Spherical Circle Planes

Abstract

In this chapter we concentrate on spherical circle planes. Just to reiterate, a spherical circle plane is a geometry whose point set is homeomorphic to the sphere and all of whose circles are subsets of the sphere that are topological circles. The axiom of joining that every spherical circle plane satisfies guarantees a unique circle that contains three given points. The derived geometry of a spherical circle plane at a point is an R²-plane (check this!). A spherical circle plane is called a flat Möbius plane or flat inversive plane if the derived geometries at all its points are flat affine planes. It is easy to show that a flat spherical circle plane is a flat Möbius plane if and only if it satisfies the so-called axiom of touching: Given a point p on a circle c and another point q, there is a uniquely determined circle d that contains both points and touches c, that is, intersects c only in p or coincides with c.