Cayley--Klein Geometries

In Chap. 12 we introduced the Poincaré half-plane as a model of the plane hyperbolic geometry. It is a remarkable fact that hyperbolic distance may be expressed by the cross-ratio ( 12.13 ), i.e. by means of a projective quantity. In the first half of 1800 J. V. Poncelet (1788–1867) discovered that some euclidean notions might be formulated in terms of projective geometry. In order to show these connections we have to consider the euclidean plane immersed into the complex projective plane.