In mathematics, a Minkowski plane (named after
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number ...
) is one of the
Benz planes (the others being
Möbius plane and
Laguerre plane In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre.
The classical Laguerr ...
).
Classical real Minkowski plane
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
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
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 P.
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
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by de ...
(not degenerated quadric of index 2).
The axioms of a Minkowski plane
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:
* 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
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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 (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
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
shows:
is isomorphic
to the geometry of the plane sections on a hyperboloid of one sheet (
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is d ...
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
*
Conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
References
*
Walter Benz
Walter Benz (May 2, 1931 Lahnstein – January 13, 2017 Ratzeburg) was a German mathematician, an expert in geometry.
Benz studied at the Johannes Gutenberg University of Mainz and received his doctoral degree in 1954, with Robert Furch as his ...
(1973) ''Vorlesungen über Geomerie der Algebren'',
Springer
Springer or springers may refer to:
Publishers
* Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag.
** Springer Nature, a multinationa ...
*
Francis Buekenhout (editor) (1995) ''Handbook of
Incidence Geometry
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incidenc ...
'',
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as '' The Lancet'', '' Cell'', the ScienceDirect collection of electronic journals, '' Trends'', t ...
{{isbn, 0-444-88355-X
External links
Benz planein the ''
Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics.
Overview
The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...
''
Lecture Note ''Planar Circle Geometries'', an Introduction to Moebius-, Laguerre- and Minkowski Planes
Planes (geometry)