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.
- ^ a b Yaglom, Isaak Moiseevitch (1968). Complex Numbers in Geometry. Academic Press. Originally published as Kompleksnye Chisla i Ikh Primenenie v Geometrii (in Russian). Moscow: Fizmatgiz. 1963
- ^ Bolt, Michael; Ferdinands, Timothy; Kavlie, Landon (2009). "The most general planar transformations that map parabolas into parabolas". Involve: A Journal of Mathematics. 2 (1): 79–88. doi:10.2140/involve.2009.2.79. ISSN 1944-4176.
- ^ Fillmore, Jay P.; Springer, Arthur (1995-03-01). "New euclidean theorems by the use of Laguerre transformations — Some geometry of Minkowski (2+1)-space". Journal of Geometry. 52 (1): 74–90. doi:10.1007/BF01406828. ISSN 1420-8997. S2CID 122511184.
- ^ Barrett, David E.; Bolt, Michael (June 2010). "Laguerre Arc Length from Distance Functions". Asian Journal of Mathematics. 14 (2): 213–234. doi:10.4310/AJM.2010.v14.n2.a3. ISSN 1093-6106.