A proof of the Koebe-Andre'ev-Thurston theorem via flow from tangency packings

Abstract

Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles. In this paper, we extend the Connelly-Gortler method to allow circles to overlap by angles up to $\pi/2$. As a result, we obtain a new proof of the general Koebe-Andre'ev-Thurston theorem for disk packings on $\mathbb{S}^2$ with overlaps and a numerical algorithm for computing them. Our development makes use of the correspondence between circles and disks on $\mathbb{S}^2$ and hyperplanes and half-spaces in the 4-dimensional Minkowski spacetime $\mathbb{R}^{1,3}$, which we illuminate in a preliminary section. Using this view we generalize a notion of convexity of circle polyhedra that has recently been used to prove the global rigidity of certain circle packings. Finally, we use this view to show that all convex circle polyhedra are infinitesimally rigid, generalizing a recent related result.