A Branched Andreev-thurston Theorem for Circle Packings of the Sphere

A Probabilistic Proof of Thurston's Conjecture on Circle Packings

Solving Beltrami Equations by Circle Packing

We use Andreev-Thurston's theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.

Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps

The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric properties. We attempt to illustrate this with a few results about uniformizations of finitely connected planar domains. For example, the following variatio...

Rigidity of Infinite (circle) Packings

The nerve of a packing is a graph that encodes its combinatorics. The vertices of the nerve correspond to the packed sets, and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect. The nerve of a circle packing and other well-behaved packings, on the sphere or in the plane, is a planar graph. It was an observation of Thurston [Thl, Chap...

Planar Lombardi Drawings for Subcubic Graphs

We prove that every planar graph with maximum degree three has a planar drawing in which the edges are drawn as circular arcs that meet at equal angles around every vertex. Our construction is based on the Koebe–Thurston–Andreev circle packing theorem, and uses a novel type of Voronoi diagram for circle packings that is invariant under Möbius transformations, defined using three-dimensional hyp...