Beltrami-Klein model

The Beltrami-Klein model for ${ℍ}^2$ is the disc 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.

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 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 $\mathrm{\ell }$ be a line in the Beltrami-Klein model that is not a diameter of the circle. The pole of $\mathrm{\ell }$ is the intersection of the Euclidean lines that are tangent (http://planetmath.org/TangentLine) to the circle at the endpoints of $\mathrm{\ell }$.

Poles are important for the following reason: Given a line $\mathrm{\ell }$ that is not a diameter of the Beltrami-Klein model, one constructs a line perpendicular to $\mathrm{\ell }$ by considering Euclidean lines passing through $P(\mathrm{\ell })$. Thus, given two disjointly parallel lines $\mathrm{\ell }$ and $m$ that are not diameters of the Beltrami-Klein model, one constructs their common perpendicular by connecting their poles.

In the above picture, $n$ is the common perpendicular of $\mathrm{\ell }$ and $m$.