1561:
1553:
1270:
207:
28:
219:
195:
1493:
is the unique principal curvature. The type of umbilic is classified by the cubic form from the cubic part and corresponding
Jacobian cubic form. Whilst principal directions are not uniquely defined at an umbilic the limits of the principal directions when following a ridge on the surface can be
1689:
A submanifold is said to be umbilic (or all-umbilic) if this condition holds at every point "p". This is equivalent to saying that the submanifold can be made totally geodesic by an appropriate conformal change of the metric of the surrounding ("ambient") manifold. For example, a surface in
677:
1576:
so each sheet of the elliptical focal surface will have three cuspidal edges which come together at the umbilic focus and then switch to the other sheet. For a hyperbolic umbilic there is a single cuspidal edge which switch from one sheet to the other.
1471:
132:
lines passing through the umbilic and hyperbolic umbilics have just one. Parabolic umbilics are a transitional case with two ridges one of which is singular. Other configurations are possible for transitional cases. These cases correspond to the
1297:
on inner deltoid: cubic umbilics. Outer circle, the birth of umbilics separates star and monstar configurations. Outer deltoid, separates monstar and lemon configuration. Diagonals and the horizontal line - symmetrical umbilics with mirror
1543:
transition. Both umbilics will be hyperbolic, one with a star pattern and one with a monstar pattern. The outer circle in the diagram, a right angle cubic form, gives these transitional cases. Symbolic umbilics are a special case of this.
169:
of the vector field is either −½ (star) or ½ (lemon, monstar). Elliptical and parabolic umbilics always have the star pattern, whilst hyperbolic umbilics can be star, lemon, or monstar. This classification was first due to
1081:
776:
938:
1195:
1259:
877:
330:
570:
1309:
810:
559:
The equivalence classes of such cubics under uniform scaling form a three-dimensional real projective space and the subset of parabolic forms define a surface – called the
365:
523:
436:
385:
1491:
1291:
830:
697:
473:
553:
1680:
1660:
567:. Taking equivalence classes under rotation of the coordinate system removes one further parameter and a cubic forms can be represent by the complex cubic form
185:. Generic genus 0 surfaces have at least four umbilics of index ½. An ellipsoid of revolution has two non-generic umbilics each of which has index 1.
1573:
128:
The three main types of umbilic points are elliptical umbilics, parabolic umbilics and hyperbolic umbilics. Elliptical umbilics have the three
1303:
943:
702:
97:
is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero
Gaussian curvature. The
1925:
893:
87:
1494:
found and these correspond to the root-lines of the cubic form. The pattern of lines of curvature is determined by the
Jacobian.
1739:
1086:
1200:
839:
1864:
1826:
245:
36:
1903:
1898:
1893:
1888:
165:
around the umbilic which typically form one of three configurations: star, lemon, and lemonstar (or monstar). The
181:
0 with isolated umbilics, e.g. an ellipsoid, the index of the principal direction vector field must be 2 by the
672:{\displaystyle z^{3}+3{\overline {\beta }}z^{2}{\overline {z}}+3\beta z{\overline {z}}^{2}+{\overline {z}}^{3}}
1466:{\displaystyle z={\tfrac {1}{2}}\kappa (x^{2}+y^{2})+{\tfrac {1}{3}}(ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3})+\ldots }
70:
Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the
129:
781:
182:
1814:
122:
118:
335:
125:
states that every smooth surface topologically equivalent to the sphere has at least two umbilics.
485:
398:
1598:
1293:—plane. The Inner deltoid give parabolic umbilics, separates elliptical and hyperbolic umbilics.
166:
17:
1602:
1590:
370:
1476:
1276:
815:
778:, the inner deltoid, elliptical forms are inside the deltoid and hyperbolic one outside. If
682:
448:
1772:
1749:
887:
531:
110:
1665:
1645:
8:
178:
171:
102:
52:
1776:
1910:
1529:
1294:
564:
155:
71:
1784:
1930:
1860:
1822:
1735:
1539:
In a generic family of surfaces umbilics can be created, or destroyed, in pairs: the
1535:
On the diagonals and the horizontal line - symmetrical umbilics with mirror symmetry
1780:
1727:
1560:
1552:
560:
84:
Does every smooth topological sphere in
Euclidean space have at least two umbilics?
48:
1852:
1745:
114:
206:
1877:
1302:
Any surface with an isolated umbilic point at the origin can be expressed as a
239:
The classification of umbilics is closely linked to the classification of real
1763:
Berry, M V; Hannay, J H (1977). "Umbilic points on
Gaussian random surfaces".
1731:
1699:
105:
every point is a flat umbilic. A closed surface topologically equivalent to a
1919:
1569:
161:
Umbilics can also be characterised by the pattern of the principal direction
98:
218:
162:
1568:
The elliptical umbilics and hyperbolic umbilics have distinctly different
194:
1580:
1269:
1810:
240:
109:
may or may not have zero umbilics, but every closed surface of nonzero
76:
47:
are points on a surface that are locally spherical. At such points the
1690:
Euclidean space is umbilic if and only if it is a piece of a sphere.
1197:. Using complex numbers the Jacobian is a parabolic cubic form when
1076:{\displaystyle F(x,y)=(x^{2}+y^{2},ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3})}
771:{\displaystyle \beta ={\tfrac {1}{3}}(2e^{i\theta }+e^{-2i\theta })}
31:
Lines of curvature on an ellipsoid showing umbilic points (red).
27:
94:
933:{\displaystyle F:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}}
106:
64:
1556:
A surface with an elliptical umbilic, and its focal surface.
1564:
A surface with a hyperbolic umbilic and its focal surface.
101:
is an example of a surface with a flat umbilic and on the
1882:
Leçons sur la théorie génerale des surfaces: Volumes I–IV
1911:
Pictures of star, lemon, monstar, and further references
1521:
Between outer circle and outer deltoid - monstar pattern
1190:{\displaystyle bx^{3}+(2c-a)x^{2}y+(d-2b)xy^{2}-cy^{3}}
1581:
Definition in higher dimension in
Riemannian manifolds
1367:
1320:
858:
713:
1722:
Berger, Marcel (2010), "The Caradéodory conjecture",
1668:
1648:
1479:
1312:
1279:
1203:
1089:
946:
896:
842:
832:
is not a cube root of unity then the cubic form is a
818:
784:
705:
685:
573:
534:
488:
451:
401:
373:
338:
248:
1497:
The classification of umbilic points is as follows:
1254:{\displaystyle \beta =-2e^{i\theta }-e^{-2i\theta }}
1261:, the outer deltoid in the classification diagram.
1083:. Up to a constant multiple this is the cubic form
872:{\displaystyle \left|\beta \right|={\tfrac {1}{3}}}
1674:
1654:
1485:
1465:
1285:
1253:
1189:
1075:
932:
871:
824:
804:
770:
691:
671:
547:
517:
467:
430:
379:
359:
324:
189:configurations of lines of curvature near umbilics
1601:is some normal vector tensor the induced metric (
387:. There are a number of possibilities including:
1917:
1859:, Cambridge University Press, pp. 198–213,
1532:of the inner deltoid - cubic (symbolic) umbilics
332:. A cubic form will have a number of root lines
325:{\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}}
78:
367:such that the cubic form is zero for all real
1809:
229:
1512:Outside inner deltoid - hyperbolic umbilics
441:Three lines, two of which are coincident: a
1762:
1501:Inside inner deltoid - elliptical umbilics
879:then two of the root lines are orthogonal.
836:which play a special role for umbilics. If
1726:, Springer, Heidelberg, pp. 389–390,
1572:. A ridge on the surface corresponds to a
59:. The name "umbilic" comes from the Latin
1504:On inner circle - two ridge lines tangent
1264:
920:
905:
55:are equal, and every tangent vector is a
51:in all directions are equal, hence, both
1851:
1805:
1803:
1559:
1551:
1268:
26:
1876:
1819:Catastrophe Theory and its Applications
528:Three coincident lines, standard model
154:elementary catastrophes of René Thom's
88:(more unsolved problems in mathematics)
14:
1918:
1847:
1845:
1843:
1841:
1839:
1837:
1721:
1800:
1682:is the mean curvature vector at
1524:Outside outer deltoid - lemon pattern
1509:On inner deltoid - parabolic umbilics
805:{\displaystyle \left|\beta \right|=1}
1834:
1702:– an anatomical term meaning
1518:On outer circle - birth of umbilics
24:
1515:Inside outer circle - star pattern
25:
1942:
1926:Differential geometry of surfaces
1605:). Equivalently, for all vectors
37:differential geometry of surfaces
1547:
679:with a single complex parameter
217:
205:
193:
174:and the names come from Hannay.
890:of the vector valued function
79:Unsolved problem in mathematics
1791:
1756:
1715:
1454:
1378:
1360:
1334:
1155:
1140:
1121:
1106:
1070:
968:
962:
950:
915:
765:
724:
354:
342:
234:
13:
1:
1709:
699:. Parabolic forms occur when
360:{\displaystyle \lambda (x,y)}
1704:of, or relating to the navel
1273:Umbilic classification, the
658:
638:
615:
595:
518:{\displaystyle x^{2}y+y^{3}}
431:{\displaystyle x^{2}y-y^{3}}
117:, has at least one umbilic.
7:
1785:10.1088/0305-4470/10/11/009
1693:
10:
1947:
230:Classification of umbilics
1857:Geometric Differentiation
1732:10.1007/978-3-540-70997-8
882:A second cubic form, the
391:Three distinct lines: an
113:, embedded smoothly into
886:is formed by taking the
380:{\displaystyle \lambda }
1599:Second fundamental form
1486:{\displaystyle \kappa }
834:right-angled cubic form
123:Constantin Carathéodory
1676:
1656:
1603:First fundamental form
1597:, the (vector-valued)
1591:Riemannian submanifold
1565:
1557:
1487:
1467:
1299:
1287:
1286:{\displaystyle \beta }
1265:Umbilic classification
1255:
1191:
1077:
934:
873:
826:
825:{\displaystyle \beta }
806:
772:
693:
692:{\displaystyle \beta }
673:
549:
519:
478:A single real line: a
469:
468:{\displaystyle x^{2}y}
432:
381:
361:
326:
119:An unproven conjecture
32:
1677:
1657:
1563:
1555:
1488:
1468:
1288:
1272:
1256:
1192:
1078:
935:
874:
827:
807:
773:
694:
674:
550:
548:{\displaystyle x^{3}}
520:
480:hyperbolic cubic form
470:
433:
393:elliptical cubic form
382:
362:
327:
183:Poincaré–Hopf theorem
39:in three dimensions,
30:
1675:{\displaystyle \nu }
1666:
1655:{\displaystyle \nu }
1646:
1593:is umbilical if, at
1477:
1310:
1277:
1201:
1087:
944:
894:
888:Jacobian determinant
840:
816:
782:
703:
683:
571:
532:
486:
449:
443:parabolic cubic form
399:
371:
336:
246:
111:Euler characteristic
53:principal curvatures
1777:1977JPhA...10.1809B
57:principal direction
1884:, Gauthier-Villars
1672:
1652:
1566:
1558:
1483:
1463:
1376:
1329:
1300:
1283:
1251:
1187:
1073:
930:
869:
867:
822:
802:
768:
722:
689:
669:
565:Christopher Zeeman
545:
515:
465:
428:
377:
357:
322:
177:For surfaces with
156:catastrophe theory
72:Gaussian curvature
33:
1741:978-3-540-70996-1
1724:Geometry revealed
1541:birth of umbilics
1375:
1328:
1306:parameterisation
866:
721:
661:
641:
618:
598:
482:, standard model
445:, standard model
395:, standard model
49:normal curvatures
16:(Redirected from
1938:
1885:
1870:
1869:
1853:Porteous, Ian R.
1849:
1832:
1831:
1807:
1798:
1795:
1789:
1788:
1760:
1754:
1752:
1719:
1681:
1679:
1678:
1673:
1661:
1659:
1658:
1653:
1492:
1490:
1489:
1484:
1472:
1470:
1469:
1464:
1453:
1452:
1437:
1436:
1412:
1411:
1393:
1392:
1377:
1368:
1359:
1358:
1346:
1345:
1330:
1321:
1292:
1290:
1289:
1284:
1260:
1258:
1257:
1252:
1250:
1249:
1228:
1227:
1196:
1194:
1193:
1188:
1186:
1185:
1170:
1169:
1133:
1132:
1102:
1101:
1082:
1080:
1079:
1074:
1069:
1068:
1053:
1052:
1028:
1027:
1009:
1008:
993:
992:
980:
979:
939:
937:
936:
931:
929:
928:
923:
914:
913:
908:
878:
876:
875:
870:
868:
859:
853:
831:
829:
828:
823:
811:
809:
808:
803:
795:
777:
775:
774:
769:
764:
763:
742:
741:
723:
714:
698:
696:
695:
690:
678:
676:
675:
670:
668:
667:
662:
654:
648:
647:
642:
634:
619:
611:
609:
608:
599:
591:
583:
582:
561:umbilic bracelet
554:
552:
551:
546:
544:
543:
524:
522:
521:
516:
514:
513:
498:
497:
474:
472:
471:
466:
461:
460:
437:
435:
434:
429:
427:
426:
411:
410:
386:
384:
383:
378:
366:
364:
363:
358:
331:
329:
328:
323:
321:
320:
305:
304:
280:
279:
261:
260:
221:
209:
197:
80:
45:umbilical points
21:
1946:
1945:
1941:
1940:
1939:
1937:
1936:
1935:
1916:
1915:
1878:Darboux, Gaston
1873:
1867:
1850:
1835:
1829:
1808:
1801:
1797:Porteous, p 208
1796:
1792:
1771:(11): 1809–21.
1761:
1757:
1742:
1720:
1716:
1712:
1696:
1667:
1664:
1663:
1647:
1644:
1643:
1633:
1583:
1550:
1478:
1475:
1474:
1448:
1444:
1432:
1428:
1407:
1403:
1388:
1384:
1366:
1354:
1350:
1341:
1337:
1319:
1311:
1308:
1307:
1278:
1275:
1274:
1267:
1236:
1232:
1220:
1216:
1202:
1199:
1198:
1181:
1177:
1165:
1161:
1128:
1124:
1097:
1093:
1088:
1085:
1084:
1064:
1060:
1048:
1044:
1023:
1019:
1004:
1000:
988:
984:
975:
971:
945:
942:
941:
924:
919:
918:
909:
904:
903:
895:
892:
891:
857:
843:
841:
838:
837:
817:
814:
813:
785:
783:
780:
779:
750:
746:
734:
730:
712:
704:
701:
700:
684:
681:
680:
663:
653:
652:
643:
633:
632:
610:
604:
600:
590:
578:
574:
572:
569:
568:
539:
535:
533:
530:
529:
509:
505:
493:
489:
487:
484:
483:
456:
452:
450:
447:
446:
422:
418:
406:
402:
400:
397:
396:
372:
369:
368:
337:
334:
333:
316:
312:
300:
296:
275:
271:
256:
252:
247:
244:
243:
237:
232:
225:
222:
213:
210:
201:
198:
153:
146:
139:
115:Euclidean space
91:
90:
85:
82:
23:
22:
15:
12:
11:
5:
1944:
1934:
1933:
1928:
1914:
1913:
1908:
1907:
1906:
1901:
1896:
1891:
1872:
1871:
1865:
1833:
1827:
1799:
1790:
1755:
1740:
1713:
1711:
1708:
1707:
1706:
1695:
1692:
1671:
1651:
1629:
1625:) =
1582:
1579:
1574:cuspidal edges
1570:focal surfaces
1549:
1546:
1537:
1536:
1533:
1527:
1526:
1525:
1522:
1519:
1516:
1510:
1507:
1506:
1505:
1482:
1462:
1459:
1456:
1451:
1447:
1443:
1440:
1435:
1431:
1427:
1424:
1421:
1418:
1415:
1410:
1406:
1402:
1399:
1396:
1391:
1387:
1383:
1380:
1374:
1371:
1365:
1362:
1357:
1353:
1349:
1344:
1340:
1336:
1333:
1327:
1324:
1318:
1315:
1282:
1266:
1263:
1248:
1245:
1242:
1239:
1235:
1231:
1226:
1223:
1219:
1215:
1212:
1209:
1206:
1184:
1180:
1176:
1173:
1168:
1164:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1131:
1127:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1100:
1096:
1092:
1072:
1067:
1063:
1059:
1056:
1051:
1047:
1043:
1040:
1037:
1034:
1031:
1026:
1022:
1018:
1015:
1012:
1007:
1003:
999:
996:
991:
987:
983:
978:
974:
970:
967:
964:
961:
958:
955:
952:
949:
927:
922:
917:
912:
907:
902:
899:
865:
862:
856:
852:
849:
846:
821:
801:
798:
794:
791:
788:
767:
762:
759:
756:
753:
749:
745:
740:
737:
733:
729:
726:
720:
717:
711:
708:
688:
666:
660:
657:
651:
646:
640:
637:
631:
628:
625:
622:
617:
614:
607:
603:
597:
594:
589:
586:
581:
577:
557:
556:
542:
538:
526:
512:
508:
504:
501:
496:
492:
476:
464:
459:
455:
439:
425:
421:
417:
414:
409:
405:
376:
356:
353:
350:
347:
344:
341:
319:
315:
311:
308:
303:
299:
295:
292:
289:
286:
283:
278:
274:
270:
267:
264:
259:
255:
251:
236:
233:
231:
228:
227:
226:
223:
216:
214:
211:
204:
202:
199:
192:
190:
151:
144:
137:
86:
83:
77:
9:
6:
4:
3:
2:
1943:
1932:
1929:
1927:
1924:
1923:
1921:
1912:
1909:
1905:
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1886:
1883:
1879:
1875:
1874:
1868:
1866:0-521-00264-8
1862:
1858:
1854:
1848:
1846:
1844:
1842:
1840:
1838:
1830:
1828:0-273-01029-8
1824:
1820:
1816:
1812:
1806:
1804:
1794:
1786:
1782:
1778:
1774:
1770:
1766:
1759:
1751:
1747:
1743:
1737:
1733:
1729:
1725:
1718:
1714:
1705:
1701:
1698:
1697:
1691:
1687:
1685:
1669:
1649:
1641:
1637:
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1592:
1588:
1578:
1575:
1571:
1562:
1554:
1548:Focal surface
1545:
1542:
1534:
1531:
1528:
1523:
1520:
1517:
1514:
1513:
1511:
1508:
1503:
1502:
1500:
1499:
1498:
1495:
1480:
1460:
1457:
1449:
1445:
1441:
1438:
1433:
1429:
1425:
1422:
1419:
1416:
1413:
1408:
1404:
1400:
1397:
1394:
1389:
1385:
1381:
1372:
1369:
1363:
1355:
1351:
1347:
1342:
1338:
1331:
1325:
1322:
1316:
1313:
1305:
1296:
1280:
1271:
1262:
1246:
1243:
1240:
1237:
1233:
1229:
1224:
1221:
1217:
1213:
1210:
1207:
1204:
1182:
1178:
1174:
1171:
1166:
1162:
1158:
1152:
1149:
1146:
1143:
1137:
1134:
1129:
1125:
1118:
1115:
1112:
1109:
1103:
1098:
1094:
1090:
1065:
1061:
1057:
1054:
1049:
1045:
1041:
1038:
1035:
1032:
1029:
1024:
1020:
1016:
1013:
1010:
1005:
1001:
997:
994:
989:
985:
981:
976:
972:
965:
959:
956:
953:
947:
925:
910:
900:
897:
889:
885:
880:
863:
860:
854:
850:
847:
844:
835:
819:
799:
796:
792:
789:
786:
760:
757:
754:
751:
747:
743:
738:
735:
731:
727:
718:
715:
709:
706:
686:
664:
655:
649:
644:
635:
629:
626:
623:
620:
612:
605:
601:
592:
587:
584:
579:
575:
566:
562:
540:
536:
527:
510:
506:
502:
499:
494:
490:
481:
477:
462:
457:
453:
444:
440:
423:
419:
415:
412:
407:
403:
394:
390:
389:
388:
374:
351:
348:
345:
339:
317:
313:
309:
306:
301:
297:
293:
290:
287:
284:
281:
276:
272:
268:
265:
262:
257:
253:
249:
242:
220:
215:
208:
203:
196:
191:
188:
187:
186:
184:
180:
175:
173:
168:
164:
159:
157:
150:
143:
136:
131:
126:
124:
120:
116:
112:
108:
104:
100:
99:monkey saddle
96:
89:
75:
74:is positive.
73:
68:
66:
62:
58:
54:
50:
46:
42:
38:
29:
19:
1881:
1856:
1818:
1815:Stewart, Ian
1793:
1768:
1764:
1758:
1723:
1717:
1703:
1688:
1683:
1639:
1635:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1594:
1586:
1584:
1567:
1540:
1538:
1496:
1301:
883:
881:
833:
558:
479:
442:
392:
238:
176:
163:vector field
160:
148:
141:
134:
127:
92:
69:
60:
56:
44:
40:
34:
1811:Poston, Tim
241:cubic forms
235:Cubic forms
1920:Categories
1899:Volume III
1821:, Pitman,
1765:J. Phys. A
1710:References
1304:Monge form
1904:Volume IV
1894:Volume II
1880:(1896) ,
1700:umbilical
1670:ν
1650:ν
1481:κ
1461:…
1332:κ
1298:symmetry.
1281:β
1247:θ
1238:−
1230:−
1225:θ
1211:−
1205:β
1172:−
1147:−
1116:−
916:→
848:β
820:β
790:β
761:θ
752:−
739:θ
707:β
687:β
659:¯
639:¯
627:β
616:¯
596:¯
593:β
416:−
375:λ
340:λ
61:umbilicus
1931:Surfaces
1889:Volume I
1855:(2001),
1817:(1978),
1694:See also
1662:, where
1585:A point
1473:, where
884:Jacobian
41:umbilics
1773:Bibcode
1750:2724440
1638:,
1621:,
1609:,
212:Monstar
172:Darboux
35:In the
18:Umbilic
1863:
1825:
1748:
1738:
224:Lemon
95:sphere
1617:, II(
1589:in a
1530:Cusps
1295:Cusps
179:genus
167:index
130:ridge
107:torus
103:plane
65:navel
1861:ISBN
1823:ISBN
1736:ISBN
812:and
200:Star
147:and
93:The
1781:doi
1728:doi
1686:.
1613:at
563:by
121:of
67:).
43:or
1922::
1836:^
1813:;
1802:^
1779:.
1769:10
1767:.
1746:MR
1744:,
1734:,
940:,
158:.
140:,
1787:.
1783::
1775::
1753:.
1730::
1684:p
1642:)
1640:V
1636:U
1634:(
1631:p
1627:g
1623:V
1619:U
1615:p
1611:V
1607:U
1595:p
1587:p
1458:+
1455:)
1450:3
1446:y
1442:d
1439:+
1434:2
1430:y
1426:x
1423:c
1420:3
1417:+
1414:y
1409:2
1405:x
1401:b
1398:3
1395:+
1390:3
1386:x
1382:a
1379:(
1373:3
1370:1
1364:+
1361:)
1356:2
1352:y
1348:+
1343:2
1339:x
1335:(
1326:2
1323:1
1317:=
1314:z
1244:i
1241:2
1234:e
1222:i
1218:e
1214:2
1208:=
1183:3
1179:y
1175:c
1167:2
1163:y
1159:x
1156:)
1153:b
1150:2
1144:d
1141:(
1138:+
1135:y
1130:2
1126:x
1122:)
1119:a
1113:c
1110:2
1107:(
1104:+
1099:3
1095:x
1091:b
1071:)
1066:3
1062:y
1058:d
1055:+
1050:2
1046:y
1042:x
1039:c
1036:3
1033:+
1030:y
1025:2
1021:x
1017:b
1014:3
1011:+
1006:3
1002:x
998:a
995:,
990:2
986:y
982:+
977:2
973:x
969:(
966:=
963:)
960:y
957:,
954:x
951:(
948:F
926:2
921:R
911:2
906:R
901::
898:F
864:3
861:1
855:=
851:|
845:|
800:1
797:=
793:|
787:|
766:)
758:i
755:2
748:e
744:+
736:i
732:e
728:2
725:(
719:3
716:1
710:=
665:3
656:z
650:+
645:2
636:z
630:z
624:3
621:+
613:z
606:2
602:z
588:3
585:+
580:3
576:z
555:.
541:3
537:x
525:.
511:3
507:y
503:+
500:y
495:2
491:x
475:.
463:y
458:2
454:x
438:.
424:3
420:y
413:y
408:2
404:x
355:)
352:y
349:,
346:x
343:(
318:3
314:y
310:d
307:+
302:2
298:y
294:x
291:c
288:3
285:+
282:y
277:2
273:x
269:b
266:3
263:+
258:3
254:x
250:a
152:4
149:D
145:5
142:D
138:4
135:D
81::
63:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
