Minkowski plane


In mathematics, a Minkowski plane is one of the Benz planes.

Classical real Minkowski plane

Applying the pseudo-euclidean distance on two points we get the geometry of hyperbolas, because a pseudo-euclidean circle is a hyperbola with midpoint.
By a transformation of coordinates,, the pseudo-euclidean distance can be rewritten as. The hyperbolas then have asymptotes parallel to the non-primed coordinate axes.
The following completion homogenizes the geometry of hyperbolas:
  • the set of points:
  • the set of cycles
The incidence structure is called the classical real Minkowski plane.
The set of points consists of, two copies of and the point.
Any line is completed by point, any hyperbola by the two points .
Two points can not be connected by a cycle if and only if or.
We define:
Two points, are -parallel if and -parallel if.

Both these relations are equivalence relations on the set of points.
Two points are called parallel if
or.
From the definition above we find:
Lemma:
  • For any pair of non parallel points, there is exactly one point with.
  • For any point and any cycle there are exactly two points with.
  • For any three points,,, pairwise non parallel, there is exactly one cycle that contains.
  • For any cycle, any point and any point and there exists exactly one cycle such that, i.e. touches at point.
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.

Axioms of a Minkowski plane

Let be an incidence structure with the set of points, the set of cycles and two equivalence relations and on set. For we define:
and.
An equivalence class or is called -generator and -generator, respectively.

Two points are called parallel if or.
An incidence structure is called Minkowski plane if the following axioms hold:

  • C1: For any pair of non parallel points there is exactly one point with.
  • C2: For any point and any cycle there are exactly two points with.
  • C3: For any three points, pairwise non parallel, there is exactly one cycle which contains.
  • C4: For any cycle, any point and any point and there exists exactly one cycle such that, i.e., touches at point.
  • C5: Any cycle contains at least 3 points. There is at least one cycle and a point not in.
For investigations the following statements on parallel classes are advantageous.
  • C1′: For any two points, we have.
  • C2′: For any point and any cycle we have:.
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.
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:
  • if and only if
  • if and only if.
Hence and.

Finite Minkowski-planes

For finite Minkowski-planes we get from C1′, C2′:
This gives rise of the definition:

For a finite Minkowski plane and a cycle of 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 by an arbitrary field then we get in any case a Minkowski plane.
Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane.
Theorem : For the Minkowski plane the following is true:
Theorem : 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 : 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 in projective 3-space over field.
Remark: There are a lot of Minkowski planes that are not miquelian. 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.