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:

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.

- ,
- ,
- ,
- the function is a bijection between the real line and the interval .

If and are real numbers, we define

- .
- and .

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.

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.

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.

- with equality if and only if .
- .
- 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,

The disc together with distance is called the
*hyperbolic plane.*

is a partition of

The curve is called a