Möbius Transformation

A transformation of the form

$$w = f(z) = {az+b\over cz+d},$$

where $a$, $b$, $c$, $d \in \Bbb{C}$ and

$$ad-bc \not = 0,$$

is a Conformal Transformation and is called a Möbius transformation. It is linear in both $w$ and $z$.

Every Möbius transformation except $f(z) = z$ 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 $\alpha$ and $\alpha_S$. Define $\beta \equiv f(\alpha)$, restricting $\beta$ as required. Compute $\beta_S$. $f(\alpha_S)$ then equals $\beta_S$ since the Möbius transformation preserves symmetry (the Symmetry Principle). Plug in $\alpha$ and $\alpha_S$ into the general Möbius transformation and set equal to $\beta$ and $\beta_S$. Without loss of generality, let $c=1$ and solve for $a$ and $b$ in terms of $\beta$. Plug back into the general expression to obtain a Möbius transformation.

See also Symmetry Principle