801:
1783:
1779:
1764:
1760:
1756:
1752:
1734:
1729:
1723:
1720:
1719:
1682:
1678:
1663:
1659:
1641:
1637:
1622:
1618:
1614:
1610:
1608:
1605:
1604:
1594:strict sublevel
1582:lower level set
1539:
1535:
1520:
1516:
1498:
1494:
1479:
1475:
1471:
1467:
1449:
1444:
1438:
1435:
1434:
1428:
1407:
1399:
1396:critical points
1383:
1371:
1367:
1355:
1332:
1296:
1292:
1290:
1287:
1286:
1265:
1261:
1257:
1255:
1252:
1251:
1234:
1230:
1228:
1225:
1224:
1170:
1166:
1164:
1161:
1160:
1138:
1135:
1134:
1118:
1115:
1114:
1086:
1083:
1082:
1047:
1044:
1043:
1006:
1003:
1002:
976:
972:
952:
949:
948:
947:. For example,
928:
925:
924:
898:
894:
892:
889:
888:
869:
865:
856:
852:
850:
827:
824:
823:
820:
756:
737:
726:
723:
717:
714:
708:
705:
699:
679:
676:
670:
667:
661:
604:
600:
585:
581:
563:
559:
544:
540:
536:
532:
514:
510:
508:
505:
504:
497:
482:
478:
464:
463:
462:
461:
454:
446:
442:
436:
429:
419:
411:
408:
407:
406:
402:
395:
388:
377:
371:
368:
360:
359:
355:
348:
337:
331:
328:
320:
319:
315:
304:
298:
295:
284:
283:
282:
281:
274:
266:
262:
256:
249:
243:
239:
235:
229:
223:
216:
210:
202:
196:
189:
179:
171:
168:
167:
166:
162:
155:
148:
137:
131:
128:
120:
119:
115:
108:
97:
91:
88:
80:
79:
75:
64:
58:
55:
42:
35:
28:
23:
22:
15:
12:
11:
5:
2157:
2147:
2146:
2130:
2129:
2078:
2050:
2049:
2047:
2044:
2043:
2042:
2037:
2032:
2025:
2022:
1994:
1993:
1981:
1977:
1974:
1971:
1966:
1962:
1958:
1955:
1952:
1947:
1943:
1939:
1936:
1933:
1930:
1925:
1921:
1917:
1914:
1911:
1906:
1902:
1898:
1894:
1859:superlevel set
1855:
1854:
1842:
1838:
1835:
1832:
1827:
1823:
1819:
1816:
1813:
1808:
1804:
1800:
1797:
1794:
1791:
1786:
1782:
1778:
1775:
1772:
1767:
1763:
1759:
1755:
1751:
1748:
1745:
1742:
1737:
1732:
1728:
1713:
1712:
1700:
1696:
1693:
1690:
1685:
1681:
1677:
1674:
1671:
1666:
1662:
1658:
1655:
1652:
1649:
1644:
1640:
1636:
1633:
1630:
1625:
1621:
1617:
1613:
1570:
1569:
1557:
1553:
1550:
1547:
1542:
1538:
1534:
1531:
1528:
1523:
1519:
1515:
1512:
1509:
1506:
1501:
1497:
1493:
1490:
1487:
1482:
1478:
1474:
1470:
1466:
1463:
1460:
1457:
1452:
1447:
1443:
1427:
1424:
1404:local extremum
1376:
1375:
1374:at that point.
1360:differentiable
1331:
1328:
1302:
1299:
1295:
1272:
1268:
1264:
1260:
1237:
1233:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1173:
1169:
1148:
1145:
1142:
1122:
1102:
1099:
1096:
1093:
1090:
1063:
1060:
1057:
1054:
1051:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
990:
987:
984:
979:
975:
971:
968:
965:
962:
959:
956:
932:
912:
909:
906:
901:
897:
872:
868:
864:
859:
855:
849:
846:
843:
840:
837:
834:
831:
819:
816:
776:implicit curve
755:
752:
721:
712:
703:
674:
665:
641:
640:
629:
622:
618:
615:
612:
607:
603:
599:
596:
593:
588:
584:
580:
577:
574:
571:
566:
562:
558:
555:
552:
547:
543:
539:
535:
531:
528:
525:
522:
517:
513:
486:real variables
440:
434:
427:
410:
409:
400:
393:
386:
375:
369:
362:
361:
353:
346:
335:
329:
322:
321:
313:
302:
296:
289:
288:
287:
286:
285:
260:
254:
247:
237:
233:
227:
221:
214:
208:
200:
194:
187:
170:
169:
160:
153:
146:
135:
129:
122:
121:
113:
106:
95:
89:
82:
81:
73:
62:
56:
49:
48:
47:
46:
45:
26:
9:
6:
4:
3:
2:
2156:
2145:
2142:
2141:
2139:
2125:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2089:
2082:
2074:
2070:
2066:
2062:
2055:
2051:
2041:
2038:
2036:
2033:
2031:
2028:
2027:
2021:
2019:
2015:
2011:
2007:
2003:
1999:
1979:
1975:
1972:
1964:
1960:
1956:
1953:
1950:
1945:
1941:
1934:
1931:
1923:
1919:
1915:
1912:
1909:
1904:
1900:
1892:
1884:
1883:
1882:
1880:
1876:
1872:
1868:
1864:
1860:
1840:
1836:
1833:
1825:
1821:
1817:
1814:
1811:
1806:
1802:
1795:
1792:
1784:
1780:
1776:
1773:
1770:
1765:
1761:
1753:
1749:
1743:
1735:
1730:
1726:
1718:
1717:
1716:
1698:
1694:
1691:
1683:
1679:
1675:
1672:
1669:
1664:
1660:
1653:
1650:
1642:
1638:
1634:
1631:
1628:
1623:
1619:
1611:
1603:
1602:
1601:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1555:
1551:
1548:
1540:
1536:
1532:
1529:
1526:
1521:
1517:
1510:
1507:
1499:
1495:
1491:
1488:
1485:
1480:
1476:
1468:
1464:
1458:
1450:
1445:
1441:
1433:
1432:
1431:
1423:
1421:
1417:
1413:
1405:
1397:
1393:
1389:
1380:
1365:
1361:
1353:
1351:
1347:
1346:
1341:
1336:
1325:
1320:
1316:
1300:
1297:
1293:
1270:
1266:
1262:
1258:
1235:
1231:
1222:
1217:
1197:
1194:
1191:
1185:
1179:
1171:
1167:
1146:
1143:
1140:
1120:
1097:
1094:
1091:
1081:
1077:
1058:
1055:
1052:
1029:
1026:
1020:
1017:
1014:
1008:
985:
977:
973:
969:
963:
960:
957:
946:
930:
907:
899:
895:
870:
866:
862:
857:
853:
847:
841:
838:
835:
829:
815:
813:
809:
805:
801:
797:
793:
787:
785:
781:
777:
769:
765:
760:
751:
749:
744:
740:
735:
734:
720:
711:
702:
697:
696:
691:
690:
682:
673:
664:
659:
655:
654:
649:
648:
627:
620:
616:
613:
605:
601:
597:
594:
591:
586:
582:
575:
572:
564:
560:
556:
553:
550:
545:
541:
533:
529:
523:
515:
511:
503:
502:
501:
495:
491:
487:
477:
473:
469:
457:
450:
443:
433:
426:
422:
415:
399:
392:
385:
381:
374:
366:
352:
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341:
334:
326:
312:
308:
301:
293:
277:
270:
263:
253:
246:
240:
226:
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213:
207:
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186:
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175:
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105:
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94:
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1878:
1874:
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1866:
1862:
1858:
1857:is called a
1856:
1714:
1597:
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1589:
1585:
1581:
1577:
1574:sublevel set
1573:
1572:is called a
1571:
1429:
1394:outside the
1388:hypersurface
1381:
1377:
1348:
1339:
1218:
1133:centered at
1113:with radius
1080:metric space
887:A level set
821:
788:
773:
768:trefoil knot
745:
738:
733:hypersurface
730:
718:
709:
700:
693:
686:
680:
671:
662:
657:
653:contour line
651:
644:
642:
471:
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66:
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18:Level curves
1412:singularity
764:co-ordinate
743:variables.
500:, that is:
468:mathematics
2046:References
1715:Similarly
1414:such as a
1001:, because
784:isosurface
695:isosurface
2108:0025-5610
2014:convexity
2010:non-empty
2006:boundness
1954:…
1932:∣
1913:…
1873:). And a
1834:≥
1815:…
1793:∣
1774:…
1673:…
1651:∣
1632:…
1549:≤
1530:…
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1489:…
1451:−
1183:↦
1144:∈
970:∈
804:isochrone
595:…
573:∣
554:…
472:level set
458:= 1, 2, 3
278:= 1, 2, 3
2138:Category
2124:10043417
2030:Epigraph
2024:See also
2008:of some
1392:manifold
1364:gradient
818:Examples
808:isoquant
796:isotherm
678:. When
494:constant
2116:1819784
1596:set of
1350:Theorem
1343:angles.
689:surface
658:isoline
2122:
2114:
2106:
2004:, the
1586:trench
1390:and a
1362:, the
1076:sphere
945:circle
800:isogon
792:isobar
741:> 3
731:level
687:level
645:level
625:
496:value
242:where
230:+ ⋯ +
2120:S2CID
2067:(1).
2000:. By
1592:). A
1418:or a
1078:in a
748:fiber
647:curve
488:is a
474:of a
445:) in
437:, …,
257:, …,
197:, …,
2104:ISSN
1973:>
1692:<
1420:cusp
1366:of
810:and
716:and
692:(or
669:and
470:, a
451:+ 1)
416:− 1)
271:+ 1)
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176:− 1)
2096:doi
2069:doi
1881:is
1877:of
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1600:is
1588:of
1584:or
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656:or
490:set
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1980:}
1976:c
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1965:n
1961:x
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621:}
617:c
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