Concept
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 , 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 that the plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder (see below). Originally the classical Laguerre plane was defined as the geometry of the oriented lines and circles in the real Euclidean plane (see ). Here we prefer the parabola model of the classical Laguerre plane. We define: the set of points, the set of cycles. The incidence structure is called classical Laguerre plane. The point set is plus a copy of (see figure). Any parabola/line gets the additional point . Points with the same x-coordinate cannot be connected by curves . Hence we define: Two points are parallel () if or there is no cycle containing and . For the description of the classical real Laguerre plane above two points are parallel if and only if . is an equivalence relation, similar to the parallelity of lines. The incidence structure has the following properties: Lemma: For any three points , pairwise not parallel, there is exactly one cycle containing . For any point and any cycle there is exactly one point such that . For any cycle , any point and any point that is not parallel to there is exactly one cycle through with , i.e. and touch each other at . Similar to the sphere model of the classical Moebius plane there is a cylinder model for the classical Laguerre plane: is isomorphic to the geometry of plane sections of a circular cylinder in . The following mapping is a projection with center that maps the x-z-plane onto the cylinder with the equation , axis and radius The points (line on the cylinder through the center) appear not as images.
