Ideal 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 Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles.

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.

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).

Three ideal triangles in the Poincaré disk model creating an ideal pentagon
Two ideal triangles in the Poincaré half-plane model
Dimensions related to an ideal triangle and its incircle, depicted in the Beltrami–Klein model (left) and the Poincaré disk model (right)
The δ-thin triangle condition used in δ-hyperbolic space