| Abstract: | We present a common axiomatic characterization of Cayley-Klein geometries over fields of characteristic \({\neq 2}\). To this end the axiom system of Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg1973) for plane absolute geometry, which allows a common axiomatization of Euclidean, hyperbolic and elliptic geometry, is generalized. The notion of plane absolute geometry is broadened in several aspects. The most important one is that the principle of duality holds: the dual of a Cayley-Klein geometry is also a Cayley-Klein geometry. The various Cayley-Klein geometries are singled out by additional axioms like the Euclidean or hyperbolic parallel axiom or their dual statements. |
