Minkowski plane

Applying the pseudo-euclidean distance

(instead of the euclidean distance) we get the geometry of hyperbolas, because a pseudo-euclidean circle

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.

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).

be an incidence structure with the set

(For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.)

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 Lemma — For a Minkowski plane

the following is true Analogously to Möbius and Laguerre planes we get the connection to the linear geometry via the residues.

and call it the residue at point P. For the classical Minkowski plane

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

an incidence structure with two equivalence relations

The minimal model of a Minkowski plane can be established over the set

This gives rise of the definition: For a finite Minkowski plane

Simple combinatorial considerations yield Lemma — For a finite Minkowski plane

the following is true: We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace

Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane ⁠

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

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

Remark: There are a lot of Minkowski planes that are not miquelian (s. weblink below).

Because any quadratic set of index 2 in projective 3-space is a quadric (see quadratic set).