In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.
Because the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any hyperbolic triangle, this fact is important in the study of δ-hyperbolic space.
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 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 i.e. it does not preserve angles.
Real ideal triangle groupEdit
The Poincaré disk model tiled with ideal triangles
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 (Schwartz 2001).
^Thurston, Dylan (Fall 2012). "274 Curves on Surfaces, Lecture 5" (PDF). Retrieved 23 July 2013.
^ ab"What is the radius of the inscribed circle of an ideal triangle". Retrieved 9 December 2015.
Schwartz, Richard Evan (2001). "Ideal triangle groups, dented tori, and numerical analysis". Annals of Mathematics. Ser. 2. 153 (3): 533–598. arXiv:math.DG/0105264. doi:10.2307/2661362. JSTOR 2661362. MR 1836282.