Ideal triangle

Three ideal triangles in the circular disk model

Two ideal triangles in the half-space model

In hyperbolic geometry , an ideal triangle is a hyperbolic triangle whose three corners are in the ideal boundary .

More generally, ideal triangles can be defined in manifolds of non-positive cutting curvature.

properties

• All ideal triangles are congruent to each other.
• The interior angles of an ideal triangle are all zero.
• An ideal triangle has an infinitely large diameter .
• An ideal triangle is not contained in any larger hyperbolic triangle.

The inscribed circle of an ideal triangle in the Beltrami-Klein model and in the circular disk model

In hyperbolic geometry, i.e. in spaces with a curvature of section −1, the following still applies:

• Ideal triangles have the area . ${\displaystyle \pi }$
• The inscribed circle of an ideal triangle has the radius

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

The distance between an inner point and the edge of the ideal triangle is at most with equality only for the inscribed center. ${\displaystyle r}$

• The points of contact of the inscribed circle with the edge of the ideal triangle form an equilateral triangle of side length with the golden section . ${\displaystyle d=\ln \left({\frac {{\sqrt {5}}+1}{{\sqrt {5}}-1}}\right)=2\ln \varphi \approx 0.962}$ ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$
• The distance from one edge point to another edge side is at most , with equality only for the contact points of the inscribed circle. ${\displaystyle a=\ln \left(1+{\sqrt {2}}\right)\approx 0.881}$

Thin triangles and Gromov hyperbolicity

δ-thin triangles in the definition of δ-hyperbolic spaces

For each side of an ideal triangle lies in the neighborhood of the union of the other two sides. Because every other hyperbolic triangle is contained in an ideal triangle, it follows from this the δ-hyperbolicity of the hyperbolic plane and of all higher-dimensional hyperbolic spaces . ${\displaystyle \delta =\ln({\sqrt {2}}+1)}$ ${\displaystyle \delta }$

Models

In the circular disk model of the hyperbolic plane, a hyperbolic triangle is bordered by three circles that intersect the circle at infinity at three right angles.

In the half-space model , an ideal triangle is bordered by three semicircles or straight lines perpendicular to each other and on the real straight line.

In the Beltrami-Klein model , an ideal triangle is a Euclidean triangle inscribed in the circle at infinity. The Euclidean interior angles are not zero because this model does not conform .

Ideal triangular groups

Real ideal triangle groups

The ideal triangular group ${\displaystyle (\infty ,\infty ,\infty )}$

A real ideal triangle group is generated by the reflections of the hyperbolic plane on the three sides of an ideal triangle mirroring group . It is isomorphic to the free product . ${\displaystyle \mathbb {Z} _{2}*\mathbb {Z} _{2}}$

Complex ideal triangle group

→ Main article : Complex ideal triangle group

A complex ideal triangle group is the reflection group created by the complex reflections of the complex hyperbolic plane on the three sides of an ideal triangle.

literature

• Richard Evan Schwartz : Ideal triangle groups, dented tori, and numerical analysis , Annals of Mathematics 153, 533-598 (2001)