In mathematics, a Möbius plane (named after
August Ferdinand Möbius
August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer.
Early life and education
Möbius was born in Schulpforta, Electorate of Saxony, and was descended on hi ...
) is one of the
Benz plane
In mathematics, a Benz plane is a type of 2-dimensional geometrical structure, named after the German mathematician Walter Benz. The term was applied to a group of objects that arise from a common axiomatization of certain structures and split int ...
s: Möbius plane,
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 Laguerre ...
and
Minkowski plane. The classical example is based on the geometry of lines and circles in the real
affine plane.
A second name for Möbius plane is inversive plane. It is due to the existence of ''inversions'' in the classical Möbius plane. An inversion is an
involutory mapping which leaves the points of a circle or line fixed (see below).
Relation to affine planes
Affine planes are systems of points and lines that satisfy, amongst others, the property that two points determine exactly one line. This concept can be generalized to systems of points and circles, with each circle being determined by three non-collinear points. However, three
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
points determine a line, not a circle. This drawback can be removed by adding a
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
to every line. If we call both circles and such completed lines ''cycles'', we get an
incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
in which every three points determine exactly one cycle.
In an affine plane the parallel relation between lines is essential. In the geometry of cycles, this relation is generalized to the ''touching'' relation. Two cycles ''touch'' each other if they have just one point in common. This is true for two
tangent circles
In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tange ...
or a line that is
tangent to a circle. Two completed lines touch if they have only the point at infinity in common, so they are parallel. The touching relation has the property
*for any cycle
, point
on
and any point
not on
there is exactly one cycle
containing points
and touching
(at point
).
These properties essentially define an ''axiomatic Möbius plane''. But the classical Möbius plane is not the only geometrical structure that satisfies the properties of an axiomatic Möbius plane. A simple further example of a Möbius plane can be achieved if one replaces the real numbers by
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s. The usage of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (instead of the real numbers) does not lead to a Möbius plane, because in the complex affine plane the curve
is not a circle-like curve, but a hyperbola-like one. Fortunately there are a lot of
fields (numbers) together with suitable
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s that lead to Möbius planes (see below). Such examples are called ''miquelian'', because they fulfill
Miquel's theorem
Miquel's theorem is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circle ...
. All these miquelian Möbius planes can be described by space models. The classical real Möbius plane can be considered as the geometry of circles on the unit sphere. The essential advantage of the space model is that any cycle is just a circle (on the sphere).
Classical real Möbius plane
We start from the real affine plane
with the
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
and get the real
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
:
is the ''point'' set, the ''lines'' are described by equations
or
and a ''circle'' is a set of points that fulfills an equation
:
.
The geometry of lines and circles of the euclidean plane can be homogenized (similar to the projective completion of an affine plane) by embedding it into the incidence structure
:
with
:
, the ''set of points'', and
:
the ''set of cycles''.
:
is called ''classical real Möbius plane''.
Within the new structure the completed lines play no special role anymore. Obviously
has the following properties.
*For any set of three points
there is exactly one cycle
which contains
.
*For any cycle
, any point
and
there exists exactly one cycle
with:
and
, i.e.
and
''touch'' each other at point
.
:
can be described using the
complex numbers.
represents point
:
:
, and
:
::
(
is the conjugate number of
.)
The advantage of this description is, that one checks easily that the following permutations of
map cycles on cycles.
: (1)
with
(rotation + dilatation)
: (2)
with
(translation)
: (3)
(reflection at
)
: (4)
(reflection or inversion through the real axis)
Considering
as
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
over
one recognizes
that the mappings
generate the group
(s.
PGL(2,C),
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
). The geometry
is a homogeneous structure, ''i.e.'', its
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
is
transitive. Hence from (4) we get: For any cycle there exists an
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
. For example:
is the inversion which fixes the unit circle
. This property gives rise to the alternate name inversive plane.
Similar to the space model of a
desarguesian projective plane there exists a
space model for the geometry
which omits the formal difference between cycles defined by lines and cycles defined by circles: The geometry
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the geometry of circles on a sphere. The isomorphism can be performed by 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 ...
. For example:
''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''
(PDF; 891 kB), S. 60.
:
is a projection with center and maps
*the x-y-plane onto the sphere with equation , midpoint and radius .
*the ''circle'' with equation into the plane . That means, the image of a circle is a plane section of the sphere and hence a circle (on the sphere) again. The corresponding planes do ''not contain'' center .
*the ''line'' into the plane . So, the image of a line is a circle (on the sphere) through point but not containing point .
Axioms of a Möbius plane
The incidental behavior of the classical real Möbius plane gives reason to the following definition of an axiomatic
Möbius plane.
An incidence structure with ''point set'' and ''set of cycles''
is called ''Möbius plane'' if the following axioms hold:
: A1: For any three points there is exactly one cycle that contains .
: A2: For any cycle , any point and there exists exactly one cycle with: and ( and ''touch'' each other at point ).
: A3: Any cycle contains at least three points. There is at least one cycle.
Four points are ''concyclic'' if there is a cycle with .
One should not expect that the axioms above define the classical real Möbius plane. There are a lot of examples of axiomatic Möbius planes which are different from the classical one (see below). Similar to the minimal model of an affine plane one find the ''minimal model'' of a Möbius plane. It consists of points:
.
Hence: .
The connection between the classical Möbius plane and the real affine plane can be found in a similar way between the minimal model of a Möbius plane and the minimal model of an affine plane. This strong connection is typical for Möbius planes and affine planes (see below).
For a Möbius plane and we define structure
and call it the ''residue at point P''.
For the classical model the residue at point is the underlying real affine plane. The essential meaning of the residue shows the following theorem.
Theorem:
Any residue of a Möbius plane is an affine plane.
This theorem allows to use the plenty results on affine planes for investigations on Möbius planes and gives rise to an equivalent definition of a Möbius plane:
Theorem:
An incidence structure is a Möbius plane if and only if the following
property is fulfilled
:A': For any point the residue is an affine plane.
For finite Möbius planes, i.e. , we have (similar to affine planes):
* Any two cycles of a Möbius plane have the same number of points.
This gives reason for the following definition:
For a finite Möbius plane and a cycle the integer is called ''order'' of .
From combinatorics we get
*Let be a Möbius plane of order . Then a) any residue is an affine plane of order , b) , c)
Miquelian Möbius planes
Looking for further examples of Möbius planes it seems promising to generalize the classical construction starting with a quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
on an affine plane over a field for defining circles. But, just to replace the real numbers by any field and to keep the classical quadratic form for describing the circles does not work in general. For details one should look into the lecture note below. So, only for ''suitable pairs'' of fields and quadratic forms one gets Möbius planes . They are (as the classical model) characterized by huge homogeneity and the following theorem of Miquel.
Theorem (Miquel):
For the Möbius plane the following is true:
If for any 8 points
which can be assigned to the vertices
of a cube such that the points in 5 faces correspond to concyclical
quadruples than the sixth quadruple of points is concyclical, too.
The converse is true, too.
Theorem (Chen):
Only a Möbius plane satisfies the Theorem of Miquel.
Because of the last Theorem a Möbius plane is called a ''miquelian Möbius plane''.
Remark: The ''minimal model'' of a Möbius plane is miquelian. It is isomorphic to the Möbius plane
:: with (field ) and .
::(For example, the unit circle is the point set .)
Remark: If we choose the field of complex numbers, there is ''no suitable'' quadratic form at all.
::The choice (the field of rational numbers) and is suitable.
::The choice (the field of rational numbers) and is suitable, too.
Remark: A 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 sphere (nondegenerate 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 de ...
of index 1) in projective 3-space over field .
Remark: A proof of Miquel's theorem for the classical (real) case can be found here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a ...
. It is elementary and based on the theorem of an inscribed angle
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an in ...
.
Remark: There are many Möbius planes which are ''not miquelian'' (see weblink below). The class which is most similar to miquelian Möbius planes are the ovoidal Möbius planes. An ovoidal Möbius plane is the geometry of the plane sections of an ovoid
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either o ...
. An ovoid is a quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Definition of a qu ...
and bears the same geometric properties as a sphere in a projective 3-space: 1) a line intersects an ovoid in none, one or two points and 2) at any point of the ovoid the set of the tangent lines form a plane, the ''tangent plane''. A simple ovoid in real 3-space can be constructed by glueing together two suitable halves of different ellipsoids, such that the result is not a quadric. Even in the finite case there exist ovoids (see quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Definition of a qu ...
). Ovoidal Möbius planes are characterized by the bundle theorem In Euclidean geometry, the bundle theorem is a statement about six circles and eight points in the Euclidean plane. In general incidence geometry, it is a similar property that a Möbius plane may or may not satisfy. According to Kahn's Theorem, it ...
.
Finite Möbius planes and block designs
A block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
with the parameters of the one-point extension of a finite affine plane of order , i.e. a --design, is a Möbius plane of order .
These finite block designs satisfy the axioms defining a Möbius plane, when a circle is interpreted as a block of the design.
The only known finite values for the order of a Möbius plane are prime or prime powers. The only known finite Möbius planes are constructed within finite projective geometries.
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 ...
*Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
References
*W. Benz, ''Vorlesungen über Geometrie 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 ...
(1973)
*F. Buekenhout (ed.), ''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 ''inciden ...
'', 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'', ...
(1995)
*P. Dembowski, ''Finite Geometries'', Springer-Verlag (1968)
External links
Möbius plane
in the ''Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structu ...
''
Benz plane
in the ''Encyclopedia of Mathematics''
Lecture Note ''Planar Circle Geometries', an Introduction to Möbius-, Laguerre- and Minkowski Planes
{{DEFAULTSORT:Mobius plane
Classical geometry
Incidence geometry