Möbius Transformation
A transformation of the form
where
, 
, 
, 
and 
is a Conformal Transformation and is called a Möbius transformation. It is linear in both
and 
.Every Möbius transformation except 
has one or two Fixed Points. The Möbiustransformation sends Circles and lines to Circles or lines. Möbius transformationspreserve symmetry. The Cross-Ratio is invariant under a Möbius transformation. A Möbius transformation is acomposition of translations, rotations, magnifications, and inversions.
To determine a particular Möbius transformation, specify the map of three points which preserve orientation. Aparticular Möbius transformation is then uniquely determined. To determine a general Möbius transformation, pick twosymmetric points 
and 
. Define 
, restricting 
as required. Compute
. 
then equals since the Möbius transformation preserves symmetry (the SymmetryPrinciple). Plug in and into the general Möbius transformation and set equal to and. Without loss of generality, let 
and solve for and in terms of . Plug back into thegeneral expression to obtain a Möbius transformation.
- Church-Turing Thesis
- Chu Space
- Chu-Vandermonde Identity
- Chvátal's Art Gallery Theorem
- Chvátal's Theorem
- ci
- Ci
- Cigarettes
- Cin
- Circle
- Circle Caustic
- Circle-Circle Intersection
- Circle Cutting
- Circle Evolute
- Circle Inscribing
- Circle Involute
- Circle Involute Pedal Curve
- Circle Lattice Points
- Circle Lattice Theorem
- Circle Map
- Circle Method
- Circle Negative Pedal Curve
- Circle Notation
- Circle Order
- Circle Orthotomic