Circle Packings with Generalized Branching

Attempts to build a discrete theory for rational maps on the sphere via circle packing have foundered on discretization effects in locating branch points. The authors remove this impediment by introducing generalized branch points. A generalized branch point need no longer be attached to an individual circle, but with the help of chaperones and other devices, can be positioned anywhere that the geometry requires. The effects will be illustrated in images and videos as we use generalized branching to fix flaws in discrete Ahlfors and Weierstrasse functions.