Ideal triangle
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.
Properties
Ideal triangles have the following properties:- All ideal triangles are congruent to each other.
- The interior angles of an ideal triangle are all zero.
- An ideal triangle has infinite perimeter.
- An ideal triangle is the largest possible triangle in hyperbolic geometry.
- Any ideal triangle has area π.
Distances in an ideal triangle
- The inscribed circle to an ideal triangle has radius
- The inscribed circle meets the triangle in three points of tangency, forming an equilateral contact triangle with side length where is the golden ratio.
- The distance from any point on a side of the triangle to another side of the triangle is equal or less than, with equality only for the points of tangency described above.
Thin triangle condition
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.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 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. 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.