Möbius Transformations

Möbius transformations (also called linear fractional transformations) are maps from the complex plane to itself of the form
$f(z) = \frac{a z + b}{c z + d}$
where $$a, b, c, d \in \mathbf{C}$$ and $$a d – b c \neq 0$$. This definition extends to functions on the Riemann sphere, where the geometry of such transformations becomes more apparent. The following short video shows off a remarkable correspondence between symmetries of the sphere and Möbius transformations.