# ideal triangle

In hyperbolic geometry, an ideal triangle is a set of three lines which connect three distinct points on the boundary of the model of hyperbolic geometry.

Below is an example of an ideal triangle in the Beltrami-Klein model:

Below is an example of an ideal triangle in the Poincaré disc model:

Below are some examples of ideal triangles in the upper half plane model:

speaking, none of these figures are triangles in hyperbolic geometry; however, ideal triangles are useful for proving that, given $r\in\mathbb{R}$ with $0, there is a triangle in hyperbolic geometry whose angle sum in radians is equal to $r$.

Title ideal triangle IdealTriangle 2013-03-22 17:08:26 2013-03-22 17:08:26 Wkbj79 (1863) Wkbj79 (1863) 5 Wkbj79 (1863) Definition msc 51M10 msc 51-00 LimitingTriangle