In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry.
An inversion of the Möbius plane with respect to any circle is an involution which fixes the points on the circle and exchanges the points in the interior and exterior, the center of the circle exchanged with the point at infinity. In inversive geometry a straight line is considered to be a generalized circle containing the point at infinity; inversion of the plane with respect to a line is a Euclidean reflection.
More generally, a Möbius plane is an incidence structure with the same incidence relationships as the classical Möbius plane. It is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane.
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 points determine a line, not a circle. This drawback can be removed by adding a point at infinity to every line. If we call both circles and such completed lines cycles, we get an incidence structure 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 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
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 numbers. The usage of complex numbers (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 forms that lead to Möbius planes (see below). Such examples are called miquelian, because they fulfill Miquel's theorem. 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).
We start from the real affine plane with the quadratic form and get the real Euclidean plane: 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
Within the new structure the completed lines play no special role anymore. Obviously has the following properties.
complex numbers. represents point :
( is the conjugate number of .)
The advantage of this description is, that one checks easily that the following permutations of map cycles on cycles.
Considering as projective line over one recognizes that the mappings generate the group (s. PGL(2,C), Möbius transformation). The geometry is a homogeneous structure, i.e., its automorphism group is transitive. Hence from (4) we get: For any cycle there exists an inversion. 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 to the geometry of circles on a sphere. The isomorphism can be performed by a suitable stereographic projection. For example: ^{ [1]}
is a projection with center and maps
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:
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
For finite Möbius planes, i.e. , we have (similar to affine planes):
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
Looking for further examples of Möbius planes it seems promising to generalize the classical construction starting with a quadratic form 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
Remark: If we choose the field of complex numbers, there is no suitable quadratic form at all.
Remark: A stereographic projection shows: is isomorphic to the geometry of the plane
Remark: A proof of Miquel's theorem for the classical (real) case can be found here. It is elementary and based on the theorem of an inscribed angle.
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 ovoid is a quadratic set 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). Ovoidal Möbius planes are characterized by the bundle theorem.
A block design 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.