On elementary circle-geometry in Cayley–Klein planes

Abstract

This contribution is sort of an addendum to a recently published paper on circle-geometries in Cayley–Klein planes, see Martini and Spirova (Publicationes Mathematicae Debrecen 72:371–383, 2008b), as it deals with further generalisations and extensions of the author’s results to circle-geometries in all Cayley–Klein planes. The main methods in this paper are the interpretation of planar figures in space and the dualizing according to the duality principle of projective spaces. There are, in principle, only three types of \({\mathbb{R}^2}\)-ring structures and, thus, only three types of corresponding circle-geometries, see Benz (Geometrie der Algebren, 1973a). Therefore, each generalisation to non-Euclidean planes must turn out to be just another representation of the classical Euclidean cases. This point of view gives more insight into why some elementary geometric theorems remain valid when changing the place of action from the Euclidean plane to non-Euclidean circle planes and makes explicit proofs of such elementary geometric theorems in non-Euclidean circle planes superfluous. We believe that even the Euclidean cases of circle-geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle-geometries might also be of interest for their own. For example, among the planar Cayley–Klein geometries the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated, similar to the isotropic Mobius geometry, by suitable projections of the Blaschke cylinder.