In mathematics, aMinkowski plane (named afterHermann Minkowski) is one of theBenz planes (the others beingMöbius plane andLaguerre plane).

Applying thepseudo-euclidean distance${\displaystyle d(P_1,P_2)=(x_1^{\prime }-x_2^{\prime }{)}^2-(y_1^{\prime }-y_2^{\prime }{)}^2}$ on two points${\displaystyle P_i=(x_i^{\prime },y_i^{\prime })}$ (instead of the euclidean distance) we get the geometry ofhyperbolas, because a pseudo-euclidean circle${\displaystyle \{P\in {\mathbb{R}}^2\mid d(P,M)=r\}}$ is ahyperbola with midpoint⁠${\displaystyle M}$⁠.

By a transformation of coordinates⁠${\displaystyle x_i=x_i^{\prime }+y_i^{\prime }}$⁠,⁠${\displaystyle y_i=x_i^{\prime }-y_i^{\prime }}$⁠, the pseudo-euclidean distance can be rewritten as⁠${\displaystyle d(P_1,P_2)=(x_1-x_2)(y_1-y_2)}$⁠. The hyperbolas then haveasymptotes parallel to the non-primed coordinate axes.

The following completion (see Möbius and Laguerre planes)homogenizes the geometry of hyperbolas:

• the set ofpoints:$${\displaystyle \mathcal{P}:={\left(\mathbb{R}\cup \left\{\mathrm{\infty }\right\}\right)}^2={\mathbb{R}}^2\cup \left(\left\{\mathrm{\infty }\right\}\times \mathbb{R}\right)\cup \left(\mathbb{R}\times \left\{\mathrm{\infty }\right\}\right)\cup \left\{\left(\mathrm{\infty },\mathrm{\infty }\right)\right\},\mathrm{\infty }\notin \mathbb{R},}$$
• the set ofcycles

Theincidence structure${\displaystyle (\mathcal{P},\mathcal{Z},\in )}$ is called theclassical real Minkowski plane.

The set of points consists of⁠${\displaystyle {\mathbb{R}}^2}$⁠, two copies of${\displaystyle \mathbb{R}}$ and the point⁠${\displaystyle (\mathrm{\infty },\mathrm{\infty })}$⁠.

Any line${\displaystyle y=ax+b,a\ne 0}$ is completed by point⁠${\displaystyle (\mathrm{\infty },\mathrm{\infty })}$⁠, any hyperbola${\displaystyle y=\frac{a}{x-b}+c,a\ne 0}$ by the two points${\displaystyle (b,\mathrm{\infty }),(\mathrm{\infty },c)}$ (see figure).

Two points${\displaystyle (x_1,y_1)\ne (x_2,y_2)}$ can not be connected by a cycle if and only if${\displaystyle x_1=x_2}$ or⁠${\displaystyle y_1=y_2}$⁠.

We define:Two points${\displaystyle P_1}$,${\displaystyle P_2}$ are(+)-parallel (⁠${\displaystyle P_1{\parallel }_{+}{P}_2}$⁠) if${\displaystyle x_1=x_2}$ and(−)-parallel (⁠${\displaystyle P_1{\parallel }_{-}{P}_2}$⁠) if⁠${\displaystyle y_1=y_2}$⁠.
Both these relations areequivalence relations on the set of points.

Two points${\displaystyle P_1,P_2}$ are calledparallel (⁠${\displaystyle P_1\parallel P_2}$⁠) if${\displaystyle P_1{\parallel }_{+}{P}_2}$ or⁠${\displaystyle P_1{\parallel }_{-}{P}_2}$⁠.

From the definition above we find:

Lemma:

• For any pair of non parallel points⁠${\displaystyle A}$⁠,${\displaystyle B}$ there is exactly one point${\displaystyle C}$ with⁠${\displaystyle A{\parallel }_{+}C{\parallel }_{-}B}$⁠.
• For any point${\displaystyle P}$ and any cycle${\displaystyle z}$ there are exactly two points${\displaystyle A,B\in z}$ with⁠${\displaystyle A{\parallel }_{+}P{\parallel }_{-}B}$⁠.
• For any three points⁠${\displaystyle A}$⁠,⁠${\displaystyle B}$⁠,⁠${\displaystyle C}$⁠, pairwise non parallel, there is exactly one cycle${\displaystyle z}$ that contains⁠${\displaystyle A,B,C}$⁠.
• For any cycle⁠${\displaystyle z}$⁠, any point${\displaystyle P\in z}$ and any point${\displaystyle Q,P\nparallel Q}$ and${\displaystyle Q\notin z}$ there exists exactly one cycle${\displaystyle z^{\prime }}$ such that⁠${\displaystyle z\cap z^{\prime }=\{P\}}$⁠, i.e.${\displaystyle z}$touches${\displaystyle z^{\prime }}$ at point⁠${\displaystyle P}$⁠.

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 inprojective 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of ahyperboloid of one sheet (not degenerated quadric of index 2).

Let${\displaystyle \left(\mathcal{P},\mathcal{Z};{\parallel }_{+},{\parallel }_{-},\in \right)}$ be an incidence structure with the set${\displaystyle \mathcal{P}}$ of points, the set${\displaystyle \mathcal{Z}}$ of cycles and two equivalence relations${\displaystyle {\parallel }_{+}}$ ((+)-parallel) and${\displaystyle {\parallel }_{-}}$ ((−)-parallel) on set${\displaystyle \mathcal{P}}$. For${\displaystyle P\in \mathcal{P}}$ we define:${\displaystyle {\overline{P}}_{+}:=\left\{Q\in \mathcal{P}\mid Q{\parallel }_{+}P\right\}}$ and${\displaystyle {\overline{P}}_{-}:=\left\{Q\in \mathcal{P}\mid Q{\parallel }_{-}P\right\}}$.An equivalence class${\displaystyle {\overline{P}}_{+}}$ or${\displaystyle {\overline{P}}_{-}}$ 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${\displaystyle A,B}$ are calledparallel (${\displaystyle A\parallel B}$) if${\displaystyle A{\parallel }_{+}B}$ or${\displaystyle A{\parallel }_{-}B}$.

An incidence structure${\displaystyle \mathfrak{M}:=(\mathcal{P},\mathcal{Z};{\parallel }_{+},{\parallel }_{-},\in )}$ is calledMinkowski plane if the following axioms hold:

• C1: For any pair of non parallel points${\displaystyle A,B}$ there is exactly one point${\displaystyle C}$ with${\displaystyle A{\parallel }_{+}C{\parallel }_{-}B}$.
• C2: For any point${\displaystyle P}$ and any cycle${\displaystyle z}$ there are exactly two points${\displaystyle A,B\in z}$ with${\displaystyle A{\parallel }_{+}P{\parallel }_{-}B}$.
• C3: For any three points${\displaystyle A,B,C}$, pairwise non parallel, there is exactly one cycle${\displaystyle z}$ which contains${\displaystyle A,B,C}$.
• C4: For any cycle${\displaystyle z}$, any point${\displaystyle P\in z}$ and any point${\displaystyle Q,P\nparallel Q}$ and${\displaystyle Q\notin z}$ there exists exactly one cycle${\displaystyle z^{\prime }}$ such that${\displaystyle z\cap z^{\prime }=\{P\}}$, i.e.,${\displaystyle z}$touches${\displaystyle z^{\prime }}$ at point${\displaystyle P}$.
• C5: Any cycle contains at least 3 points. There is at least one cycle${\displaystyle z}$ and a point${\displaystyle P}$ not in${\displaystyle z}$.

For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous.

• C1′: For any two points⁠${\displaystyle A}$⁠,${\displaystyle B}$ we have⁠${\displaystyle \left|{\overline{A}}_{+}\cap {\overline{B}}_{-}\right|=1}$⁠.
• C2′: For any point${\displaystyle P}$ and any cycle${\displaystyle z}$ we have:⁠${\displaystyle \left|{\overline{P}}_{+}\cap z\right|=1=\left|{\overline{P}}_{-}\cap z\right|}$⁠.

First consequences of the axioms are

Analogously to Möbius and Laguerre planes we get the connection to the lineargeometry via the residues.

For a Minkowski plane${\displaystyle \mathfrak{M}=(\mathcal{P},\mathcal{Z};{\parallel }_{+},{\parallel }_{-},\in )}$ and${\displaystyle P\in \mathcal{P}}$ we define the local structure$${\displaystyle {\mathfrak{A}}_P:=(\mathcal{P}\setminus \overline{P},\{z\setminus \{\overline{P}\}\mid P\in z\in \mathcal{Z}\}\cup \{E\setminus \overline{P}\mid E\in \mathcal{E}\setminus \{{\overline{P}}_{+},{\overline{P}}_{-}\}\},\in )}$$and call it theresidue at point P.

For the classical Minkowski plane${\displaystyle {\mathfrak{A}}_{(\mathrm{\infty },\mathrm{\infty })}}$ is the real affine plane⁠${\displaystyle {\mathbb{R}}^2}$⁠.

An immediate consequence of axioms C1 to C4 and C1′, C2′ are the following two theorems.

Theminimal model of a Minkowski plane can be established over the set${\displaystyle \overline{K}:=\{0,1,\mathrm{\infty }\}}$ of three elements:$${\displaystyle \mathcal{P}:={\overline{K}}^2}$$

Parallel points:

• ${\displaystyle (x_1,y_1){\parallel }_{+}(x_2,y_2)}$ if and only if${\displaystyle x_1=x_2}$
• ${\displaystyle (x_1,y_1){\parallel }_{-}(x_2,y_2)}$ if and only if⁠${\displaystyle y_1=y_2}$⁠.

Hence${\displaystyle \left|\mathcal{P}\right|=9}$ and⁠${\displaystyle \left|\mathcal{Z}\right|=6}$⁠.

For finite Minkowski-planes we get from C1′, C2′:

This gives rise of thedefinition:
For a finite Minkowski plane${\displaystyle \mathfrak{M}}$ and a cycle${\displaystyle z}$ of${\displaystyle \mathfrak{M}}$ we call the integer${\displaystyle n=\left|z\right|-1}$ theorder of${\displaystyle \mathfrak{M}}$.

Simple combinatorial considerations yield

We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace${\displaystyle \mathbb{R}}$ by an arbitraryfield${\displaystyle K}$ then we getin any case a Minkowski plane⁠${\displaystyle \mathfrak{M}(K)=(\mathcal{P},\mathcal{Z};{\parallel }_{+},{\parallel }_{-},\in )}$⁠.

Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane⁠${\displaystyle \mathfrak{M}(K)}$⁠.

Theorem (Miquel): For the Minkowski plane${\displaystyle \mathfrak{M}(K)}$ the following is true:

If for any 8 pairwise not parallel points${\displaystyle P_1,...,P_8}$ which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples, then the sixth quadruple of points is concyclical, too.

(For a better overview in the figure there are circles drawn instead of hyperbolas.)

Theorem (Chen): Only a Minkowski plane${\displaystyle \mathfrak{M}(K)}$ satisfies the theorem of Miquel.

Because of the last theorem${\displaystyle \mathfrak{M}(K)}$ is called amiquelian Minkowski plane.

Remark: Theminimal model of a Minkowski plane is miquelian.

It is isomorphic to the Minkowski plane${\displaystyle \mathfrak{M}(K)}$ with${\displaystyle K=\mathrm{GF}(2)}$ (field⁠${\displaystyle \{0,1\}}$⁠).

An astonishing result is

Theorem (Heise): Any Minkowski plane ofeven order is miquelian.

Remark: A suitablestereographic projection shows:${\displaystyle \mathfrak{M}(K)}$ is isomorphicto the geometry of the plane sections on a hyperboloid of one sheet (quadric of index 2) in projective 3-space over field${\displaystyle K}$.

Remark: There are a lot of Minkowski planes that arenot miquelian (s. weblink below). But there are no "ovoidal Minkowski" planes, in difference to Möbius and Laguerre planes. Because anyquadratic set of index 2 in projective 3-space is a quadric (seequadratic set).

• Conformal geometry

• Walter Benz (1973)Vorlesungen über Geomerie der Algebren,Springer
• Francis Buekenhout (editor) (1995)Handbook ofIncidence Geometry,ElsevierISBN 0-444-88355-X

• Benz plane in theEncyclopedia of Mathematics
• Lecture NotePlanar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes

• Planes (geometry)

• Articles with short description
• Short description matches Wikidata