![]() | ||
In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane.
Contents
Classical real Minkowski plane
Applying the pseudo-euclidean distance
By a transformation of coordinates
The following completion (see Möbius and Laguerre planes) homogenizes the geometry of hyperbolas:
The incidence structure
The set of points consists of
Any line
Two points
We define: Two points
Both these relations are equivalence relations on the set of points.
Two points
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).
The axioms of a Minkowski plane
Let be
Two points
An incidence structure
For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous.
C1′: For any two pointsFirst consequences of the axioms are
Lemma: For a Minkowski plane
Analogously to Möbius and Laguerre planes we get the connection to the linear geometry via the residues.
For a Minkowski plane
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.
Theorem: For a Minkowski plane
Theorem: Let be
Minimal model
The minimal model of a Minkowski plane can be established over the set
Parallel points:
Hence:
Finite Minkowski-planes
For finite Minkowski-planes we get from C1′, C2′:
Lemma: Let be
This gives rise of the definition:
For a finite Minkowski plane
Simple combinatorial considerations yield
Lemma: For a finite Minkowski plane
Miquelian Minkowski planes
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
(For a better overview in the figure there are circles drawn instead of hyperbolas.)
Theorem (Chen): Only a Minkowski plane
Because of the last theorem
Remark: The minimal model of a Minkowski plane is miquelian.
It is isomorphic to the Minkowski planeAn astonishing result is
Theorem (Heise): Any Minkowski plane of even order is miquelian.
Remark: A suitable stereographic projection shows:
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).