Möbius transformations

A light at the top of the sphere throws its grid onto the table. Spin, raise, slide or tip the sphere, and each time the grid on the table changes by a Möbius transformation T(z) = (az + b)/(cz + d). Press Reset and move one control at a time: spinning about the vertical is a rotation, elliptic, fixed at the origin; raising the sphere is an enlargement, hyperbolic; sliding it is a translation, parabolic, with no fixed point on the table; tipping it is elliptic too, fixing ±1. The value of (a + d)² in the corner decides the type: between 0 and 4 elliptic, exactly 4 parabolic, above 4 hyperbolic, anything else loxodromic. Spin and raise together and it is loxodromic: (a + d)² usually picks up an imaginary part, and the grid turns as it grows.