Circle Geometries Modeled in Projective Lines over $$ {{\mathbb{R}}^2} $$ -rings

Abstract

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of \( {{\mathbb{R}}^2} \)-ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. 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 in themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Mobius geometry by suitable projections of the Blaschke cylinder.