3 - Spherical Circle Planes

Summary

Flat Möbius planes were first investigated by Wölk [1966] and Strambach [1967c]. Later, Strambach [1970d], [1972], [1973], [1974a], [1974b] studied the more general spherical circle planes. For more information about Möbius planes and, in particular, finite Möbius planes, we refer to Dembowski [1968], Delandtsheer [1995], Hering [1965], Mäurer [1967], Wilker [1981] and the references given there.

A spherical circle plane is a point–circle geometry whose point set is (homeomorphic to) S² and whose circles are topological circles on S². Furthermore, the Axiom of Joining B1 (see p. 7) is satisfied, that is, any three distinct points are contained in exactly one of the circles. A spherical circle plane is a flat Möbius plane if, in addition, the Axiom of Touching B2 is satisfied, that is, for each circle C and any two distinct points p, q with p ∈ C there is precisely one circle through p and q that touches C at p geometrically, that is, intersects C only at the point p or coincides with C.

Models of the Classical Flat Möbius Plane

In this first section we describe a number of models of the classical flat Möbius plane. For detailed information about most of these models see Benz [1973].