Minkowski plane

A Minkowski plane, named after Hermann Minkowski, in the classical case is an incidence structure, which essentially describes the geometry of given by an equation of the form of hyperbolas and straight lines in the real plane view. Points with the same x -or y- coordinates are not connected, hence they are referred ( )- parallel and ( -)- parallel.

Obviously, the following applies: By 3 pairwise non-parallel points is exactly a hyperbola. However: A straight line is uniquely determined already by 2 points. Two hyperbolas can intersect in two points or one point touch ( common tangent ) or avoid. As with Möbius and Laguerre planes obtained simpler geometric relationships when homogenizing the geometry of the hyperbolas / line through addition of other points: One hyperbola Add to that the two points of a straight line to the point added, calling the so extended hyperbola / straight Cycles (see picture). The new incidence structure now has properties similar to a Möbius or Laguerre plane ( see section axioms ) and has (even as Möbius and Laguerre planes) a spatial model: The classical Minkowski plane is isomorphic to the geometry of the plane sections of a single- hyperboloid (see picture) in the real projective space. A single-shell hyperboloid is a quadric, the straights and nondegenerate projective conics contains.

In addition to these geometric models of classical real Minkowski plane, there is the representation over the ring of abnormal - complex numbers ( analogous to the description of the classical Möbius plane over the complex numbers ). An abnormally - complex number has (such as a complex number ) with the shape though.

A Minkowski plane is a 3 - Benz levels: Möbius plane Laguerre plane and Minkowski plane. The classical Möbius plane is the geometry of circles and the classical Laguerre plane of the geometry of parabolas.

The name Minkowski plane is due to the Minkowski metric with which one describes pseudoeuklidische "circles" ( hyperbolas ).

The axioms of a Minkowski plane

It is a structure with the amount of incidence of points, the amount of Cycles and two equivalence relations ( ( )- parallel ) and ( (-)- parallel) to the set of points. For a point, we define: and. An equivalence class is called or ( )- generating or (-)- generating. ( In the spatial model of the classical Minkowski plane is a generatrix of a straight line on the hyperboloid. ) Two points are called parallel () if or applies.

An incidence structure is called Minkowski plane if the following axioms hold:

For investigations of a Minkowski plane, to the following (C1 ) or are (C2 ) equivalent statements of advantage.

Analogous to Möbius and Laguerre planes here are the following local structures affine planes.

For a Minkowski space, and we define

And call this incidence structure derivative at point P.

In classical real Minkowski plane is the real affine plane (see first picture).

A direct consequence of the axioms (C1 ) - ( C4) and (C1 ' ), ( C2') is:

Sentence: For a Minkowski plane each lead is an affine plane.

Hence the alternative definition

Sentence: It is an incidence structure with two equivalence relations and on the set of points.

The minimal model

The minimal model of a Minkowski plane can be defined over the set of 3 elements:

So is the number of points and the Cycles.

For finite Minkowski planes follows from (C1 ' ), ( C2'):

• It is a finite Minkowski plane, that is,. For each pair of Zykeln and each pair of generators is considered.

This gives rise to the following definition: For a finite Minkowski plane and one of Cycles we call the natural number of the order.

Simple combinatorial considerations arise:

• For a finite Minkowski plane applies:

The classic real Minkowski plane

The formal definition of the classical real Minkowski plane specifies the homogenization of the geometry of the hyperbolas described in the introduction:

The incidence structure is called classical real Minkowski plane.

The set of points consists of two copies of and the point. Each line is the point, each hyperbola supplemented by the two points (see Figure 1 ).

Two points can not be connected by a Cycles iff or. We define: Two points are ( )- parallel () if, and ( -)- parallel (), if applicable. Both relations are equivalence relations on the MNGE the points. Two points are called parallel () when or applies.

• The incidence structure defined here satisfies the axioms of a Minkowski plane.

Like the classic Möbius and Laguerre - level, there is a spatial model for the classical real Minkowski plane. However, an affine quadric to describe is not enough:

• The classical Minkowski plane is isomorphic to the geometry of the plane sections of a single- hyperboloid in 3-dimensional real projective space.

Miquel Minkowski planes

The most important non-classical Minkowski planes is obtained by the simple substitution of the real numbers in the classical model by any body. The incidence structure obtained is a Minkowski plane for each body. They are characterized by the appropriate version of the theorem of Miquel Analog to the Möbius and Laguerre planes:

Set ( Miquel ): For a Minkowski plane applies:

The meaning of the sentence by Miquel shows the following set of Chen:

Set ( CHEN ): Only a Minkowski plane satisfies the set of Miquel.

Because in this sentence is a Miquel Minkowski plane.

Note: The minimal model of a Minkowski plane is miquelsch.

An amazing result of

Set ( Heise ): Each Minkowski plane of even order is miquelsch.

Note: A suitable stereographic projection shows is isomorphic to the geometry of the plane sections on a hyperboloid ( quadric of index 2 ) in 3-dimensional projective space over.

Not Miquel Minkowski planes

There are numerous non- Miquel Minkowski planes (see weblink circle geometries ). But: There are no ovoidalen Minkowski planes ( unlike Möbius and Laguerre planes), because a quadratic amount of index 2 in 3-dimensional projective space is already a quadric (see quadratic set ). Many non Miquel examples are obtained by a generalization of the context of the hyperbolas / line with the fractional linear mappings ( projective group PGL (2, K) ). But the blose replacement of hyperbolas in the classical model by the similar curves does not provide Miquel Minkowski planes.