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'Beltrami-Klein model'
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| Title of object: |
Beltrami-Klein model |
| Canonical Name: |
BeltramiKleinModel2 |
| Type: |
Definition |
| Created on: |
2007-05-20 14:19:37 |
| Modified on: |
2007-05-20 14:26:06 |
| Classification: |
msc:51-00, msc:51M10 |
| Defines: |
pole |
| Synonyms: |
Beltrami-Klein model=Klein-Beltrami model
Beltrami-Klein model=Klein model |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{pstricks}
\usepackage{amsthm} |
Content:
The \emph{Beltrami-Klein model} for $\mathbb{H}^2$ is the disc $\{(x,y) \in \mathbb{R}^2 : x^2+y^2<1 \}$ in which a point is similar to the Euclidean point and a line is defined to be a chord (excluding its endpoints) of the (circular) boundary.
\begin{center}
\begin{pspicture}(-2,-2)(2,2)
\pscircle[linestyle=dashed](0,0){2}
\psline{o-o}(-2,0)(1.414,1.414)
\end{pspicture}
\end{center}
The Beltrami-Klein model has the advantage that lines in the model resemble Euclidean lines; however, it has the drawback that it is not angle preserving. That is, the Euclidean measure of an angle within the model is not necessarily the measure of the angle in hyperbolic geometry.
Some points outside of the Beltrami-Klein model are important for constructions within the model. The following is an example of such:
Let $\ell$ be a line in the Beltrami-Klein model that is not a diameter of the circle. The \emph{pole} of $\ell$ is the intersection of the Euclidean lines that are tangent to the circle at the endpoints of $\ell$.
\begin{center}
\begin{pspicture}(-3,-2)(3,5)
\pscircle[linestyle=dashed](0,0){2}
\psline{o-o}(-2,0)(1.414,1.414)
\rput[a](0,0.8){$\ell$}
\psline{<->}(-2,-2)(-2,5)
\psline{<->}(-2.7172,5)(2.828,0.1)
\psdots(-2,4.828)
\rput[l](-2,5){$P(\ell)$}
\rput[b](-2,-2){.}
\rput[b](-2.7172,5){.}
\rput[b](2.828,0){.}
\end{pspicture}
\end{center}
Poles are important for the following reason: Given two disjointly parallel lines $\ell$ and $m$ that are not diameters of the Beltrami-Klein model, one constructs their common perpendicular by connecting their poles.
\begin{center}
\begin{pspicture}(-3,-3)(3,6)
\pscircle[linestyle=dashed](0,0){2}
\psline{o-o}(-2,0)(1.414,1.414)
\rput[a](0,0.8){$\ell$}
\psline{o-o}(-1.2,-1.6)(1.2,-1.6)
\rput[b](1,-1.6){$m$}
\psline{<->}(-2,-2.5)(-2,5)
\psline{<->}(-2.7172,5)(2.828,0.1)
\psline{<->}(-2.4,1.2)(0.3,-2.4)
\psline{<->}(-0.3,-2.4)(2.4,1.2)
\psdots(-2,4.828)(0,-2)
\rput[l](-2,5){$P(\ell)$}
\rput[l](0,-2){$P(m)$}
\rput[b](-2,-2.5){.}
\rput[b](-2.7172,5){.}
\rput[b](2.828,0.1){.}
\psline{<->}(-2.2,5.7)(0.2,-2.7)
\rput[b](-2.2,5.7){$n$}
\end{pspicture}
\end{center}
In the above picture, $n$ is the common perpendicular of $\ell$ and $m$. |
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