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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:
{2} \psline{o-o}(-... ...2) \psline{o-o}(0,2)(1.732,-1) \psline{o-o}(-1.732,-1)(1.732,-1) \end{pspicture}](http://images.planetmath.org/cache/objects/9447/js/img1.png)
Below is an example of an ideal triangle in the Poincaré disc model:
{2} \psarc{o-o}(0,... ...641,2){3.4641}{300}{360} \psarc{o-o}(3.4641,2){3.4641}{180}{240} \end{pspicture}](http://images.planetmath.org/cache/objects/9447/js/img2.png)
Below are some examples of ideal triangles in the upper half plane model:
![\begin{pspicture}(-2,-0.1)(4,4) \psline[linestyle=dashed]{<->}(-2,0)(4,0) \pslin... ...}(-1,0)(-1,4) \psline{o->}(3,0)(3,4) \psarc{o-o}(1,0){2}{0}{180} \end{pspicture}](http://images.planetmath.org/cache/objects/9447/js/img3.png)
![\begin{pspicture}(-5,-0.1)(5,4) \psline[linestyle=dashed]{<->}(-5,0)(5,0) \psarc... ...{0}{180} \psarc{o-o}(2,0){2}{0}{180} \psarc{o-o}(0,0){4}{0}{180} \end{pspicture}](http://images.planetmath.org/cache/objects/9447/js/img4.png)
Strictly speaking, none of these figures are triangles in hyperbolic geometry; however, ideal triangles are useful for proving that, given $r \in \mathbb{R}$ with $0<r<\pi$ , there is a triangle in hyperbolic geometry whose angle sum in radians is equal to $r$ .
ideal triangle is owned by Warren Buck.
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