Inhyperbolic geometry anideal triangle is ahyperbolic triangle whose three vertices all areideal points. Ideal triangles are also sometimes calledtriply asymptotic triangles ortrebly asymptotic triangles. The vertices are sometimes calledideal vertices. All ideal triangles arecongruent.

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.

In the standard hyperbolic plane (a surface where the constantGaussian curvature is −1) we also have the following properties:

• Any ideal triangle has area π.^[1]

• Theinscribed circle to an ideal triangle has radius

${\displaystyle r=\ln \sqrt{3}=\frac{1}{2}\ln 3=\mathrm{artanh}\frac{1}{2}=2\mathrm{artanh}(2-\sqrt{3})=}$${\displaystyle =\mathrm{arsinh}\frac{1}{3}\sqrt{3}=\mathrm{arcosh}\frac{2}{3}\sqrt{3}\approx 0.549}$ .^[2]

The distance from any point in the triangle to the closest side of the triangle is less than or equal to the radiusr above, with equality only for the center of the inscribed circle.

• The inscribed circle meets the triangle in three points of tangency, forming an equilateralcontact triangle with side length${\displaystyle d=\ln \left(\frac{\sqrt{5}+1}{\sqrt{5}-1}\right)=2\ln \phi \approx 0.962}$^[2] where${\displaystyle \phi =\frac{1+\sqrt{5}}{2}}$ is thegolden ratio.

A circle with radiusd around a point inside the triangle will meet or intersect at least two sides of the triangle.

• The distance from any point on a side of the triangle to another side of the triangle is equal or less than${\displaystyle a=\ln \left(1+\sqrt{2}\right)\approx 0.881}$, with equality only for the points of tangency described above.

a is also thealtitude of theSchweikart triangle.

Because the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for anyhyperbolic triangle. This fact is important in the study ofδ-hyperbolic space.

In thePoincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles.

In thePoincaré half-plane model, an ideal triangle is modeled by anarbelos, the figure between three mutually tangentsemicircles.

In theBeltrami–Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that iscircumscribed 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 notconformal i.e. it does not preserve angles.

The Poincaré disk model tiled with ideal triangles

The ideal (∞ ∞ ∞)triangle group

Another ideal tiling

The real idealtriangle group is thereflection group generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic to thefree product of three order-two groups (Schwartz 2001).

• 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.

• Hyperbolic geometry
• Types of triangles

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