Möbius transformation

A Möbius transformation, sometimes called Möbius mapping or ( broken) called linear function called, in mathematics, a conformal mapping of the Riemann sphere to itself is named after August Ferdinand Möbius.

Discrete groups of Möbius transformations are called Kleinian groups.

The general formula of Möbius transformation is given by

Where are complex numbers that satisfy.

Every Möbius transformation can be continued to a unique isometry of the three-dimensional hyperbolic space.

• 8.1 isometries

Illustration

By expanding the complex plane by a point at infinity, the image under the Riemann number sphere is defined for the value that is mapped to. in turn is mapped to on, otherwise on himself

The inverse map is given by

.

As with true turn is a Möbius transformation.

Common applications for the illustration example in the context of signal processing in the bilinear transformation which produces a reference to the system or between analog, digital and continuous systems, discrete systems.

Elementary types

A Mobius transformation can be obtained by a suitable composition of transformations of three elementary types:

• Shift (translation ): The shift of the vector is described by the illustration.
• Drehstreckung: With the complex number ( with ) describes the mapping a stretching by a factor combined with a rotation through the angle.
• Overthrow ( inversion ) The inversion is described by the illustration. For a grid, the inversion can be illustrated as follows:

The real axis ( including the point infinity) and the imaginary axis (as well ) are thereby mapped to itself. The other vertical and horizontal lines are converted to groups, wherein said straight lines are transformed into smaller circles with increasing distance from the origin.

Since all lines through the " point at infinity " run, all these circles through the origin. Conversely, all circuits are containing the origin transformed to a degree - all other channels are transformed back to circles.

Composition by elemental types

A Möbius transformation can now be means of the representation

Build as follows:

The group of Möbius transformations

The set of all Möbius transformations forms a group: the sequential execution of two Möbius transformations is in fact again a Mobius transformation, as well as the inverse mapping is a Möbius transformations such. This group is a Lie group and is isomorphic to: Every complex 2 × 2 matrix with determinant equal to 0 gives a Möbius transformation, and two such matrices provide exactly the same transformation is then when they are complex multiples of each other. Since is complex four-dimensional and a dimension is divided out, the group of Möbius transformations has over the dimension 3.

• See also Klein group.

Determining a transformation by three points

For three given points z1, z2, z3 on the Riemann sphere and their image points w1, w2, w3 can be a Möbius map f ( z) with f ( zi) = wi for i = 1,2,3, see.

An easy way is first z1, z2, z3 represent 0, 1, ∞ by

And the resultant matrix

And w1, w2, w3 to 0, 1, ∞ through. It arises for the corresponding matrix for f:

Möbius transformation as automorphism of the Riemann number sphere

This kind of transformations is important in the theory of functions, since each bijective conformal mapping of the complex plane ( with infinity) onto itself is a Möbius transformation. Equivalent to the statement that every bijective conformal self-map of the Riemann number sphere is a Möbius transformation.

For this reason, the group of Möbius transformations is precisely the isometry group of the hyperbolic three- space: This has as a boundary at infinity the Riemann number sphere. An isometry of hyperbolic space corresponds uniquely to a conformal bijective self-map of the edge at infinity and vice versa.

The relationship between boundary at infinity and hyperbolic space can be seen most easily in the upper half-space model.

According to the isometries of the hyperbolic plane is obtained as conformal mappings of the compactified real line. These are the real Möbius transformations as above, but with defined. In other words, there are those Möbius transformations which the real line - mapped to itself - and hence the upper half of the complex plane.

Kleinian and Fuchsian groups

Discrete subgroups of is called Kleinian groups, discrete subgroups of Fuchsian groups as.

The limit set of a Kleinian group Γ is a subset of the Riemann number sphere, defined as the average of the edge at infinity with the completion of a train Γx, where x is a point of hyperbolic space, and the definition of the limit set of x is independent of the chosen point.

A Klein ( Fuchsian ) group is called Klein ( Fuchsian ) group first kind if the limit set is all ( or most ). Otherwise, it is a Klein ( Fuchsian ) group 2 Article

The Klein ( Fuchsian ) group 1 type include, in particular the so-called lattice in (respectively), ie discrete subgroups Γ for which there is a fundamental domain finite volume in the three - (or two - ) dimensional hyperbolic space. ( Equivalent: for which the quotient space of three or two-dimensional hyperbolic space by Γ finite volume. )

Transitivitätseigenschaften

A Möbius transformation is uniquely determined by the fact that it defines three pairwise different values ​​of the function for three pairwise distinct complex numbers (or infinite).

The group of Möbius transformations operated sharply triply transitive on the Riemann number sphere.

Geometric properties

In addition to the conformity of Möbius transformations and preserve the cross-ratio, the circular relationship is another geometric invariant, that is, circles on the Riemann number sphere are mapped under these pictures on circles on the sphere; but in general not pointwise. An interesting decision criterion provides a set of function theory: The three different points of the sphere passes exactly one circle. If and only a point P is located on that particular circle, when the double ratio of the four points is real-valued, or assumes a value of infinity. The point P is then and only then one of the three given if the cross ratio is 0, 1 or is infinite.

Isometries

The length of sustaining Moebiustransformations the complex plane given by the elementary isometric displacements (translations ) and rotation, that is, by using and wherein and are complex numbers.

The isometries on the Riemannian ball number can be generated by the π -periodic rotation

And the rotation, again with and complex. Of the fixed points and are, i.e., the number of ball rotates around the axis shared by and. The fixed points of 0 and ∞. By repeated application of all isometrics can be produced on the Riemann sphere. The distance -preserving rotations around the axis given by 1 and -1 are given for example by

The group of isometries has dimension 3 over the field of real numbers. This applies both to the isometries of the plane as well as for the isometries of the Riemannian ball number.