Möbius transformation
A Möbius transformation is a bijection on the extended complex plane given by
where and
It can be shown that the inverse, and composition of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition.
The geometric interpretation of the Möbius group is that it is the group of automorphisms of the Riemann sphere.
Any Möbius map can be composed from the elementary transformations - dilations, translations and inversions. If we define a line to be a circle passing through then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.
Title | Möbius transformation |
Canonical name | MobiusTransformation |
Date of creation | 2013-03-22 12:23:19 |
Last modified on | 2013-03-22 12:23:19 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 21 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 30D99 |
Synonym | fractional linear transformation |
Synonym | linear fractional transformation |
Related topic | ProofOfConformalMobiusCircleMapTheorem |
Related topic | AutomorphismsOfUnitDisk |
Related topic | UnitDiskUpperHalfPlaneConformalEquivalenceTheorem |
Related topic | InversionOfPlane |