Ideal triangle

In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all lie on the circle at infinity. In the Poincaré disk model an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. In the hyperbolic metric any two ideal triangles are congruent. Ideal triangles are also sometimes called "triply-asymptotic triangles" or "trebly-asymptotic triangles".

Properties

* The interior angles of an ideal triangle are all zero.
* Any ideal triangle has area π and infinite perimeter.
* The inscribed circle to an ideal triangle meets the triangle in three points of tangency, forming an equilateral triangle with side length:$4lnfrac\{1+sqrt\; 5\}\{2\}approx\; 1.925.$ [http://www.cabri.net/abracadabri/GeoNonE/GeoHyper/KBModele/Biss3KB.html]
* Any point in the triangle is within constant distance of some two sides of the triangle.

Real ideal triangle group

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

*cite journal
author = Schwartz, Richard Evan
title = Ideal triangle groups, dented tori, and numerical analysis
journal = Ann. of Math., Ser. 2
volume = 153
year = 2001
issue = 3
pages = 533–598
id = arxiv | archive = math.DG | id = 0105264