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2013-09-17T10:45:20Z

124.43.115.10:

[[Image:Ideal circles.GIF|thumb|right|200px|Three ideal triangles in the Poincaré disk model]]
[[Image:IdealTriangle HalfPlane.jpg|thumb|right|200px|Two ideal triangles in the upper half-plane model]]

In [[hyperbolic geometry]] an '''ideal triangle''' is a [[hyperbolic triangle#Hyperbolic geometry|hyperbolic triangle]] whose three vertices all lie on the circle at infinity. These vertices can be called '''ideal vertices'''. In the hyperbolic metric, any two ideal triangles are [[Congruence (geometry)|congruent]]. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''.

== Models ==
In the [[Poincaré disk model]] of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. In the [[Poincaré half-plane model]], an ideal triangle is modeled by an [[arbelos]], the figure between three mutually tangent [[semicircle]]s. And in the [[Beltrami–Klein model]] of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is [[circumscribed]] 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 [[conformal map|conformal]] i.e. it does not preserve angles.

== Properties ==
In the standard hyperbolic plane (with [[Gaussian curvature]] -1 at every point):
* The interior angles of an ideal triangle are all zero.
* Any ideal triangle has area π.<ref name="Thurston 2012">{{cite web | url=http://math.berkeley.edu/~qchu/Notes/274/Lecture5.pdf | title=274 Curves on Surfaces, Lecture 5 | date=Fall 2012 | accessdate=23 July 2013 | author=Thurston, Dylan}}</ref>
* Any ideal triangle has infinite perimeter.
* The [[inscribed circle]] to an ideal triangle meets the triangle in three points of tangency, forming an equilateral triangle with side length
::<math>d=4\ln\varphi\approx 1.925,</math>
:where <math>\varphi=\frac{1+\sqrt 5}{2}</math> is the [[golden ratio]].<ref name="Biss3KB">{{cite web | url=http://www.cabri.net/abracadabri/GeoNonE/GeoHyper/KBModele/Biss3KB.html | title=Modèle hyperbolique de Klein - Beltrami | accessdate=23 July 2013}}</ref>
* The distance from any point in the triangle to the second-closest side of the triangle is less than or equal to ''d'', with equality only for the three equilateral triangle vertices described above. The same inequality holds for hyperbolic triangles more generally; in a non-ideal triangle, the distance to the second-closest side is strictly less than ''d''.
If the curvature is −''K'' everywhere rather than −1, the areas above should be multiplied by 1/''K'' and the lengths and distances should be multiplied by 1/√''K''.

== Real ideal triangle group ==
{| class=wikitable align=right width=480
|+ The Poincaré disk model tiled with ideal triangles
|- align=center valign=top
|[[File:H2checkers iii.png|240px]]<BR>The ideal (∞ ∞ ∞) [[triangle group]]
|[[File:Ideal-triangle hyperbolic tiling.svg|240px]]<BR>Another ideal tiling
|}
The real ideal [[triangle group]] is the [[reflection group]] generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic to the free product of three order-two groups (Schwarz 2001).

== References ==
{{reflist|2}}

== Bibliography ==
*{{cite journal
| doi = 10.2307/2661362
| author = Schwartz, Richard Evan
| title = Ideal triangle groups, dented tori, and numerical analysis
| journal = Annals of Mathematics | series = Ser. 2
| volume = 153
| year = 2001
| issue = 3
| pages = 533–598
| arxiv = math.DG/0105264
| mr = 1836282
| jstor = 2661362}}

[[Category:Hyperbolic geometry]]

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