Benz plane

Remarks

The three types differ in their parallelism of points. A Möbius plane has no parallel points, a Laguerre plane has one parallelism, and a Minkowski plane has two. In each type, three pairwise non-parallel points lie on exactly one circle, and a touching axiom holds.

Example: the real plane with a point $\infty$ added, with circles and lines through $\infty$ as circles, is a Möbius plane, hence a Benz plane.

Non-example: a projective plane, with its lines as circles, is not a Benz plane. Three non-collinear points lie on no line, so the axiom on three points fails for each of the three types.

The classical Benz planes are the geometries of plane sections of a sphere, a cylinder and a hyperboloid of one sheet.