The Beltrami-Klein Model

   This model is often referred to as the Klein model, because of the extensive work done by the German mathematician Felix Klein in geometry with this model. Fix, once and for all, a circle in with center O and radius OR. Let

Identify the points of with the points in .

A chord  of is the Euclidean segment AB joining two points . is an open chord. These open chords represent the lines of the hyperbolic plane. To say that P lies on (A,B) means , the Euclidean line, and .

It is easy to see that this model satisfies the Hyperbolic Axiom.

 
Figure 17.1: Parallel lines in the Klein model

Here, . The lines and are two lines that do not intersect k in , as is the line . The difference is that and are limiting parallel to k in different directions, while and k are hyperparallel, admitting a common perpendicular. Recall that the points of do not belong to the hyperbolic plane. They are called ideal points of .  The points outside of are called ultraideal points. 

By saying that the lines k and admit a common perpendicular raises the question about how one defines congruence of segments and angles. It is not obvious, for it would seem that lines must be of finite extent, no line measuring more than twice the diameter of . If this were the case, we would not have a model for hyperbolic geometry, as the congruence axioms, Archimedes' axiom, and Dedekind's axiom definitely would not hold. We must define a different method for measuring the length of segments and the measure of angles in this model.

I will at this point only mention the method for measuring the length of segments and the definition of right angles. The remaining measurement will follow from work that we do later, and will be noted then.

Let and let denote the endpoints of the chord through A and B. Let denote the Euclidean distance from A to B, or the length of the segment AB. Define the Klein distance 

 
Figure 17.2: Length in the Klein model

Note then that as B approaches Q, approaches 0, thus

This allows us to see then that we can find a segment of any length on any ray (Axiom C-1). Also, it will follow that Dedekind's and Archimedes' axioms are valid.

Our technique for measuring angles will be introduced at a later time. It depends on the model for hyperbolic geometry due to Poincaré. It is not the way in which angles are measured in Euclidean geometry. For this reason the Klein model is not a conformal  model of geometry. We can talk about right angles, without too much difficulty. Let l and m be two lines in the Klein model of the hyperbolic plane, or K-lines. in the Klein sense if,

1. l or m is a diameter and l and m are perpendicular in the Euclidean sense, or
2. if neither l or m is a diameter, do the following: Let and be the tangents to at the ends of l. Since l is not a diameter, . Put . P(l) is called the pole of l. m is perpendicular to l in the Klein sense if and only if the Euclidean line extending m passes through P(l). 

All of this will be verified shortly.