Chapter 24 - Topological Circle Geometries

Abstract

This chapter focuses on topological circle geometries. The geometry of all circles on the real 2-sphere in 3-dimensional Euclidean space (i.e. the intersection of the sphere with Euclidean planes) has since long been investigated under various aspects. Another model of the same geometry is obtained by stereographic projection from one point of the sphere onto a plane not passing through the point of projection. This leads to the geometry of Euclidean lines extended by an infinite point and Euclidean circles in the real plane. Similarly, one considers extensions of the geometry of Euclidean lines and Euclidean parabolae with given direction of axis or Euclidean hyperbolae with given direction of asymptotes. An algebraic description of these geometries leads to chain geometries. It discusses the planar situation and takes a more geometric point of view. An axiomatization is given and thus a generalization of these geometries which leads to so-called circle planes. Since affine and projective planes occur as derived incidence structures, the structure theory and classification of topological locally compact connected finite-dimensional affine and projective planes plays a crucial role.