In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane).

Classical real Minkowski 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 M. By a transformation of coordinates x_i = x'_i + y'i, y_i = x'i - y'i, the pseudo-euclidean distance can be rewritten as d(P_1,P_2) = (x_1 - x_2) (y_1 - y_2). 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 incidence structure is called the classical real Minkowski plane. The set of points consists of \R^2, two copies of \R and the point (\infty,\infty). Any line is completed by point (\infty,\infty), any hyperbola by the two points (see figure). Two points can not be connected by a cycle if and only if x_1 = x_2 or y_1 = y_2. We define: Two points P_1, P_2 are (+)-parallel (P_1 \parallel+ P_2) if x_1 = x_2 and (−)-parallel (P_1\parallel- P_2) if y_1 = y_2. Both these relations are equivalence relations on the set of points. Two points P_1,P_2 are called parallel (P_1*parallel** P_2) if or P_1 *parallel**- P_2. From the definition above we find: Lemma: 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).

Axioms of a Minkowski plane

Let be an incidence structure with the set \mathcal P of points, the set \mathcal Z of cycles and two equivalence relations \parallel_+ ((+)-parallel) and \parallel_- ((−)-parallel) on set \mathcal P. For we define: and. An equivalence class or is called (+)-generator and (−)-generator, respectively. (For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.) Two points A, B are called parallel if or. An incidence structure is called Minkowski plane if the following axioms hold: For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous. First consequences of the axioms are Analogously to Möbius and Laguerre planes we get the connection to the linear geometry via the residues. For a Minkowski plane and we define the local structure and call it the residue at point P. For the classical Minkowski plane is the real affine plane \R^2. An immediate consequence of axioms C1 to C4 and C1′, C2′ are the following two theorems.

Minimal model

The minimal model of a Minkowski plane can be established over the set of three elements: Parallel points: Hence and \mathcal Z \right.

Finite Minkowski-planes

For finite Minkowski-planes we get from C1′, C2′: This gives rise of the definition: For a finite Minkowski plane \mathfrak M and a cycle z of \mathfrak M we call the integer the order of. Simple combinatorial considerations yield

Miquelian Minkowski planes

We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace \R by an arbitrary field K then we get in any case a Minkowski plane {\mathfrak M}(K)=({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in). Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane \mathfrak M (K). Theorem (Miquel): For the Minkowski plane the following is true: (For a better overview in the figure there are circles drawn instead of hyperbolas.) Theorem (Chen): Only a Minkowski plane satisfies the theorem of Miquel. Because of the last theorem is called a miquelian Minkowski plane. Remark: The minimal model of a Minkowski plane is miquelian. An astonishing result is Theorem (Heise): Any Minkowski plane of even order is miquelian. Remark: A suitable stereographic projection shows: is isomorphic to the geometry of the plane sections on a hyperboloid of one sheet (quadric of index 2) in projective 3-space over field K. Remark: There are a lot of Minkowski planes that are not miquelian (s. weblink below). But there are no "ovoidal Minkowski" planes, in difference to Möbius and Laguerre planes. Because any quadratic set of index 2 in projective 3-space is a quadric (see quadratic set).