In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane).

Classical real Minkowski plane

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classical Minkowski plane: 2d/3d-model

Applying the pseudo-euclidean distance   on two points   (instead of the euclidean distance) 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 (see Möbius and Laguerre planes) 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   (see figure).

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 (not degenerated quadric of index 2).

Axioms of a Minkowski plane

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Let   be an incidence structure with the set   of points, the set   of cycles and two equivalence relations   ((+)-parallel) and   ((−)-parallel) on set  . For   we define:   and  . An equivalence class   or   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   are called parallel ( ) if   or  .

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

 
Minkowski-axioms-c1-c2
 
Minkowski-axioms-c3-c4
  • 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 (equivalent to C1, C2 respectively) are advantageous.

  • C1′: For any two points  ,   we have  .
  • C2′: For any point   and any cycle   we have:  .

First consequences of the axioms are

Lemma — For a Minkowski plane   the following is true

  1. Any point is contained in at least one cycle.
  2. Any generator contains at least 3 points.
  3. 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   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.

Theorem —  For a Minkowski plane   any residue is an affine plane.

Theorem —  Let be   an incidence structure with two equivalence relations   and   on the set   of points (see above).

Then,   is a Minkowski plane if and only if for any point   the residue   is an affine plane.

Minimal model

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Minkowski plane: 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

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

Lemma — Let be   a finite Minkowski plane, i.e.  . For any pair of cycles   and any pair of generators   we have:  .

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

Lemma — For a finite Minkowski plane   the following is true:

  1. Any residue (affine plane) has order  .
  2.  ,
  3.  .

Miquelian Minkowski planes

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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 of Miquel

Theorem (Miquel): For the Minkowski plane   the following is true:

If for any 8 pairwise not parallel points   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   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.

It is isomorphic to the Minkowski plane   with   (field  ).

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). 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).

See also

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References

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