Composition of Möbius transformations

Every Möbius transformation is the composition of mappings of the type:

\[ \begin{aligned} z &\mapsto az \qquad \text{(rotation & dilation)} \\ z &\mapsto z + b \quad \text{(translation)} \\ z &\mapsto \frac{1}{z} \qquad \text{(inversion)} \\ \end{aligned} \]

which all map circles and lines to circles or lines.

More at Analysis of a Complex Kind.