29:
829:
1115:
243:
94:
808:
468:
97:
The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere. In this example, the curvature of a 2D-surface is mapped onto a 1D unit circle.
533:
644:
617:
380:
58:
1025:
1146:
619:, the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.
1018:
987:
832:
A surface with a parabolic line and its Gauss map. A ridge passes through the parabolic line giving rise to a cusp on the Gauss map.
290:
80:
397:
860:. Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface. Cusps occur when:
200:
1141:
268:
192:
933:
921:
1161:
1119:
1011:
868:
1151:
1100:
390:. In this case a point on the submanifold is mapped to its oriented tangent subspace. One can also map to its oriented
41:
836:
The Gauss map reflects many properties of the surface: when the surface has zero
Gaussian curvature, (that is along a
477:
253:
51:
45:
37:
1090:
474:, this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as
264:
17:
272:
257:
813:
803:{\displaystyle \iint _{R}\pm |N_{u}\times N_{v}|\ du\,dv=\iint _{R}K|X_{u}\times X_{v}|\ du\,dv=\iint _{S}K\ dA}
62:
208:
114:
1045:
587:
343:
102:
1034:
196:
181:
857:
547:
1070:
845:
841:
636:
222:
204:
110:
1156:
1065:
1060:
983:
958:
1080:
875:
632:
577:
1075:
937:
925:
853:
849:
471:
199:. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The
130:
1095:
1085:
1055:
837:
566:
226:
212:
828:
1135:
338:
304:
929:
918:
327:
242:
153:
118:
977:
538:
Finally, the notion of Gauss map can be generalized to an oriented submanifold
961:
188:
122:
982:. Research Notes in Mathematics. Vol. 55. London: Pitman Publisher Ltd.
966:
976:
Thomas
Banchoff; Terence Gaffney; Clint McCrory; Daniel Dreibelbis (1982).
817:
1003:
218:
Gauss first wrote a draft on the topic in 1825 and published in 1827.
187:
The Gauss map can be defined (globally) if and only if the surface is
93:
827:
16:
This article is about differential geometry. For other uses, see
1007:
334:
the Gauss map can also be defined, and its target space is the
236:
22:
910:, English translation. Hewlett, New York: Raven Press (1965).
463:{\displaystyle {\tilde {G}}_{k,n}\cong {\tilde {G}}_{n-k,n}}
917:, (1982) Research Notes in Mathematics 55, Pitman, London.
639:. This is the original interpretation given by Gauss.
535:), hence this is consistent with the definition above.
874:
at the closure of the set of inflection points of the
125:
to the surface at that point. Namely, given a surface
647:
627:
The area of the image of the Gauss map is called the
590:
480:
400:
346:
802:
611:
527:
462:
374:
901:Disquisitiones generales circa superficies curvas
311:as a map from a hypersurface to the unit sphere
50:but its sources remain unclear because it lacks
528:{\displaystyle {\tilde {G}}_{1,n}\cong S^{n-1}}
1019:
557:. In that case, the Gauss map then goes from
8:
271:. Unsourced material may be challenged and
1026:
1012:
1004:
816:links total curvature of a surface to its
908:General investigations of curved surfaces
848:and these cusps were studied in depth by
782:
768:
754:
748:
735:
726:
717:
703:
689:
683:
670:
661:
652:
646:
603:
598:
589:
513:
494:
483:
482:
479:
442:
431:
430:
414:
403:
402:
399:
360:
349:
348:
345:
291:Learn how and when to remove this message
203:determinant of the Gauss map is equal to
117:that maps each point in the surface to a
81:Learn how and when to remove this message
92:
913:Banchoff, T., Gaffney T., McCrory C.,
572:. The target space for the Gauss map
7:
269:adding citations to reliable sources
864:The surface has a bi-tangent plane
612:{\displaystyle M=\mathbf {R} ^{n}}
394:subspace; these are equivalent as
375:{\displaystyle {\tilde {G}}_{k,n}}
14:
1147:Differential geometry of surfaces
303:The Gauss map can be defined for
172:) is a unit vector orthogonal to
1114:
1113:
599:
241:
221:There is also a Gauss map for a
27:
382:, i.e. the set of all oriented
211:of the Gauss map is called the
180:. The Gauss map is named after
930:Dan Dreibelbis' online version
755:
727:
690:
662:
488:
470:via orthogonal complement. In
436:
408:
354:
1:
882:There are two types of cusp:
840:) the Gauss map will have a
580:built on the tangent bundle
1178:
18:Gauss map (disambiguation)
15:
1109:
1101:Gauss's law for magnetism
1041:
631:and is equivalent to the
136:, the Gauss map is a map
871:crosses a parabolic line
844:. This fold may contain
36:This article includes a
1091:Gauss's law for gravity
979:Cusps of Gauss Mappings
932:(accessed 2023-07-01),
915:Cusps of Gauss Mappings
546:in an oriented ambient
322:For a general oriented
65:more precise citations.
833:
824:Cusps of the Gauss map
804:
613:
561:to the set of tangent
529:
464:
376:
156:) such that for each
98:
1142:Differential geometry
1046:Gauss composition law
831:
805:
614:
530:
465:
377:
164:, the function value
103:differential geometry
96:
1162:Carl Friedrich Gauss
1035:Carl Friedrich Gauss
928:<--broken link;
814:Gauss–Bonnet theorem
645:
588:
584:. In the case where
478:
398:
344:
265:improve this section
197:Euler characteristic
191:, in which case its
1152:Riemannian geometry
942:Koenderink, J. J.,
548:Riemannian manifold
1071:Gaussian curvature
959:Weisstein, Eric W.
946:, MIT Press (1990)
936:2008-08-02 at the
924:2008-08-02 at the
834:
800:
637:Gaussian curvature
609:
525:
460:
372:
205:Gaussian curvature
99:
38:list of references
1129:
1128:
1066:Gaussian brackets
876:asymptotic curves
793:
761:
696:
491:
472:Euclidean 3-space
439:
411:
357:
301:
300:
293:
225:, which computes
91:
90:
83:
1169:
1117:
1116:
1081:Gaussian surface
1028:
1021:
1014:
1005:
1000:
998:
996:
972:
971:
888:hyperbolic cusps
842:fold catastrophe
809:
807:
806:
801:
791:
787:
786:
759:
758:
753:
752:
740:
739:
730:
722:
721:
694:
693:
688:
687:
675:
674:
665:
657:
656:
633:surface integral
618:
616:
615:
610:
608:
607:
602:
578:Grassmann bundle
534:
532:
531:
526:
524:
523:
505:
504:
493:
492:
484:
469:
467:
466:
461:
459:
458:
441:
440:
432:
425:
424:
413:
412:
404:
381:
379:
378:
373:
371:
370:
359:
358:
350:
296:
289:
285:
282:
276:
245:
237:
86:
79:
75:
72:
66:
61:this article by
52:inline citations
31:
30:
23:
1177:
1176:
1172:
1171:
1170:
1168:
1167:
1166:
1132:
1131:
1130:
1125:
1105:
1076:Gaussian period
1037:
1032:
994:
992:
990:
975:
957:
956:
953:
938:Wayback Machine
926:Wayback Machine
896:
878:of the surface.
854:Terence Gaffney
850:Thomas Banchoff
826:
778:
744:
731:
713:
679:
666:
648:
643:
642:
629:total curvature
625:
623:Total curvature
597:
586:
585:
565:-planes in the
509:
481:
476:
475:
429:
401:
396:
395:
347:
342:
341:
297:
286:
280:
277:
262:
246:
235:
233:Generalizations
131:Euclidean space
87:
76:
70:
67:
56:
42:related reading
32:
28:
21:
12:
11:
5:
1175:
1173:
1165:
1164:
1159:
1154:
1149:
1144:
1134:
1133:
1127:
1126:
1124:
1123:
1110:
1107:
1106:
1104:
1103:
1098:
1093:
1088:
1086:Gaussian units
1083:
1078:
1073:
1068:
1063:
1061:Gauss's method
1058:
1056:Gauss notation
1053:
1048:
1042:
1039:
1038:
1033:
1031:
1030:
1023:
1016:
1008:
1002:
1001:
988:
973:
952:
951:External links
949:
948:
947:
940:
919:online version
911:
906:Gauss, K. F.,
904:
899:Gauss, K. F.,
895:
892:
880:
879:
872:
865:
838:parabolic line
825:
822:
799:
796:
790:
785:
781:
777:
774:
771:
767:
764:
757:
751:
747:
743:
738:
734:
729:
725:
720:
716:
712:
709:
706:
702:
699:
692:
686:
682:
678:
673:
669:
664:
660:
655:
651:
624:
621:
606:
601:
596:
593:
567:tangent bundle
522:
519:
516:
512:
508:
503:
500:
497:
490:
487:
457:
454:
451:
448:
445:
438:
435:
428:
423:
420:
417:
410:
407:
369:
366:
363:
356:
353:
315: ⊆
299:
298:
249:
247:
240:
234:
231:
227:linking number
213:shape operator
89:
88:
46:external links
35:
33:
26:
13:
10:
9:
6:
4:
3:
2:
1174:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1139:
1137:
1122:
1121:
1112:
1111:
1108:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1047:
1044:
1043:
1040:
1036:
1029:
1024:
1022:
1017:
1015:
1010:
1009:
1006:
991:
989:0-273-08536-0
985:
981:
980:
974:
969:
968:
963:
960:
955:
954:
950:
945:
941:
939:
935:
931:
927:
923:
920:
916:
912:
909:
905:
902:
898:
897:
893:
891:
889:
885:
884:elliptic cusp
877:
873:
870:
866:
863:
862:
861:
859:
858:Clint McCrory
855:
851:
847:
843:
839:
830:
823:
821:
819:
815:
810:
797:
794:
788:
783:
779:
775:
772:
769:
765:
762:
749:
745:
741:
736:
732:
723:
718:
714:
710:
707:
704:
700:
697:
684:
680:
676:
671:
667:
658:
653:
649:
640:
638:
634:
630:
622:
620:
604:
594:
591:
583:
579:
575:
571:
568:
564:
560:
556:
553:of dimension
552:
549:
545:
542:of dimension
541:
536:
520:
517:
514:
510:
506:
501:
498:
495:
485:
473:
455:
452:
449:
446:
443:
433:
426:
421:
418:
415:
405:
393:
389:
385:
367:
364:
361:
351:
340:
337:
333:
329:
325:
320:
318:
314:
310:
306:
305:hypersurfaces
295:
292:
284:
274:
270:
266:
260:
259:
255:
250:This section
248:
244:
239:
238:
232:
230:
228:
224:
219:
216:
214:
210:
206:
202:
198:
194:
190:
185:
183:
182:Carl F. Gauss
179:
175:
171:
167:
163:
159:
155:
151:
147:
143:
139:
135:
132:
128:
124:
120:
116:
112:
108:
104:
95:
85:
82:
74:
64:
60:
54:
53:
47:
43:
39:
34:
25:
24:
19:
1118:
1050:
993:. Retrieved
978:
965:
943:
914:
907:
900:
887:
883:
881:
835:
820:properties.
811:
641:
628:
626:
581:
573:
569:
562:
558:
554:
550:
543:
539:
537:
391:
387:
383:
339:Grassmannian
335:
331:
323:
321:
316:
312:
308:
302:
287:
278:
263:Please help
251:
220:
217:
209:differential
195:is half the
186:
177:
173:
169:
165:
161:
157:
149:
145:
141:
137:
133:
126:
106:
100:
77:
68:
57:Please help
49:
1096:Gauss's law
962:"Gauss Map"
944:Solid Shape
818:topological
386:-planes in
328:submanifold
154:unit sphere
119:unit vector
63:introducing
1136:Categories
894:References
207:, and the
189:orientable
123:orthogonal
1051:Gauss map
967:MathWorld
780:∬
742:×
715:∬
677:×
659:±
650:∬
518:−
507:≅
489:~
447:−
437:~
427:≅
409:~
355:~
252:does not
107:Gauss map
71:July 2011
1157:Surfaces
1120:Category
934:Archived
922:Archived
336:oriented
281:May 2020
201:Jacobian
121:that is
115:function
995:4 March
635:of the
273:removed
258:sources
152:is the
148:(where
111:surface
59:improve
986:
903:(1827)
792:
760:
695:
392:normal
193:degree
105:, the
869:ridge
846:cusps
576:is a
113:is a
109:of a
44:, or
997:2016
984:ISBN
886:and
856:and
812:The
256:any
254:cite
223:link
330:of
307:in
267:by
176:at
160:in
129:in
101:In
1138::
964:.
890:.
867:A
852:,
582:TM
570:TM
319:.
229:.
215:.
184:.
144:→
140::
48:,
40:,
1027:e
1020:t
1013:v
999:.
970:.
798:A
795:d
789:K
784:S
776:=
773:v
770:d
766:u
763:d
756:|
750:v
746:X
737:u
733:X
728:|
724:K
719:R
711:=
708:v
705:d
701:u
698:d
691:|
685:v
681:N
672:u
668:N
663:|
654:R
605:n
600:R
595:=
592:M
574:N
563:k
559:X
555:n
551:M
544:k
540:X
521:1
515:n
511:S
502:n
499:,
496:1
486:G
456:n
453:,
450:k
444:n
434:G
422:n
419:,
416:k
406:G
388:R
384:k
368:n
365:,
362:k
352:G
332:R
326:-
324:k
317:R
313:S
309:R
294:)
288:(
283:)
279:(
275:.
261:.
178:p
174:X
170:p
168:(
166:N
162:X
158:p
150:S
146:S
142:X
138:N
134:R
127:X
84:)
78:(
73:)
69:(
55:.
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
