Beltrami-Klein model

The Beltrami-Klein model for 2 is the disc {(x,y)2:x2+y2<1} in which a point is similarMathworldPlanetmathPlanetmath to the Euclidean point and a line is defined to be a chord (excluding its endpointsMathworldPlanetmath) 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 be a line in the Beltrami-Klein model that is not a diameterMathworldPlanetmathPlanetmath of the circle. The pole of is the intersection of the Euclidean lines that are tangent ( to the circle at the endpoints of .


Poles are important for the following reason: Given a line that is not a diameter of the Beltrami-Klein model, one constructs a line perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath to by considering Euclidean lines passing through P(). Thus, given two disjointly parallel lines 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 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