Cylindrical circle planes and, in particular, flat Laguerre planes were first investigated by Groh [1968], [1969]. For information about general Laguerre planes we refer to Delandtsheer [1995], Hartmann [1982b], Kleinewillinghöfer [1980], and references given there.

A cylindrical circle plane is a point–circle geometry whose point set is (homeomorphic to) the cylinder S1 × R. Its point set is equipped with one nontrivial parallelism. The parallel classes of this parallelism are the verticals (generators) on the cylinder. The circles of the cylindrical circle plane are graphs of continuous functions S1 → R that form a system of topological circles on the cylinder such that the Axiom of Joining B1 (see p. 7) is satisfied, that is, any three pairwise nonparallel points determine exactly one curve in the system. A cylindrical circle plane is a flat Laguerre plane if and only if it also satisfies Axiom B2.

Cylindrical circle planes are just one of the infinitely many types of tubular circle planes that we will investigate in the last chapter of this book. On the other hand, among these different types of circle planes only the cylindrical circle planes allow an exposition that parallels those of the spherical and toroidal circle planes. Also, we will see in the next chapter that flat Laguerre planes are really the most important, the best understood, and the most general among the nested flat circle planes of rank 3.