Laguerre plane

In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre.

The classical Laguerre plane is an incidence structure that describes the incidence behaviour of the curves ${\displaystyle y=ax^{2}+bx+c}$ , i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve ${\displaystyle y=ax^{2}+bx+c}$ the point ${\displaystyle (\infty ,a)}$ is added. A further advantage of this completion is that the plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder (see below).