Abstract
We use three-dimensional hyperbolic geometry to define a form of power diagram for systems of circles in the plane that is invariant under Mobius transformations. By applying this construction to circle packings derived from the Koebe---Andreev---Thurston circle packing theorem, we show that every planar graph of maximum degree three has a planar Lombardi drawing (a drawing in which the edges are drawn as circular arcs, meeting at equal angles at each vertex). We use circle packing to construct planar Lombardi drawings of a special class of 4-regular planar graphs, the medial graphs of polyhedral graphs, and we show that not every 4-regular planar graph has a planar Lombardi drawing. We also use these power diagrams to characterize the graphs formed by two-dimensional soap bubble clusters (in equilibrium configurations) as being exactly the 3-regular bridgeless planar multigraphs, and we show that soap bubble clusters in stable equilibria must in addition be 3-connected.
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