Interactive Course on Projective Geometry

Appendix 2

In this chapter we study the relation between projective and metric geometry. Specifically, we consider the problem, posed by Hilbert, of determining all the metrics on open convex subsets of the projective plane such that projective line segments are geodesics.

We start by considering 2-dimensional normed spaces and Hilbert geometries. The latter are a natural generalization of the Cayley-Klein model of hyperbolic geometry. To construct more general examples, we follow Busemann in defining the length of a line segment as the area of the set of projective lines intersecting it. Changing the way in which we measure areas, we obtain different notions of distance. We shall also sketch a proof, due to R. Ambartzumian, of the fact that any (continuous) notion of distance for which projective line segments are geodesics comes from Busemann's construction.