page_title: Laguerre plane
section_title: Finite Laguerre planes
hierarchical_section_title: Laguerre plane / Finite Laguerre planes
context_page_description: In mathematics, a Laguerre plane is one of the Benz planes: the Möbius plane, Laguerre plane and Minkowski plane, named after the French mathematician Edmond Nicolas Laguerre.
  Essentially the classical Laguerre plane is an incidence structure that describes the incidence behaviour of the curves, i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve the point is added. A further advantage of this completion is: The plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder.
context_section_description: The following incidence structure is a minimal model of a Laguerre plane:
  Hence  and 
  For finite Laguerre planes, i.e. , we get:
  Lemma:
  For any cycles  and any generator  of a finite Laguerre plane
   we have:
  .
  For a finite Laguerre plane  and a cycle  the integer  is called order of .
  From combinatorics we get
  Lemma:
  Let  be a Laguerre—plane of order . Then
  a) any residue  is an affine plane of order   b)   c)
caption: minimal model of a Laguerre plane (only 4 of 8 cycles are shown)