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 $m$.

Title	Beltrami-Klein model
Canonical name	BeltramiKleinModel
Date of creation	2013-03-22 17:06:37
Last modified on	2013-03-22 17:06:37
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	18
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 51M10
Classification	msc 51-00
Synonym	Klein-Beltrami model
Synonym	Klein model
Related topic	ConvertingBetweenTheBeltramiKleinModelAndThePoincareDiscModel
Defines	pole