Models and theorems of the classical circle planes (Q1313467)

The author gives an account of the classical circle planes over an arbitrary field \(F\) with special emphasis on the field of real numbers. These circle planes are usually defined as the geometries of plane sections of a quadric surface in \(PG(3,F)\). The author studies their stereographic projections onto the affine plane over \(F\) and also their embeddings into the Lie quadric in \(PG(4,F)\). This embedding natural leads to the notion of oriented circle or cycle. The author gives some classical and some quite new theorems on circle geometries, e.g. he solves the problem of Apollonius and he proves an unpublished result of Searby on chains of cycles. The case char \(F=2\) requires special considerations, which are not dealt with in all details.