f(z) = (az+b)/(cz+d) — maps circles and lines to circles and lines
A Möbius transformation f(z) = (az+b)/(cz+d) with ad−bc ≠ 0 is a conformal bijection of the Riemann sphere. Every circle or line maps to a circle or line. These transformations form a group (composition = matrix multiplication of [[a,b],[c,d]]). They generate all hyperbolic geometry (Poincaré disk model uses Möbius maps preserving the unit disk).