Minkowski plane

In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane). Applying the pseudo-euclidean distance on two points (instead of the euclidean distance) we get the geometry of hyperbolas, because a pseudo-euclidean circle is a hyperbola with midpoint . By a transformation of coordinates , , the pseudo-euclidean distance can be rewritten as . The hyperbolas then have asymptotes parallel to the non-primed coordinate axes. The following completion (see Möbius and Laguerre planes) homogenizes the geometry of hyperbolas: the set of points: the set of cycles The incidence structure is called the classical real Minkowski plane. The set of points consists of , two copies of and the point . Any line is completed by point , any hyperbola by the two points (see figure). Two points can not be connected by a cycle if and only if or . We define: Two points are (+)-parallel () if and (−)-parallel () if . Both these relations are equivalence relations on the set of points. Two points are called parallel () if or . From the definition above we find: Lemma: For any pair of non parallel points there is exactly one point with . For any point and any cycle there are exactly two points with . For any three points , , , pairwise non parallel, there is exactly one cycle that contains . For any cycle , any point and any point and there exists exactly one cycle such that , i.e. touches at point P. Like the classical Möbius and Laguerre planes Minkowski planes can be described as the geometry of plane sections of a suitable quadric. But in this case the quadric lives in projective 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of a hyperboloid of one sheet (not degenerated quadric of index 2). Let be an incidence structure with the set of points, the set of cycles and two equivalence relations ((+)-parallel) and ((−)-parallel) on set . For we define: and . An equivalence class or is called (+)-generator and (−)-generator, respectively.