A transformation of the form
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.
See also Symmetry Principle