1420:
851:
1564:
Recently, quasi-conformal geometry has attracted attention from different fields, such as applied mathematics, computer vision and medical imaging. Computational quasi-conformal geometry has been developed, which extends the quasi-conformal theory into a discrete setting. It has found various
638:
690:
242:
1109:
425:
975:
1344:
1524:
2040:
Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in
Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich,
2014:
Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in
Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich,
1280:
846:{\displaystyle (1+|\mu |)^{2}\textstyle {\left|{\frac {\partial f}{\partial z}}\right|^{2}},\qquad (1-|\mu |)^{2}\textstyle {\left|{\frac {\partial f}{\partial z}}\right|^{2}}.}
536:
301:
679:
1960:
163:
998:
1720:
1836:
320:
2053:
2027:
1674:
1412:
The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact.
1456:
856:
The eigenvalues represent, respectively, the squared length of the major and minor axis of the ellipse obtained by pulling back along
1948:
1921:
1777:
1808:
Berichte über die
Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Classe
523:
881:
2078:
1666:
1357:
A homeomorphism is 1-quasiconformal if and only if it is conformal. Hence the identity map is always 1-quasiconformal. If
2095:
2073:
86:
2105:
480:
74:
1293:
1912:, Die Grundlehren der mathematischen Wissenschaften, vol. 126 (2nd ed.), Berlin–Heidelberg–New York:
1718:(1977), "Quasiconformal mappings, with applications to differential equations, function theory and topology",
2068:
1490:
2100:
1584:
1460:
633:{\displaystyle \left|{\frac {\partial f}{\partial z}}\right|^{2}\left|\,dz+\mu (z)\,d{\bar {z}}\right|^{2}}
1955:
1594:
1574:
1238:
1765:
267:
1878:
1589:
1803:
1484:
646:
261:
142:
237:{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=\mu (z){\frac {\partial f}{\partial z}},}
1987:
145:
1866:
2049:
2023:
1944:
1917:
1896:
1773:
1769:
1670:
1631:
153:
2041:
2015:
2003:
1979:
1969:
1935:
1886:
1853:
1815:
1791:
1729:
1704:
1688:
1647:
1639:
1621:
1612:
1579:
51:
31:
2060:
2034:
1999:
1931:
1849:
1787:
1743:
1700:
1684:
1104:{\displaystyle K=\sup _{z\in D}|K(z)|={\frac {1+\|\mu \|_{\infty }}{1-\|\mu \|_{\infty }}}}
2057:
2031:
2007:
1995:
1983:
1939:
1927:
1913:
1857:
1845:
1819:
1795:
1783:
1739:
1708:
1697:
1692:
1680:
1651:
1643:
1122:
133:. There are a variety of equivalent definitions, depending on the required smoothness of
1828:
1882:
1463:
from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that
1455:
Of central importance in the theory of quasiconformal mappings in two dimensions is the
1287:
1419:
2089:
1549:
496:
469:
312:
47:
1958:(1938), "On the solutions of quasi-linear elliptic partial differential equations",
1734:
1871:
Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences
1607:
1409:-quasiconformal. The set of 1-quasiconformal maps forms a group under composition.
1350:≠ 0, this is an example of a quasiconformal homeomorphism that is not smooth. If
1752:
1660:
1715:
1565:
important applications in medical image analysis, computer vision and graphics.
682:
420:{\displaystyle ds^{2}=\Omega (z)^{2}\left|\,dz+\mu (z)\,d{\bar {z}}\right|^{2},}
54:
domains which to first order takes small circles to small ellipses of bounded
17:
1900:
1635:
1761:
1665:, University Lecture Series, vol. 38 (2nd ed.), Providence, R.I.:
1891:
981:
484:
78:
1806:(1928), "Über einige Extremalprobleme der konformen Abbildung. I, II.",
1991:
1626:
1459:, proved by Lars Ahlfors and Lipman Bers. The theorem generalizes the
55:
530:
of the usual
Euclidean metric. The resulting metric is then given by
97:
at every point maps circles to ellipses with eccentricity bounded by
1974:
522:
Without appeal to an auxiliary metric, consider the effect of the
2045:
2019:
1867:"Planar morphometry, shear and optimal quasi-conformal mappings"
1414:
452:′ equipped with the standard Euclidean metric. The function
307:). This equation admits a geometrical interpretation. Equip
505:). When μ is zero almost everywhere, any homeomorphism in
970:{\displaystyle K(z)={\frac {1+|\mu (z)|}{1-|\mu (z)|}}.}
1431:
460:. More generally, the continuous differentiability of
444:) precisely when it is a conformal transformation from
800:
726:
1542:) and satisfies the corresponding Beltrami equation (
1493:
1296:
1241:
1224:
are both quasiconformal and have constant dilatation
1001:
884:
693:
649:
539:
323:
270:
166:
643:which, relative to the background Euclidean metric
1518:
1338:
1274:
1103:
969:
845:
673:
632:
419:
295:
236:
1961:Transactions of the American Mathematical Society
1610:(1935), "Zur Theorie der Überlagerungsflächen",
1297:
1009:
271:
1534:to the unit disk which is in the Sobolev space
1526:. Then there is a quasiconformal homeomorphism
1721:Bulletin of the American Mathematical Society
464:can be replaced by the weaker condition that
8:
1837:Notices of the American Mathematical Society
1501:
1494:
1089:
1082:
1065:
1058:
152:is quasiconformal provided it satisfies the
1757:–Dimensional Quasiconformal (QCf) Mappings
1552:. As with Riemann's mapping theorem, this
1339:{\displaystyle \max(1+s,{\frac {1}{1+s}})}
1973:
1890:
1865:Jones, Gareth Wyn; Mahadevan, L. (2013),
1733:
1625:
1504:
1492:
1315:
1295:
1266:
1261:
1252:
1251:
1240:
1149: o γ : γ ∈
1092:
1068:
1049:
1041:
1024:
1012:
1000:
956:
939:
926:
909:
900:
883:
831:
807:
801:
794:
785:
777:
757:
733:
727:
720:
711:
703:
692:
660:
659:
648:
624:
608:
607:
603:
581:
569:
545:
538:
408:
392:
391:
387:
365:
353:
331:
322:
282:
274:
269:
211:
182:
181:
167:
165:
448:equipped with this metric to the domain
39:
1829:"What Is ... a Quasiconformal Mapping?"
1519:{\displaystyle \|\mu \|_{\infty }<1}
43:
1560:Computational quasi-conformal geometry
1354:= 0, this is simply the identity map.
860:the unit circle in the tangent plane.
1114:and is called the dilatation of
93:-quasiconformal if the derivative of
7:
1910:Quasiconformal mappings in the plane
1125:is as follows. If there is a finite
1121:A definition based on the notion of
304:
157:
1908:Lehto, O.; Virtanen, K. I. (1973),
1662:Lectures on quasiconformal mappings
1556:is unique up to 3 real parameters.
1401:′-quasiconformal. The inverse of a
1275:{\displaystyle z\mapsto z\,|z|^{s}}
27:Homeomorphism between plane domains
1505:
1457:measurable Riemann mapping theorem
1451:Measurable Riemann mapping theorem
1093:
1069:
818:
810:
744:
736:
556:
548:
340:
222:
214:
178:
170:
77:-preserving homeomorphism between
25:
1696:, (reviews of the first edition:
1405:-quasiconformal homeomorphism is
479:) of functions whose first-order
1827:Heinonen, Juha (December 2006),
1467:is a simply connected domain in
1418:
1172:-quasiconformal for some finite
296:{\displaystyle \sup |\mu |<1}
1735:10.1090/S0002-9904-1977-14390-5
1129:such that for every collection
767:
46:, is a (weakly differentiable)
1333:
1300:
1290:) and has constant dilatation
1262:
1253:
1245:
1145:times the extremal length of {
1042:
1038:
1032:
1025:
957:
953:
947:
940:
927:
923:
917:
910:
894:
888:
791:
786:
778:
768:
717:
712:
704:
694:
665:
613:
600:
594:
513:) that is a weak solution of (
397:
384:
378:
350:
343:
283:
275:
208:
202:
187:
1:
1772:/ Abacus Press, p. 553,
1667:American Mathematical Society
674:{\displaystyle dzd{\bar {z}}}
1475:, and suppose that μ :
2074:Encyclopedia of Mathematics
1544:
515:
501:
440:
87:continuously differentiable
2122:
1659:Ahlfors, Lars V. (2006) ,
481:distributional derivatives
434:) > 0. Then
2069:"Quasi-conformal mapping"
1751:Caraman, Petru (1974) ,
1282:is quasiconformal (here
260:for some complex valued
2067:Zorich, V. A. (2001) ,
1585:Pseudoanalytic function
1461:Riemann mapping theorem
1389:′-quasiconformal, then
1137:the extremal length of
1956:Morrey, Charles B. Jr.
1892:10.1098/rspa.2012.0653
1575:Isothermal coordinates
1520:
1340:
1276:
1105:
971:
847:
675:
634:
421:
297:
238:
36:quasiconformal mapping
1916:, pp. VIII+258,
1766:Tunbridge Wells, Kent
1521:
1471:that is not equal to
1341:
1277:
1235:> −1 then the map
1192:> 1 then the maps
1106:
972:
848:
676:
635:
422:
298:
239:
129:′ are two domains in
1814:: 367–376, 497–502,
1760:(revised ed.),
1550:distributional sense
1491:
1373:-quasiconformal and
1294:
1239:
999:
882:
691:
647:
537:
495:is required to be a
321:
268:
164:
1943:(also available as
1883:2013RSPSA.46920653J
1595:Tissot's indicatrix
1485:Lebesgue measurable
1180:is quasiconformal.
262:Lebesgue measurable
146:partial derivatives
141:is assumed to have
2096:Conformal mappings
1877:(2153): 20120653,
1627:10.1007/BF02420945
1516:
1430:. You can help by
1336:
1272:
1101:
1023:
967:
843:
842:
841:
671:
630:
417:
293:
234:
2054:978-3-03719-055-5
2028:978-3-03719-029-6
1844:(11): 1334–1335,
1804:Grötzsch, Herbert
1770:Editura Academiei
1676:978-0-8218-3644-6
1590:Teichmüller space
1448:
1447:
1331:
1161:-quasiconformal.
1099:
1008:
962:
863:Accordingly, the
825:
751:
668:
616:
563:
400:
258:
257:
229:
194:
190:
154:Beltrami equation
81:in the plane. If
61:Intuitively, let
16:(Redirected from
2113:
2106:Complex analysis
2081:
2010:
1977:
1942:
1903:
1894:
1860:
1833:
1822:
1798:
1746:
1737:
1728:(6): 1083–1100,
1695:
1654:
1629:
1613:Acta Mathematica
1580:Quasiregular map
1525:
1523:
1522:
1517:
1509:
1508:
1443:
1440:
1422:
1415:
1345:
1343:
1342:
1337:
1332:
1330:
1316:
1281:
1279:
1278:
1273:
1271:
1270:
1265:
1256:
1110:
1108:
1107:
1102:
1100:
1098:
1097:
1096:
1074:
1073:
1072:
1050:
1045:
1028:
1022:
980:The (essential)
976:
974:
973:
968:
963:
961:
960:
943:
931:
930:
913:
901:
852:
850:
849:
844:
837:
836:
835:
830:
826:
824:
816:
808:
799:
798:
789:
781:
763:
762:
761:
756:
752:
750:
742:
734:
725:
724:
715:
707:
680:
678:
677:
672:
670:
669:
661:
639:
637:
636:
631:
629:
628:
623:
619:
618:
617:
609:
574:
573:
568:
564:
562:
554:
546:
519:) is conformal.
491:. In this case,
426:
424:
423:
418:
413:
412:
407:
403:
402:
401:
393:
358:
357:
336:
335:
302:
300:
299:
294:
286:
278:
252:
243:
241:
240:
235:
230:
228:
220:
212:
195:
193:
192:
191:
183:
176:
168:
158:
38:, introduced by
32:complex analysis
30:In mathematical
21:
2121:
2120:
2116:
2115:
2114:
2112:
2111:
2110:
2086:
2085:
2066:
1975:10.2307/1989904
1954:
1924:
1914:Springer Verlag
1907:
1864:
1831:
1826:
1802:
1780:
1750:
1714:
1677:
1658:
1606:
1603:
1571:
1562:
1500:
1489:
1488:
1453:
1444:
1438:
1435:
1428:needs expansion
1320:
1292:
1291:
1260:
1237:
1236:
1186:
1123:extremal length
1088:
1075:
1064:
1051:
997:
996:
932:
902:
880:
879:
817:
809:
803:
802:
790:
743:
735:
729:
728:
716:
689:
688:
645:
644:
580:
576:
575:
555:
547:
541:
540:
535:
534:
456:is then called
364:
360:
359:
349:
327:
319:
318:
266:
265:
250:
221:
213:
177:
169:
162:
161:
107:
40:Grötzsch (1928)
28:
23:
22:
15:
12:
11:
5:
2119:
2117:
2109:
2108:
2103:
2101:Homeomorphisms
2098:
2088:
2087:
2084:
2083:
2064:
2038:
2012:
1968:(1): 126–166,
1952:
1922:
1905:
1862:
1824:
1800:
1778:
1748:
1712:
1675:
1656:
1620:(1): 157–194,
1602:
1599:
1598:
1597:
1592:
1587:
1582:
1577:
1570:
1567:
1561:
1558:
1515:
1512:
1507:
1503:
1499:
1496:
1487:and satisfies
1452:
1449:
1446:
1445:
1425:
1423:
1335:
1329:
1326:
1323:
1319:
1314:
1311:
1308:
1305:
1302:
1299:
1288:complex number
1269:
1264:
1259:
1255:
1250:
1247:
1244:
1185:
1182:
1112:
1111:
1095:
1091:
1087:
1084:
1081:
1078:
1071:
1067:
1063:
1060:
1057:
1054:
1048:
1044:
1040:
1037:
1034:
1031:
1027:
1021:
1018:
1015:
1011:
1007:
1004:
992:) is given by
978:
977:
966:
959:
955:
952:
949:
946:
942:
938:
935:
929:
925:
922:
919:
916:
912:
908:
905:
899:
896:
893:
890:
887:
875:is defined by
854:
853:
840:
834:
829:
823:
820:
815:
812:
806:
797:
793:
788:
784:
780:
776:
773:
770:
766:
760:
755:
749:
746:
741:
738:
732:
723:
719:
714:
710:
706:
702:
699:
696:
667:
664:
658:
655:
652:
641:
640:
627:
622:
615:
612:
606:
602:
599:
596:
593:
590:
587:
584:
579:
572:
567:
561:
558:
553:
550:
544:
428:
427:
416:
411:
406:
399:
396:
390:
386:
383:
380:
377:
374:
371:
368:
363:
356:
352:
348:
345:
342:
339:
334:
330:
326:
292:
289:
285:
281:
277:
273:
256:
255:
246:
244:
233:
227:
224:
219:
216:
210:
207:
204:
201:
198:
189:
186:
180:
175:
172:
106:
103:
44:Ahlfors (1935)
26:
24:
18:Quasiconformal
14:
13:
10:
9:
6:
4:
3:
2:
2118:
2107:
2104:
2102:
2099:
2097:
2094:
2093:
2091:
2080:
2076:
2075:
2070:
2065:
2062:
2059:
2055:
2051:
2047:
2043:
2039:
2036:
2033:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1976:
1971:
1967:
1963:
1962:
1957:
1953:
1950:
1949:0-387-03303-3
1946:
1941:
1937:
1933:
1929:
1925:
1923:3-540-03303-3
1919:
1915:
1911:
1906:
1902:
1898:
1893:
1888:
1884:
1880:
1876:
1872:
1868:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1838:
1830:
1825:
1821:
1817:
1813:
1810:(in German),
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1781:
1779:0-85626-005-3
1775:
1771:
1767:
1763:
1759:
1758:
1754:
1749:
1745:
1741:
1736:
1731:
1727:
1723:
1722:
1717:
1713:
1710:
1706:
1702:
1699:
1694:
1690:
1686:
1682:
1678:
1672:
1668:
1664:
1663:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1628:
1623:
1619:
1616:(in German),
1615:
1614:
1609:
1608:Ahlfors, Lars
1605:
1604:
1600:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1572:
1568:
1566:
1559:
1557:
1555:
1551:
1547:
1546:
1541:
1537:
1533:
1529:
1513:
1510:
1497:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1450:
1442:
1433:
1429:
1426:This section
1424:
1421:
1417:
1416:
1413:
1410:
1408:
1404:
1400:
1396:
1393: o
1392:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1355:
1353:
1349:
1327:
1324:
1321:
1317:
1312:
1309:
1306:
1303:
1289:
1285:
1267:
1257:
1248:
1242:
1234:
1229:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1183:
1181:
1179:
1175:
1171:
1167:
1162:
1160:
1156:
1152:
1148:
1144:
1140:
1136:
1133:of curves in
1132:
1128:
1124:
1119:
1117:
1085:
1079:
1076:
1061:
1055:
1052:
1046:
1035:
1029:
1019:
1016:
1013:
1005:
1002:
995:
994:
993:
991:
987:
983:
964:
950:
944:
936:
933:
920:
914:
906:
903:
897:
891:
885:
878:
877:
876:
874:
870:
866:
861:
859:
838:
832:
827:
821:
813:
804:
795:
782:
774:
771:
764:
758:
753:
747:
739:
730:
721:
708:
700:
697:
687:
686:
685:
684:
662:
656:
653:
650:
625:
620:
610:
604:
597:
591:
588:
585:
582:
577:
570:
565:
559:
551:
542:
533:
532:
531:
529:
525:
520:
518:
517:
512:
508:
504:
503:
498:
497:weak solution
494:
490:
488:
482:
478:
474:
471:
470:Sobolev space
467:
463:
459:
455:
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394:
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381:
375:
372:
369:
366:
361:
354:
346:
337:
332:
328:
324:
317:
316:
315:
314:
313:metric tensor
310:
306:
290:
287:
279:
264:μ satisfying
263:
254:
247:
245:
231:
225:
217:
205:
199:
196:
184:
173:
160:
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136:
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128:
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116:
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104:
102:
100:
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92:
89:, then it is
88:
84:
80:
76:
72:
69: →
68:
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59:
57:
53:
49:
48:homeomorphism
45:
42:and named by
41:
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1432:adding to it
1427:
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2020:10.4171/029
1141:is at most
871:at a point
683:eigenvalues
458:μ-conformal
438:satisfies (
75:orientation
2090:Categories
2008:0018.40501
1984:62.0565.02
1940:0267.30016
1858:1142.30322
1820:54.0378.01
1796:0342.30015
1709:1103.30001
1693:1103.30001
1652:0012.17204
1644:61.0365.03
1601:References
1184:Properties
865:dilatation
468:be in the
143:continuous
105:Definition
2079:EMS Press
1901:1364-5021
1762:București
1636:0001-5962
1548:) in the
1506:∞
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1498:μ
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1086:μ
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775:−
745:∂
737:∂
709:μ
666:¯
614:¯
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557:∂
549:∂
398:¯
376:μ
341:Ω
311:with the
305:Bers 1977
280:μ
223:∂
215:∂
200:μ
188:¯
179:∂
171:∂
79:open sets
1569:See also
1439:May 2012
1377: :
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982:supremum
524:pullback
430:where Ω(
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109:Suppose
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65: :
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2000:1501936
1992:1989904
1932:0344463
1879:Bibcode
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