128:
681:
669:
592:
43:
31:
260:
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648:
by the boundary circle. Note that in the
Beltrami-Klein model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami-Klein model, unlike the Poincaré disk and half-plane models, is not
564:
154:
264:
488:
356:
849:
721:
602:
Because the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any
255:{\displaystyle r=\ln {\sqrt {3}}={\frac {1}{2}}\ln 3=\operatorname {artanh} {\frac {1}{2}}=2\operatorname {artanh} (2-{\sqrt {3}})=}
511:
699:
generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic to the
329:{\displaystyle =\operatorname {arsinh} {\frac {1}{3}}{\sqrt {3}}=\operatorname {arcosh} {\frac {2}{3}}{\sqrt {3}}\approx 0.549}
622:
of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles.
448:
1379:
626:
47:
1384:
972:
952:
340:
The distance from any point in the triangle to the closest side of the triangle is less than or equal to the radius
947:
904:
879:
127:
641:
132:
1007:
932:
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The distance from any point on a side of the triangle to another side of the triangle is equal or less than
957:
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136:
35:
1358:
1298:
937:
82:
607:
596:
1242:
1012:
942:
884:
680:
668:
438:{\displaystyle d=\ln \left({\frac {{\sqrt {5}}+1}{{\sqrt {5}}-1}}\right)=2\ln \varphi \approx 0.962}
1348:
1323:
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1288:
1247:
962:
603:
579:
62:
54:
1353:
894:
808:
790:
111:
591:
1333:
927:
835:
781:
Schwartz, Richard Evan (2001). "Ideal triangle groups, dented tori, and numerical analysis".
503:
around a point inside the triangle will meet or intersect at least two sides of the triangle.
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800:
696:
350:
349:
The inscribed circle meets the triangle in three points of tangency, forming an equilateral
145:
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42:
1338:
1318:
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909:
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of the hyperbolic plane, an ideal triangle is modeled by a
Euclidean triangle that is
1373:
1313:
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1343:
1213:
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1123:
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700:
491:
1252:
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1128:
66:
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1113:
1103:
987:
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1199:
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17:
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30:
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Dimensions related to an ideal triangle and its incircle, depicted in the
106:
An ideal triangle is the largest possible triangle in hyperbolic geometry.
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827:
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41:
29:
757:"What is the radius of the inscribed circle of an ideal triangle"
344:
above, with equality only for the center of the inscribed circle.
566:, with equality only for the points of tangency described above.
831:
110:
In the standard hyperbolic plane (a surface where the constant
559:{\displaystyle a=\ln \left(1+{\sqrt {2}}\right)\approx 0.881}
100:
The interior angles of an ideal triangle are all zero.
514:
451:
359:
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157:
114:
is −1) we also have the following properties:
1266:
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1096:
1035:
1026:
918:
870:
662:The Poincaré disk model tiled with ideal triangles
483:{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}
558:
482:
437:
328:
254:
97:All ideal triangles are congruent to each other.
93:Ideal triangles have the following properties:
843:
8:
751:
749:
633:, the figure between three mutually tangent
69:. Ideal triangles are also sometimes called
703:of three order-two groups (Schwartz 2001).
1032:
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665:
606:, this fact is important in the study of
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103:An ideal triangle has infinite perimeter.
590:
126:
712:
595:The ÎŽ-thin triangle condition used in
629:, an ideal triangle is modeled by an
7:
673:The ideal (∞ ∞ ∞)
77:. The vertices are sometimes called
722:"274 Curves on Surfaces, Lecture 5"
118:Any ideal triangle has area π.
653:i.e. it does not preserve angles.
25:
679:
667:
148:to an ideal triangle has radius
246:
230:
123:Distances in an ideal triangle
1:
720:Thurston, Dylan (Fall 2012).
65:whose three vertices all are
34:Three ideal triangles in the
75:trebly asymptotic triangles
71:triply asymptotic triangles
46:Two ideal triangles in the
38:creating an ideal pentagon
27:Type of hyperbolic triangle
1401:
81:. All ideal triangles are
657:Real ideal triangle group
627:Poincaré half-plane model
48:Poincaré half-plane model
587:Thin triangle condition
599:
560:
484:
439:
330:
256:
140:
50:
39:
783:Annals of Mathematics
685:Another ideal tiling
594:
561:
499:A circle with radius
485:
440:
331:
257:
130:
45:
33:
1083:Nonagon/Enneagon (9)
1013:Tangential trapezoid
642:BeltramiâKlein model
512:
449:
357:
265:
155:
133:BeltramiâKlein model
1380:Hyperbolic geometry
1195:Megagon (1,000,000)
963:Isosceles trapezoid
663:
620:Poincaré disk model
604:hyperbolic triangle
580:Schweikart triangle
137:Poincaré disk model
63:hyperbolic triangle
55:hyperbolic geometry
36:Poincaré disk model
1385:Types of triangles
1165:Icositetragon (24)
661:
608:ÎŽ-hyperbolic space
600:
597:ÎŽ-hyperbolic space
556:
480:
435:
326:
252:
141:
112:Gaussian curvature
51:
40:
1367:
1366:
1208:
1207:
1185:Myriagon (10,000)
1170:Triacontagon (30)
1134:Heptadecagon (17)
1124:Pentadecagon (15)
1119:Tetradecagon (14)
1058:Quadrilateral (4)
928:Antiparallelogram
734:on 9 January 2022
689:
688:
543:
478:
472:
408:
399:
384:
353:with side length
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311:
292:
285:
244:
216:
188:
175:
16:(Redirected from
1392:
1180:Chiliagon (1000)
1160:Icositrigon (23)
1139:Octadecagon (18)
1129:Hexadecagon (16)
1033:
852:
845:
838:
829:
824:
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768:
767:
765:
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743:
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727:. Archived from
726:
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697:reflection group
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351:contact triangle
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146:inscribed circle
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1114:Tridecagon (13)
1104:Hendecagon (11)
1092:
1028:
1022:
993:Right trapezoid
914:
866:
856:
805:10.2307/2661362
796:math.DG/0105264
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691:The real ideal
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135:(left) and the
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12:
11:
5:
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1319:Pseudotriangle
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1109:Dodecagon (12)
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920:Quadrilaterals
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789:(3): 533â598.
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693:triangle group
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675:triangle group
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79:ideal vertices
59:ideal triangle
26:
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6:
4:
3:
2:
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1359:Weakly simple
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1299:Infinite skew
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1234:
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1214:Star polygons
1211:
1201:
1200:Apeirogon (â)
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1144:Icosagon (20)
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986:
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978:Parallelogram
976:
974:
973:Orthodiagonal
971:
969:
966:
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961:
959:
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953:Ex-tangential
951:
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746:
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723:
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713:
706:
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702:
698:
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682:
678:
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670:
666:
656:
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646:circumscribed
643:
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632:
628:
623:
621:
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569:
553:
550:
546:
540:
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532:
528:
524:
521:
518:
515:
507:
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502:
498:
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493:
475:
469:
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461:
455:
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432:
429:
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423:
420:
417:
414:
410:
404:
401:
396:
389:
386:
381:
373:
369:
366:
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352:
348:
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343:
339:
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337:
323:
320:
315:
308:
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268:
249:
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236:
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210:
205:
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147:
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138:
134:
129:
122:
117:
116:
115:
113:
105:
102:
99:
96:
95:
94:
88:
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84:
80:
76:
72:
68:
64:
60:
56:
49:
44:
37:
32:
19:
1153:>20 sides
1088:Decagon (10)
1073:Heptagon (7)
1063:Pentagon (5)
1053:Triangle (3)
948:Equidiagonal
889:
786:
782:
775:Bibliography
760:. Retrieved
736:. Retrieved
729:the original
715:
701:free product
690:
639:
624:
617:
601:
574:is also the
571:
500:
492:golden ratio
341:
151:
109:
92:
78:
74:
70:
67:ideal points
58:
52:
18:Ideal vertex
1349:Star-shaped
1324:Rectilinear
1294:Equilateral
1289:Equiangular
1253:Hendecagram
1097:11â20 sides
1078:Octagon (8)
1068:Hexagon (6)
1043:Monogon (1)
885:Equilateral
635:semicircles
1374:Categories
1354:Tangential
1258:Dodecagram
1036:1â10 sides
1027:By number
1008:Tangential
988:Right kite
785:. Ser. 2.
762:9 December
707:References
89:Properties
1334:Reinhardt
1243:Enneagram
1233:Heptagram
1223:Pentagram
1190:65537-gon
1048:Digon (2)
1018:Trapezoid
983:Rectangle
933:Bicentric
895:Isosceles
872:Triangles
651:conformal
551:≈
525:
453:φ
430:≈
427:φ
424:
402:−
370:
321:≈
301:
275:
237:−
228:
206:
194:
168:
83:congruent
1309:Isotoxal
1304:Isogonal
1248:Decagram
1238:Octagram
1228:Hexagram
1029:of sides
958:Harmonic
859:Polygons
576:altitude
1329:Regular
1274:Concave
1267:Classes
1175:257-gon
998:Rhombus
938:Crossed
821:1836282
813:2661362
738:23 July
695:is the
640:In the
631:arbelos
625:In the
618:In the
578:of the
490:is the
139:(right)
1339:Simple
1284:Cyclic
1279:Convex
1003:Square
943:Cyclic
905:Obtuse
900:Kepler
819:
811:
614:Models
445:where
298:arcosh
272:arsinh
225:artanh
203:artanh
1314:Magic
910:Right
890:Ideal
880:Acute
809:JSTOR
791:arXiv
732:(PDF)
725:(PDF)
554:0.881
433:0.962
324:0.549
61:is a
1344:Skew
968:Kite
863:List
764:2015
740:2013
144:The
801:doi
787:153
336:.
73:or
57:an
53:In
1376::
817:MR
815:.
807:.
799:.
748:^
637:.
610:.
522:ln
421:ln
367:ln
191:ln
165:ln
85:.
865:)
861:(
851:e
844:t
837:v
823:.
803::
793::
766:.
742:.
582:.
572:a
547:)
541:2
536:+
533:1
529:(
519:=
516:a
501:d
494:.
476:2
470:5
465:+
462:1
456:=
418:2
415:=
411:)
405:1
397:5
390:1
387:+
382:5
374:(
364:=
361:d
342:r
316:3
309:3
306:2
295:=
290:3
283:3
280:1
269:=
250:=
247:)
242:3
234:2
231:(
222:2
219:=
214:2
211:1
200:=
197:3
186:2
183:1
178:=
173:3
162:=
159:r
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
