Laguerre geometry

Summary

Laguerre geometry is the geometry of oriented lines and oriented circles of the plane. Its transformations preserve oriented contact and map oriented lines to oriented lines. Points count as circles of radius $0$, and these transformations need not map points to points. A typical example moves every oriented line parallel to itself by the same distance and adds that distance to every signed radius. It is a subgeometry of Lie sphere geometry, and its abstract incidence version is the Laguerre plane; it is named after Edmond Laguerre.