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| In hyperbolic geometry, an \emph{ideal triangle} is a set of three lines which connect three distinct points on the boundary of the model of hyperbolic geometry. |
In hyperbolic geometry, an \emph{ideal triangle} is a set of three lines which connect three distinct points on the boundary of the model of hyperbolic geometry. |
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| Below is an example of an ideal triangle in the Beltrami-Klein model: |
Below is an example of an ideal triangle in the Beltrami-Klein model: |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-2,-2)(2,2) |
\begin{pspicture}(-2,-2)(2,2) |
| \pscircle[linestyle=dashed](0,0){2} |
\pscircle[linestyle=dashed](0,0){2} |
| \psline{o-o}(-1.732,-1)(0,2) |
\psline{o-o}(-1.732,-1)(0,2) |
| \psline{o-o}(0,2)(1.732,-1) |
\psline{o-o}(0,2)(1.732,-1) |
| \psline{o-o}(-1.732,-1)(1.732,-1) |
\psline{o-o}(-1.732,-1)(1.732,-1) |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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Below is an example of an ideal triangle in the Poincar\'e disc model:
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Below is an example of an ideal triangle in the \PMlinkname{Poincar\'e disc model}{PoincareDiscModel}:
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-2,-2)(2,2) |
\begin{pspicture}(-2,-2)(2,2) |
| \pscircle[linestyle=dashed](0,0){2} |
\pscircle[linestyle=dashed](0,0){2} |
| \psarc{o-o}(0,-4){3.4641}{60}{120} |
\psarc{o-o}(0,-4){3.4641}{60}{120} |
| \psarc{o-o}(-3.4641,2){3.4641}{300}{360} |
\psarc{o-o}(-3.4641,2){3.4641}{300}{360} |
| \psarc{o-o}(3.4641,2){3.4641}{180}{240} |
\psarc{o-o}(3.4641,2){3.4641}{180}{240} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| Below are some examples of ideal triangles in the upper half plane model: |
Below are some examples of ideal triangles in the upper half plane model: |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-2,-0.1)(4,4) |
\begin{pspicture}(-2,-0.1)(4,4) |
| \psline[linestyle=dashed]{<->}(-2,0)(4,0) |
\psline[linestyle=dashed]{<->}(-2,0)(4,0) |
| \psline{o->}(-1,0)(-1,4) |
\psline{o->}(-1,0)(-1,4) |
| \psline{o->}(3,0)(3,4) |
\psline{o->}(3,0)(3,4) |
| \psarc{o-o}(1,0){2}{0}{180} |
\psarc{o-o}(1,0){2}{0}{180} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-5,-0.1)(5,4) |
\begin{pspicture}(-5,-0.1)(5,4) |
| \psline[linestyle=dashed]{<->}(-5,0)(5,0) |
\psline[linestyle=dashed]{<->}(-5,0)(5,0) |
| \psarc{o-o}(-2,0){2}{0}{180} |
\psarc{o-o}(-2,0){2}{0}{180} |
| \psarc{o-o}(2,0){2}{0}{180} |
\psarc{o-o}(2,0){2}{0}{180} |
| \psarc{o-o}(0,0){4}{0}{180} |
\psarc{o-o}(0,0){4}{0}{180} |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| \PMlinkescapetext{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 $r$. |
\PMlinkescapetext{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 $r$. |
