The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers.[1][2][3][4] When studying these transformations, the dual numbers are often interpreted as representing oriented lines on the plane.[1] The Laguerre transformations map lines to lines, and include in particular all isometries of the plane.
Strictly speaking, these transformations act on the dual number projective line, which adjoins to the dual numbers a set of points at infinity. Topologically, this projective line is equivalent to a cylinder. Points on this cylinder are in a natural one-to-one correspondence with oriented lines on the plane.