Beltrami-Klein modelBeltramiKleinModel22007-05-20 14:19:372007-06-25 17:14:52Definitionnon-Euclidean geometrypole\usepackage{amssymb}
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\usepackage{amsthm}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.
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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 \PMlinkescapetext{measure} of an angle within the model is not necessarily the angle measure 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 \PMlinkname{tangent}{TangentLine} to the circle at the endpoints of $\ell$.
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\rput[a](0,0.7){$\ell$}
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\rput[l](-1.9,4.4){$P(\ell)$}
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Poles are important for the following reason: Given a line $\ell$ that is not a diameter of the Beltrami-Klein model, one constructs a line perpendicular to $\ell$ by considering Euclidean lines passing through $P(\ell)$. Thus, 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.
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\rput[l](-1.9,4.4){$P(\ell)$}
\rput[l](0.2,-2.5){$P(m)$}
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In the above picture, $n$ is the common perpendicular of $\ell$ and $m$.