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.
Ideal triangles have the following properties:
In the standard hyperbolic plane (a surface where the constant Gaussian curvature is -1) we also have the following properties:
r=ln\sqrt{3}=
| 1 | |
| 2 |
ln3=\operatorname{artanh}
| 1 | |
| 2 |
=2\operatorname{artanh}(2-\sqrt{3})=
=\operatorname{arsinh}
| 1 | |
| 3 |
\sqrt{3} =\operatorname{arcosh}
| 2 | |
| 3 |
\sqrt{3} ≈ 0.549
The distance from any point in the triangle to the closest side of the triangle is less than or equal to the radius r above, with equality only for the center of the inscribed circle.
d=ln\left(
| \sqrt5+1 | |
| \sqrt5-1 |
\right)=2ln\varphi ≈ 0.962
| \varphi= | 1+\sqrt5 |
| 2 |
A circle with radius d around a point inside the triangle will meet or intersect at least two sides of the triangle.
a=ln\left(1+\sqrt2\right) ≈ 0.881
a is also the altitude of the Schweikart triangle.
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 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.