327:
634:
520:
760:
Then any solution to the sine-Gordon equation can be used to specify a first and second fundamental form which satisfy the GaussâCodazzi equations. There is then a theorem that any such set of initial data can be used to at least locally specify an immersed surface in
205:
620:
398:
745:
In particular, for the tractroid the GaussâCodazzi equations are the sine-Gordon equation applied to the static soliton solution, so the GaussâCodazzi equations are satisfied. In these coordinates the
1280:
1157:
788:
712:
65:
129:
531:
263:
found that the volume and the surface area of the pseudosphere are finite, despite the infinite extent of the shape along the axis of rotation. For a given edge
326:
816:
881:
Beltrami, Eugenio (1868). "Saggio sulla interpretazione della geometria non euclidea" [Treatise on the interpretation of non-Euclidean geometry].
1260:
1002:
971:
907:
515:{\displaystyle (x,y)\mapsto {\big (}v(\operatorname {arcosh} y)\cos x,v(\operatorname {arcosh} y)\sin x,u(\operatorname {arcosh} y){\big )}}
100:
1033:
918:
714:
with constant negative curvature is a pseudospherical surface. The tractroid is the simplest example. Other examples include the
1209:
236:
of constant negative
Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the
1114:
1265:
343:
739:
123:. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by
685:
A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in
861:
633:
381:
to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion
902:] (in Italian). Vol. 1. Scholarly Publishing Office, University of Michigan Library. pp. 374â405.
846:
389:
211:
1244:
1079:
750:
746:
856:
836:
654:
200:{\displaystyle t\mapsto \left(t-\tanh t,\operatorname {sech} \,t\right),\quad \quad 0\leq t<\infty .}
108:
764:
688:
41:
1194:
735:
646:
642:
1214:
88:
723:
615:{\displaystyle t\mapsto {\big (}u(t)=t-\operatorname {tanh} t,v(t)=\operatorname {sech} t{\big )}}
754:
260:
233:
215:
68:
28:
1270:
1134:
1054:
1041:
1029:
1010:
998:
979:
967:
903:
826:
821:
793:
A few examples of sine-Gordon solutions and their corresponding surface are given as follows:
715:
662:
638:
1275:
1190:
1126:
930:
719:
223:
84:
1233:
1228:
674:
249:
241:
214:(the equator is a singularity), but away from the singularities, it has constant negative
938:
738:. A sketch proof starts with reparametrizing the tractroid with coordinates in which the
1142:
335:
253:
1254:
1234:
Crocheting the
Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
1025:
Great
Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences
650:
841:
1057:
1023:
992:
961:
831:
670:
311:
1238:
1130:
330:
The pseudosphere and its relation to three other models of hyperbolic geometry
1138:
997:(revised, 3rd ed.). Springer Science & Business Media. p. 345.
1062:
339:
230:
116:
1241:, History of Mathematics, University of New South Wales. YouTube. 2012 May.
1193:(2006). "Experiencing Geometry: Euclidean and Non-Euclidean with History".
99:
315:
219:
112:
75:
at each point. Its name comes from the analogy with the sphere of radius
20:
934:
963:
Low-dimensional geometry: from
Euclidean surfaces to hyperbolic knots
851:
310:
The pseudosphere is an important geometric precursor to mathematical
282:
264:
237:
921:[Treatise on the interpretation of non-Euclidean geometry].
734:
Pseudospherical surfaces can be constructed from solutions to the
325:
98:
371:
to the meridians of the pseudosphere and the vertical geodesics
271:
245:
665:
of the hyperbolic plane, the hyperboloid is referred to as a
346:
one convenient choice is the portion of the half-plane with
669:. This usage of the word is because the hyperboloid can be
107:
The same surface can be also described as the result of
1080:"The Crochet Coral Reef Keeps Spawning, Hyperbolically"
1028:(2 ed.). Courier Dover Publications. p. 154.
919:"Essai d'interprétation de la géométrie noneuclidéenne"
1281:
Surfaces of revolution of constant negative curvature
1101:, vol. 1, Princeton University Press, p. 62
767:
757:
is â1 for any solution of the sine-Gordon equations.
691:
534:
401:
229:
The name "pseudosphere" comes about because it has a
132:
44:
307:
and therefore half that of a sphere of that radius.
782:
706:
614:
514:
199:
119:. For this reason the pseudosphere is also called
59:
730:Relation to solutions to the sine-Gordon equation
742:can be rewritten as the sine-Gordon equation.
625:is the parametrization of the tractrix above.
248:the whole pseudosphere has at every point the
607:
543:
507:
422:
392:of the pseudosphere. The precise mapping is
8:
353:. Then the covering map is periodic in the
1158:"From Pseudosphere to sine-Gordon equation"
637:Deforming the pseudosphere to a portion of
753:are written in a way that makes clear the
1184:. Amer. Math. Soc & London Math. Soc.
817:Hilbert's theorem (differential geometry)
774:
770:
769:
766:
698:
694:
693:
690:
606:
605:
542:
541:
533:
506:
505:
421:
420:
400:
334:The half pseudosphere of curvature â1 is
165:
131:
51:
47:
46:
43:
645:. In the corresponding solutions to the
632:
281:just as it is for the sphere, while the
1099:Three-dimensional geometry and topology
873:
1207:Kasner, Edward; Newman, James (1940).
641:. In differential geometry, this is a
923:Annales de l'Ăcole Normale SupĂ©rieure
7:
1078:Roberts, Siobhan (15 January 2024).
649:, this deformation corresponds to a
79:, which is a surface of curvature 1/
27:is a surface with constant negative
803:Breather solution: Breather surface
673:of imaginary radius, embedded in a
191:
14:
1261:Differential geometry of surfaces
800:Moving 1-soliton: Dini's surface
783:{\displaystyle \mathbb {R} ^{3}}
707:{\displaystyle \mathbb {R} ^{3}}
60:{\displaystyle \mathbb {R} ^{3}}
1210:Mathematics and the Imagination
894:Beltrami, Eugenio (July 2010).
388:of the upper half-plane as the
178:
177:
87:in his 1868 paper on models of
1182:Sources of Hyperbolic Geometry
1115:"A new theory of complex rays"
966:. AMS Bookstore. p. 108.
862:Mathematics in the fabric arts
797:Static 1-soliton: pseudosphere
590:
584:
557:
551:
538:
502:
490:
472:
460:
442:
430:
417:
414:
402:
136:
1:
661:In some sources that use the
83:. The term was introduced by
1239:Norman Wildberger lecture 16
1247:at the virtual math museum.
994:Mathematics and Its History
361:, and takes the horocycles
1297:
1196:Aesthetics and Mathematics
917:Beltrami, Eugenio (1869).
1217:. pp. 140, 145, 155.
960:Bonahon, Francis (2009).
344:Poincaré half-plane model
218:and therefore is locally
34:A pseudosphere of radius
1245:Pseudospherical surfaces
1022:Le Lionnais, F. (2004).
991:Stillwell, John (2010).
751:second fundamental forms
681:Pseudospherical surfaces
390:universal covering space
322:Universal covering space
1131:10.1093/imamat/69.6.521
1113:Hasanov, Elman (2004),
806:2-soliton: Kuen surface
740:GaussâCodazzi equations
1180:Stillwell, J. (1996).
784:
708:
671:thought of as a sphere
658:
616:
516:
331:
201:
104:
61:
857:Surface of revolution
837:Hyperboloid structure
785:
709:
636:
617:
517:
357:direction of period 2
338:by the interior of a
329:
252:curved geometry of a
244:curved geometry of a
240:has at every point a
202:
102:
62:
1215:Simon & Schuster
1042:Chapter 40, page 154
847:SineâGordon equation
765:
736:sine-Gordon equation
689:
647:sine-Gordon equation
532:
399:
130:
42:
1266:Hyperbolic geometry
1156:Wheeler, Nicholas.
1097:Thurston, William,
1011:extract of page 345
980:Chapter 5, page 108
89:hyperbolic geometry
1202:. Springer-Verlag.
1189:Henderson, D. W.;
1119:IMA J. Appl. Math.
1084:The New York Times
1055:Weisstein, Eric W.
900:Mathematical Works
780:
755:Gaussian curvature
704:
659:
643:Lie transformation
612:
512:
332:
261:Christiaan Huygens
216:Gaussian curvature
197:
105:
57:
29:Gaussian curvature
1004:978-1-4419-6052-8
973:978-0-8218-4816-6
935:10.24033/asens.60
909:978-1-4181-8434-6
896:Opere Matematiche
720:breather surfaces
663:hyperboloid model
259:As early as 1693
16:Geometric surface
1288:
1218:
1203:
1201:
1185:
1172:
1171:
1169:
1167:
1162:
1153:
1147:
1146:
1141:, archived from
1110:
1104:
1102:
1094:
1088:
1087:
1075:
1069:
1068:
1067:
1050:
1044:
1039:
1019:
1013:
1008:
988:
982:
977:
957:
951:
949:
947:
946:
937:. Archived from
913:
890:
878:
789:
787:
786:
781:
779:
778:
773:
713:
711:
710:
705:
703:
702:
697:
653:of the static 1-
621:
619:
618:
613:
611:
610:
547:
546:
521:
519:
518:
513:
511:
510:
426:
425:
387:
380:
370:
360:
356:
352:
306:
301:
299:
298:
295:
292:
280:
269:
224:hyperbolic plane
206:
204:
203:
198:
173:
169:
85:Eugenio Beltrami
78:
66:
64:
63:
58:
56:
55:
50:
38:is a surface in
37:
1296:
1295:
1291:
1290:
1289:
1287:
1286:
1285:
1251:
1250:
1225:
1206:
1199:
1188:
1179:
1176:
1175:
1165:
1163:
1160:
1155:
1154:
1150:
1112:
1111:
1107:
1096:
1095:
1091:
1077:
1076:
1072:
1053:
1052:
1051:
1047:
1036:
1021:
1020:
1016:
1005:
990:
989:
985:
974:
959:
958:
954:
944:
942:
916:
915:
910:
893:
891:
880:
879:
875:
870:
813:
768:
763:
762:
732:
716:Dini's surfaces
692:
687:
686:
683:
675:Minkowski space
631:
530:
529:
397:
396:
382:
372:
362:
358:
354:
347:
324:
296:
293:
290:
289:
287:
286:
275:
267:
231:two-dimensional
143:
139:
128:
127:
97:
76:
45:
40:
39:
35:
17:
12:
11:
5:
1294:
1292:
1284:
1283:
1278:
1273:
1268:
1263:
1253:
1252:
1249:
1248:
1242:
1236:
1231:
1224:
1223:External links
1221:
1220:
1219:
1204:
1186:
1174:
1173:
1148:
1125:(6): 521â537,
1105:
1089:
1070:
1058:"Pseudosphere"
1045:
1034:
1014:
1003:
983:
972:
952:
908:
885:(in Italian).
872:
871:
869:
866:
865:
864:
859:
854:
849:
844:
839:
834:
829:
827:Gabriel's Horn
824:
822:Dini's surface
819:
812:
809:
808:
807:
804:
801:
798:
777:
772:
731:
728:
701:
696:
682:
679:
639:Dini's surface
630:
627:
623:
622:
609:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
545:
540:
537:
523:
522:
509:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
424:
419:
416:
413:
410:
407:
404:
323:
320:
212:singular space
208:
207:
196:
193:
190:
187:
184:
181:
176:
172:
168:
164:
161:
158:
155:
152:
149:
146:
142:
138:
135:
96:
93:
54:
49:
15:
13:
10:
9:
6:
4:
3:
2:
1293:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1262:
1259:
1258:
1256:
1246:
1243:
1240:
1237:
1235:
1232:
1230:
1227:
1226:
1222:
1216:
1212:
1211:
1205:
1198:
1197:
1192:
1187:
1183:
1178:
1177:
1159:
1152:
1149:
1145:on 2013-04-15
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1109:
1106:
1100:
1093:
1090:
1085:
1081:
1074:
1071:
1065:
1064:
1059:
1056:
1049:
1046:
1043:
1037:
1035:0-486-49579-5
1031:
1027:
1026:
1018:
1015:
1012:
1006:
1000:
996:
995:
987:
984:
981:
975:
969:
965:
964:
956:
953:
941:on 2016-02-02
940:
936:
932:
928:
925:(in French).
924:
920:
911:
905:
901:
897:
888:
884:
877:
874:
867:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
814:
810:
805:
802:
799:
796:
795:
794:
791:
775:
758:
756:
752:
748:
743:
741:
737:
729:
727:
725:
721:
717:
699:
680:
678:
676:
672:
668:
664:
656:
652:
651:Lorentz Boost
648:
644:
640:
635:
628:
626:
602:
599:
596:
593:
587:
581:
578:
575:
572:
569:
566:
563:
560:
554:
548:
535:
528:
527:
526:
499:
496:
493:
487:
484:
481:
478:
475:
469:
466:
463:
457:
454:
451:
448:
445:
439:
436:
433:
427:
411:
408:
405:
395:
394:
393:
391:
385:
379:
375:
369:
365:
350:
345:
341:
337:
328:
321:
319:
317:
313:
308:
305:
284:
279:
273:
266:
262:
257:
255:
251:
247:
243:
239:
235:
232:
227:
225:
221:
217:
213:
194:
188:
185:
182:
179:
174:
170:
166:
162:
159:
156:
153:
150:
147:
144:
140:
133:
126:
125:
124:
122:
118:
114:
110:
101:
94:
92:
90:
86:
82:
74:
70:
52:
32:
30:
26:
22:
1208:
1195:
1181:
1164:. Retrieved
1151:
1143:the original
1122:
1118:
1108:
1098:
1092:
1083:
1073:
1061:
1048:
1024:
1017:
993:
986:
962:
955:
943:. Retrieved
939:the original
926:
922:
899:
895:
886:
882:
876:
842:Quasi-sphere
792:
759:
744:
733:
724:Kuen surface
684:
667:pseudosphere
666:
660:
624:
524:
383:
377:
373:
367:
363:
348:
333:
309:
303:
277:
258:
228:
209:
120:
106:
80:
72:
33:
25:pseudosphere
24:
18:
1191:Taimina, D.
1166:24 November
929:: 251â288.
832:Hyperboloid
629:Hyperboloid
312:fabric arts
1255:Categories
1229:Non Euclid
945:2010-07-24
889:: 248â312.
883:Gior. Mat.
868:References
722:, and the
250:negatively
242:positively
115:about its
1139:1464-3634
1063:MathWorld
657:solution.
600:
573:
567:−
539:↦
497:
479:
467:
449:
437:
418:↦
342:. In the
340:horocycle
220:isometric
192:∞
183:≤
154:
148:−
137:↦
121:tractroid
117:asymptote
109:revolving
103:Tractroid
95:Tractroid
69:curvature
1271:Surfaces
811:See also
316:pedagogy
210:It is a
113:tractrix
21:geometry
1276:Spheres
655:soliton
336:covered
300:
288:
234:surface
67:having
1137:
1032:
1001:
970:
906:
892:(Also
852:Sphere
525:where
494:arcosh
464:arcosh
434:arcosh
283:volume
270:, the
265:radius
254:saddle
238:sphere
1200:(PDF)
1161:(PDF)
898:[
747:first
222:to a
1168:2022
1135:ISSN
1030:ISBN
999:ISBN
968:ISBN
904:ISBN
749:and
597:sech
570:tanh
314:and
272:area
246:dome
189:<
163:sech
151:tanh
23:, a
1127:doi
931:doi
476:sin
446:cos
386:â„ 1
351:â„ 1
285:is
274:is
71:â1/
19:In
1257::
1213:.
1133:,
1123:69
1121:,
1117:,
1082:.
1060:.
1040:,
1009:,
978:,
790:.
726:.
718:,
677:.
376:=
366:=
318:.
276:4Ï
256:.
226:.
111:a
91:.
31:.
1170:.
1129::
1103:.
1086:.
1066:.
1038:.
1007:.
976:.
950:)
948:.
933::
927:6
914:;
912:.
887:6
776:3
771:R
700:3
695:R
608:)
603:t
594:=
591:)
588:t
585:(
582:v
579:,
576:t
564:t
561:=
558:)
555:t
552:(
549:u
544:(
536:t
508:)
503:)
500:y
491:(
488:u
485:,
482:x
473:)
470:y
461:(
458:v
455:,
452:x
443:)
440:y
431:(
428:v
423:(
415:)
412:y
409:,
406:x
403:(
384:y
378:c
374:x
368:c
364:y
359:Ï
355:x
349:y
304:R
302:Ï
297:3
294:/
291:2
278:R
268:R
195:.
186:t
180:0
175:,
171:)
167:t
160:,
157:t
145:t
141:(
134:t
81:R
77:R
73:R
53:3
48:R
36:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
