Möbius plane

In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single 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.

This drawback can be removed by adding a point at infinity to every line.

In an affine plane the parallel relation between lines is essential.

Two completed lines touch if they have only the point at infinity in common, so they are parallel.

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

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).

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 (similarly to the projective completion of an affine plane) by embedding it into the incidence structure with Then

Within the new structure the completed lines play no special role anymore.

This property gives rise to the alternate name inversive plane.

and maps The incidence behavior of the classical real Möbius plane gives rise to the following definition of an axiomatic Möbius plane.

is called a Möbius plane if the following axioms hold: Four points

One should not expect that the axioms above define the classical real Möbius plane.

, we have (as with affine planes): This justifies the following definition: From combinatorics we get: Looking for further examples of Möbius planes it seems promising to generalize the classical construction starting with a quadratic form

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.

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.

Remark: The minimal model of a Möbius plane is miquelian.

the field of complex numbers, there is no suitable quadratic form at all.

is isomorphic to the geometry of the plane Remark: A proof of Miquel's theorem for the classical (real) case can be found here.

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

These finite block designs satisfy the axioms defining a Möbius plane, when a circle is interpreted as a block of the design.