Möbius plane

A Möbius plane, named after August Ferdinand Möbius, in the classical case is an incidence structure, which is essentially the geometry of straight lines and circles in the real level view describes: A straight line, a circle through three points by 2 points is uniquely determined. A straight line or circle intersects / touches a circle at 0, 1, or 2 points. To unlock the special role of the straight line, each one adds Just a common new point added, calling circles and straight lines so extended Cycles. The new incidence structure now has simpler properties:

• (A1): There is exactly one Cycles z, which contains these 3 points each to A, B, C. ( If the points lie on a straight line g, the extended straight to the searched Cycles, they do not lie on a straight line, then there exists a circle through A, B, C.)
• (A2 ): there is exactly one z Cycles ' by P and Q, touching z in P to a Cycles z, a point P on z and a point Q not on z. ( To prove this property, you have the various possibilities for z, P and Q, through play, which without effort is, however, possible (see picture). )

However, it is not expected that the geometry of the extended straight lines and circles described here is the only incidence structure having the characteristics ( A1), ( A2). If we replace the real numbers by rational numbers, so (A1 ), (A2 ) remain valid. However, when using the complex numbers (rather than real) is the validity of the (A1), (A2) is lost. That is only the use of certain number fields (see below) receives the properties ( A1), ( A2).

In addition to the formal inhomogeneous model (there are lines and circles ) is obtained using the inverse of a suitable stereo Graphical projection a homogeneous spatial model: the points of the new incidence structure are the points on the sphere surface and the Cycles are the circles on the sphere. So the classical real Möbius plane can be used as the geometry of the plane sections (circles) are considered on a ball. The proof of (A1 ) and (A2 ) requires no annoying case distinctions in the spatial model.

A Möbius plane is one of the three - Benz levels: Möbius plane Laguerre plane and Minkowski plane. The classical Laguerre plane is the geometry of the parabolas and the classical Minkowski plane, the geometry of the hyperbolas.

The axioms of a Möbius plane

Due to the incidence characteristics ( A1), ( A2) of the classical real Möbius plane defined to:

An incidence structure with the amount of points and the amount of Zykeln called Möbius plane if the following axioms are satisfied:

Four points are called konzyklisch if there is a Cycles containing.

Already mentioned above not only meets the classic real Möbius level, the axioms ( A1), ( A2), ( A3). There are very many examples of Möbius planes that are different from the classical model (see below). Similar to the minimal model of an affine or projective plane, there is also a minimal model of a Möbius plane. It consists of five points:

So is.

The close relationship of the classical Möbius plane to the real affine plane can also be seen between the minimal model of a Möbius plane and the minimal model of an affine plane. This close relationship is even typical of Möbius levels:

For a Möbius plane and we define the incidence structure and call them derivative at point P.

In the classical model, the derivative at the point is the underlying real affine plane (see below). The importance of a derivative at a point is the easy to be proved statement:

• Each derivation of a Möbius plane is an affine plane.

This feature allows the use of many results about affine planes and is also the reason for an alternative definition of a Möbius plane:

Set: An incidence structure is exactly then a Möbius plane if and only if

For finite Möbius planes, that is, Applies (similar affinity levels):

• Cycles two each containing the same number of points.

This property gives rise to the following definition

• For a finite Möbius plane and Cycles natural number is called the order of.

From combinatorial considerations, it follows

• For a finite Möbius plane of order applies:

The classic real Möbius plane

We start from the real affine plane out and get with the square shape, the real Euclidean plane: is the set of points, straight lines are described by equations or and a circle is a set of points satisfying an equation

The geometry of the straight lines and circles may be homogenized (similar to the extension of an affinity level of a projective plane ) by being embedded in the structure having the following incidence

Within the new structure, the incidence extended straight no special role is more geometric and satisfies the axioms (A1) and (A2).

The usual description of the real plane by complex numbers (z now means no Cycles! )

(The complex conjugate of. )

The great advantage of this description is the easy way to automorphisms ( permutations of that map onto Cycles Cycles ) indicated. The following pictures are automorphisms of:

Considering as the projective line over the complex numbers, we see that the images create the group of Möbius transformations. Thus, the geometry is a very homogeneous structure. For example, you can map the real axis with an automorphism on each others Cycles. Together with the illustration ( 4) yields: At each Cycles there is a mirroring, also called inversion. For example: is the inversion at the unit circle. This property causes the common name in English literature inversive plane.

Similar to the spatial model of a projective plane, there is also a spatial model of the classical Möbius plane which cancels the formal difference between circles and extended straight: The geometry is isomorphic to the geometry of circles on a sphere. The corresponding isomorphism gives a suitable Stereographic projection. For example:

Projected from the point of

• The xy-plane on the ball from the equation. This ball has the center and the radius.
• From the circle with the equation in the plane. That the image of the circle is a plane section with the ball and thus again a circle ( on the ball ). The picture is so true circle. The circular planes do not all go through the projection point.
• The straight line in the ab plane. That a straight line is mapped to a point at the reduced spherical circle in a plane through the center of projection.

Miquel Möbius levels

In the search for further examples of a Möbius plane, it is worthwhile to generalize the classical model: We assume an affine plane over a field and a suitable quadratic form on out to define circles. But simply to replace the real numbers by any body and beizuhalten the classic square shape to describe the circuits does not always work. Only for suitable pairs of body and square shape obtained Möbius levels. These are ( as the classical model ) by a high degree of homogeneity (many automorphisms ) and the following sentence awarded by Miquel.

Set ( Miquel ):

The strength of this figure shows closing in the validity of the reversal of the sentence of Miquel:

Set ( CHEN ):

Because of the last sentence is called a Möbius plane miquelsch.

Note: The minimal model of a Möbius plane miquelsch. It is isomorphic to the Möbius plane

Comment:

Comment:

Note: In the classical case can be the set of Miquel with elementary means ( circle square) prove s set of Miquel.

Ovoidale Möbius levels

There are many Möbius planes miquelsch are not (see web link). A large class of Möbius planes containing the miquel between that form ovoidalen Möbius levels. A ovoidale Möbius plane, the geometry of planar slices on a ovoid. An ovoid is a square having the same size and geometrical properties, such as a ball in the real three -dimensional space: 1) A straight line meets an ovoid in 0,1 or 2 points, 2) to cover the amount of the tangent in a point of a plane ( the tangent plane at this point ). In the real three -dimensional space, one can, for example, a hemisphere smooth stick in a suitable manner with a half of an ellipsoid to obtain an ovoid, which is not a quadric. Even in the finite case there is ovoids that are not quadrics (see quadratic set ). For the class of ovoidalen Möbius levels there is a the set of Miquel similar lock set, the tufts set (English: Bundle theorem). He characterizes the ovoidalen Möbius levels. The set of Miquel and Tufts set have a similar meaning as the sets of Pappus and Desargues projective planes for Möbius levels.