## 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 Möbius transformation sends Circles and lines to Circles or lines. Möbius transformations preserve symmetry. The Cross-Ratio is invariant under a Möbius transformation. A Möbius transformation is a composition of translations, rotations, magnifications, and inversions.

To determine a particular Möbius transformation, specify the map of three points which preserve orientation. A particular Möbius transformation is then uniquely determined. To determine a general Möbius transformation, pick two symmetric points and . Define , restricting as required. Compute . then equals since the Möbius transformation preserves symmetry (the Symmetry Principle). 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 the general expression to obtain a Möbius transformation.