Circle Packings with Generalized Branching

TL;DR

This paper introduces generalized branch points in circle packing theory, allowing for more flexible placement of branch points, which improves the discrete modeling of rational maps on the sphere.

Contribution

It presents a novel approach to discretizing rational maps by using generalized branch points, overcoming previous limitations in circle packing methods.

Findings

Generalized branch points can be positioned flexibly within circle packings.

The method improves the accuracy of discrete Ahlfors and Weierstrasse functions.

Visual demonstrations show the effectiveness of the new approach.

Abstract

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.