# The Cayley-Klein model of hyperbolic geometry

Let be a nondegenerate conic and let be the subgroup of projective transformations that leave the conic invariant. The aim of the next four exercises is to prove the following result:

Theorem 5.1   The group acts transitively on the interior and the exterior of the conic .

Because of the results in the previous section,s it is enough to consider the case where the conic is given in homogeneous coordinates by the equation .

The strategy of the proof is to find a large class of explicit linear transformations on that preserve the cone and to show that by using transformations in this class it is possible to take a line that passes through the origin and lies in the interior (resp. exterior) of the cone to any other line that passes through the origin and lies in the interior (resp. exterior) of the cone.

Exercise 5.1 (-05)   Define the hyperbolic functions

Show that
1. ,
2. ,
3. ,
4. the function is a bijection between the real line and the interval .

If and are real numbers, we define

Exercise 5.2 (05)   Show that
1. .
2. and .

Exercise 5.3 (05)   Show that if belongs to the interval , there exists a number such that applied to the vector is a nonzero multiple of the vector . Conclude that if , there exists a number such that applied to the vector is a nonzero multiple of the vector .

The previous exercise easily implies that acts transitively on the interior of . Indeed, if and are two lines in passing through the origin and inside the cone , there exist rotations and such that and lie on the plane . By the previous exercise, there exists a transformation such that . It follows that , and we are done.

Exercise 5.4 (10)   Use the same techniques as above to show that acts transitively on the exterior of .

We now pass to the study of the action of on the interior of , which we will henceforth denote as . The first thing we remark is that the action is not doubly transtive: it is in general impossible to transform a pair of distinct points into a pair of prescribed points . As before, it suffices to consider the case where the conic is a circle.

If and are two distinct points on , draw the line that joins them and intersect it with the circle to obtain two points and . Choose the notation such that belongs to the segment joining and , and belonds to the segment joining and (Fig. 9). Now define . Notice that we have chosen this precise ordering (and the definition of and ) so that this cross-ratio be greater than one. If , we define to be one.

Figure 9.

Exercise 5.5 (15)   Let be four points on . Show that there exists a map that takes to and to if and only if .

If and are two points in , define . The rest of this section is devoted to proving that is a distance function and that its geodesics are straight lines.

Exercise 5.6   Show that
1. with equality if and only if .
2. .
3. If are three collinear points, with belonging to the segment that joins and , then .

To show that is a distance function, it remains to prove the triangle inequality for three noncollinear points (Fig. 10). Since the logarithm is an increasing function, all we must do is show that , where these quantities are

Figures 10 and 11.

Using figure 11 and the invariance of cross-ratios under perspectivities, we have that

and

Therefore,

Exercise 5.7   Finish the proof of the triangle inequality by showing that

The disc together with distance is called the hyperbolic plane.

Definition 5.1   Let be a metric space and let be a continuous curve. Define the length of as

is a partition of

The curve is called a geodesic segment if its length equals the distance between its endpoints.

Exercise 5.8 (10)   Show that any line segment on is a geodesic segment for the distance .