Möbius Maps on the Riemann Sphere Lesson 8
Every Möbius map is a rigid rotation of the Riemann sphere. Grab the sphere and spin it — the induced transformation plays out on the plane below, 1/z is a 180° flip, and lines and circles trade places because on the sphere they're all just circles. Scalings and translations are the maps that need a non-rigid flow instead — the reason the sphere's rotations are only a subgroup of all Möbius maps.