Minkowski plane

Image:Minkowski-2d3d-model.svg

Applying the pseudo-euclidean distance $d(P\_1,P\_2)\; =\; (x\text{'}\_1-x\text{'}\_2)^2\; -\; (y\text{'}\_1-y\text{'}\_2)^2$ on two points $P\_i\; =\; (x\text{'}\_i,\; y\text{'}\_i)$ (instead of the euclidean distance) we get the geometry of hyperbolas, because a pseudo-euclidean circle $\backslash \{P\backslash in\; \backslash R^2\; \backslash mid\; d(P,M)=r\backslash \}$ is a hyperbola with midpoint {{tmath|1= M }}.

By a transformation of coordinates {{tmath|1= x_i = x'_i + y'_i }}, {{tmath|1= y_i = x'_i - y'_i }}, the pseudo-euclidean distance can be rewritten as {{tmath|1= 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 set of points: $$\backslash mathcal\; P\; :=\; \backslash left(\backslash R\; \backslash cup\; \backslash left\backslash \{\backslash infty\backslash right\backslash \}\backslash right)^2\; =$$

\R^2 \cup \left(\left\{\infty\right\} \times \R\right) \cup \left(\R \times \left\{\infty\right\}\right) \

\cup \left\{\left(\infty,\infty\right)\right\} \ ,

\ \infty \notin \R,

• the set of cycles $$\backslash begin\{align\}$$

\mathcal Z :={} & \left\{\left\{\left(x,y\right) \in \R^2 \mid y = ax + b\right\} \cup \left\{\left(\infty,\infty\right)\right\} \mid a,b \in \R, a\ne 0\right\}\\

& \quad \cup \left\{\left\{\left(x,y\right) \in \R^2 \mid y=\frac{a}{x-b} + c, x \ne b\right\}

\cup \left\{\left(b,\infty\right),\left(\infty,c\right)\right\} \mid a,b,c \in \R, a \ne 0\right\}.

\end{align}

The incidence structure $(\{\backslash mathcal\; P\},\{\backslash mathcal\; Z\},\backslash in)$ is called the classical real Minkowski plane.

The set of points consists of {{tmath|1= \R^2 }}, two copies of $\backslash R$ and the point {{tmath|1= (\infty,\infty) }}.

Any line $y=ax+b\; ,a\backslash ne0$ is completed by point {{tmath|1= (\infty,\infty) }}, any hyperbola $y\; =\; \backslash frac\{a\}\{x-b\}\; +\; c,\; a\backslash ne0$ by the two points $(b,\backslash infty),(\backslash infty,c)$ (see figure).

Two points $(x\_1,y\_1)\backslash ne(x\_2,y\_2)$ can not be connected by a cycle if and only if $x\_1\; =\; x\_2$ or {{tmath|1= y_1 = y_2 }}.

We define:

Two points $P\_1$, $P\_2$ are (+)-parallel ({{tmath|1= P_1 \parallel_+ P_2 }}) if $x\_1\; =\; x\_2$ and (−)-parallel ({{tmath|1= P_1\parallel_- P_2 }}) if {{tmath|1= y_1 = y_2 }}.

Both these relations are equivalence relations on the set of points.

Two points $P\_1,P\_2$ are called parallel ({{tmath|1= P_1\parallel P_2 }}) if

$P\_1\; \backslash parallel\_+\; P\_2$ or {{tmath|1= P_1 \parallel_- P_2 }}.

From the definition above we find:

Lemma:

• For any pair of non parallel points {{tmath|1= A }}, $B$ there is exactly one point $C$ with {{tmath|1= A\parallel_+ C \parallel_- B }}.
• For any point $P$ and any cycle $z$ there are exactly two points $A,B\; \backslash in\; z$ with {{tmath|1= A\parallel_+ P \parallel_- B }}.
• For any three points {{tmath|1= A }}, {{tmath|1= B }}, {{tmath|1= C }}, pairwise non parallel, there is exactly one cycle $z$ that contains {{tmath|1= A,B,C }}.
• For any cycle {{tmath|1= z }}, any point $P\; \backslash in\; z$ and any point $Q,\; P\; \backslash not\backslash parallel\; Q$ and $Q\; \backslash notin\; z$ there exists exactly one cycle $z\text{'}$ such that {{tmath|1= z \cap z' = \{P\} }}, i.e. $z$ touches $z\text{'}$ at point {{tmath|1= 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 $\backslash left(\; \{\backslash mathcal\; P\}\; ,\; \{\backslash mathcal\; Z\}\; ;\; \backslash parallel\_+\; ,\; \backslash parallel\_-\; ,\; \backslash in\; \backslash right)$ be an incidence structure with the set $\backslash mathcal\; P$ of points, the set $\backslash mathcal\; Z$ of cycles and two equivalence relations $\backslash parallel\_+$ ((+)-parallel) and $\backslash parallel\_-$ ((−)-parallel) on set $\backslash mathcal\; P$. For $P\backslash in\; \backslash mathcal\; P$ we define:

$\backslash overline\{P\}\_+\; :=\; \backslash left\backslash \{\; Q\; \backslash in\; \backslash mathcal\; P\; \backslash mid\; Q\backslash parallel\_+\; P\; \backslash right\backslash \}$ and $\backslash overline\{P\}\_-\; :=\; \backslash left\backslash \{\; Q\; \backslash in\; \backslash mathcal\; P\; \backslash mid\; Q\backslash parallel\_-\; P\; \backslash right\backslash \}$.

An equivalence class $\backslash overline\{P\}\_+$ or $\backslash overline\{P\}\_-$ 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 $A\; ,\; B$ are called parallel ($A\; \backslash parallel\; B$) if $A\; \backslash parallel\_+\; B$ or $A\; \backslash parallel\_-\; B$.

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

File:Minkowski-axioms-c1-c2.svg

File:Minkowski-axioms-c3-c4.svg

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

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

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

First consequences of the axioms are

{{math theorem | name = Lemma | math_statement = For a Minkowski plane $\{\backslash mathfrak\; M\}$ the following is true

{{ordered list | list-style-type = lower-alpha

| Any point is contained in at least one cycle.

| Any generator contains at least 3 points.

| Two points can be connected by a cycle if and only if they are non parallel.

}}

}}

Analogously to Möbius and Laguerre planes we get the connection to the linear

geometry via the residues.

For a Minkowski plane $\backslash mathfrak\; M\; =\; (\backslash mathcal\; P,\; \backslash mathcal\; Z;\; \backslash parallel\_+,\; \backslash parallel\_-,\; \backslash in)$ and $P\; \backslash in\; \backslash mathcal\; P$ we define the local structure

$$\backslash mathfrak\; A\_P:=\; (\backslash mathcal\; P\; \backslash setminus\; \backslash overline\{P\},\backslash \{z\; \backslash setminus\backslash \{\backslash overline\{P\}\backslash \}\; \backslash mid\; P\; \backslash in\; z\; \backslash in\; \backslash mathcal\; Z\backslash \}$$

\cup \{E\setminus \overline{P} \mid E \in \mathcal E \setminus \{\overline{P}_+,\overline{P}_-\}\}, \in)

and call it the residue at point P.

For the classical Minkowski plane $\backslash mathfrak\; A\_\{(\backslash infty,\backslash infty)\}$ is the real affine plane {{tmath|1= \R^2 }}.

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

{{math theorem | For a Minkowski plane $\backslash mathfrak\; M\; =\; (\; \backslash mathcal\; P\; ,\; \backslash mathcal\; Z\; ;\; \backslash parallel\_+\; ,\; \backslash parallel\; ,\; \backslash in\; )$ any residue is an affine plane.}}

{{math theorem | Let be $\backslash mathfrak\; M\; =\; (\backslash mathcal\; P,\; \backslash mathcal\; Z;\backslash parallel\_+,\backslash parallel\_-,\backslash in)$ an incidence structure with two equivalence relations $\backslash parallel\_+$ and $\backslash parallel\_-$ on the set $\backslash mathcal\; P$ of points (see above).

Then, $\backslash mathfrak\; M$ is a Minkowski plane if and only if for any point $P$ the residue $\backslash mathfrak\; A\_P$ is an affine plane.}}

File:Minkowski-minimal-model.svg

The minimal model of a Minkowski plane can be established over the set $\backslash overline\{K\}:=\backslash \{0,1,\backslash infty\backslash \}$ of three elements:

$$\backslash mathcal\; \{P\}\; :=\; \backslash overline\{K\}^2$$

$$\backslash begin\{align\}$$

\mathcal Z :\!&= \left\{ \{ (a_1,b_1),(a_2,b_2),(a_3,b_3) \} \mid \{a_1,a_2,a_3\} = \{b_1,b_2,b_3\} = \overline{K} \right\} \\

&= \{

\{ (0,0), (1,1), (\infty,\infty) \} , \; \{ (0,0), (1,\infty), (\infty,1) \}, \\

&\qquad\{ (0,1), (1,0), (\infty,\infty) \}, \; \{ (0,1), (1,\infty), (\infty,0) \}, \\

&\qquad\{ (0,\infty), (1,1), (\infty,0) \}, \; \{ (0,\infty), (1,0), (\infty,1) \} \}

\end{align}

Parallel points:

• $(x\_1,y\_1)\; \backslash parallel\_+\; (x\_2,y\_2)$ if and only if $x\_1\; =\; x\_2$
• $(x\_1,y\_1)\backslash parallel\_-\; (x\_2,y\_2)$ if and only if {{tmath|1= y_1 = y_2 }}.

Hence $\backslash left|\; \backslash mathcal\; P\; \backslash right|\; =\; 9$ and {{tmath|1= \left{{!}} \mathcal Z \right{{!}} = 6 }}.

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

{{math theorem | name = Lemma | math_statement = Let be $\backslash mathfrak\; M\; =(\backslash mathcal\; P,\; \backslash mathcal\; Z;\; \backslash parallel\_+,\; \backslash parallel\_-,\backslash in)$ a finite Minkowski plane, i.e. . For any pair of cycles $z\_1\; ,\; z\_2$ and any pair of generators $e\_1\; ,\; e\_2$ we have:

$\backslash left|\; z\_1\; \backslash right|\; =\; \backslash left|\; z\_2\; \backslash right|\; =\; \backslash left|\; e\_1\; \backslash right|\; =\; \backslash left|\; e\_2\; \backslash right|$.}}

This gives rise of the definition:

For a finite Minkowski plane $\backslash mathfrak\; M$ and a cycle $z$ of $\backslash mathfrak\; M$ we call the integer $n\; =\; \backslash left|\; z\; \backslash right|\; -\; 1$ the order of $\{\backslash mathfrak\; M\}$.

Simple combinatorial considerations yield

{{math theorem | name = Lemma | math_statement = For a finite Minkowski plane $\backslash mathfrak\; M\; =\; (\backslash mathcal\; P\; ,\; \backslash mathcal\; Z\; ;\; \backslash parallel\_+\; ,\; \backslash parallel\_-\; ,\; \backslash in\; )$ the following is true:

{{ordered list | list-style-type = lower-alpha

| Any residue (affine plane) has order {{tmath|1= n }}.

| {{tmath|1= \left{{!}} \mathcal P \right{{!}} = ( n + 1 ) ^2 }},

| {{tmath|1= \left{{!}} \mathcal Z \right{{!}} = ( n + 1) n ( n - 1 ) }}.

}}}}

We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace $\backslash R$ by an arbitrary field $K$ then we get in any case a Minkowski plane {{tmath|1= {\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 {{tmath|1= \mathfrak M (K) }}.

File:Theorem-of-miquel.svg

Theorem (Miquel): For the Minkowski plane $\backslash mathfrak\; M\; (K)$ the following is true:

: If for any 8 pairwise not parallel points $P\_1,...,P\_8$ 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 $\backslash mathfrak\; M\; (K)$ satisfies the theorem of Miquel.

Because of the last theorem $\backslash mathfrak\; M(K)$ is called a miquelian Minkowski plane.

Remark: The minimal model of a Minkowski plane is miquelian.

: It is isomorphic to the Minkowski plane $\backslash mathfrak\; M(K)$ with $K\; =\; \backslash operatorname\{GF\}(2)$ (field {{tmath|1= \{0,1\} }}).

An astonishing result is

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

Remark: A suitable stereographic projection shows: $\backslash mathfrak\; M(K)$ 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).

• Conformal geometry

{{reflist}}

• Walter Benz (1973) Vorlesungen über Geomerie der Algebren, Springer
• Francis Buekenhout (editor) (1995) Handbook of Incidence Geometry, Elsevier {{isbn|0-444-88355-X}}

• [https://encyclopediaofmath.org/wiki/Benz_plane Benz plane] in the Encyclopedia of Mathematics
• [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes]

Category:Planes (geometry)