The Graphs of Planar Soap Bubbles
David Eppstein
TL;DR
This paper characterizes the graphs formed by two-dimensional soap bubbles as exactly the 3-regular bridgeless planar multigraphs, using geometric and Moebius invariance properties.
Contribution
It provides a complete characterization of soap bubble graphs combining local curvature conditions and Moebius invariance, linking geometry with graph theory.
Findings
Soap bubble graphs are exactly 3-regular bridgeless planar multigraphs.
The characterization remains invariant under Moebius transformations.
A Moebius-invariant power diagram of circles is used in the analysis.
Abstract
We characterize the graphs formed by two-dimensional soap bubbles as being exactly the 3-regular bridgeless planar multigraphs. Our characterization combines a local characterization of soap bubble graphs in terms of the curvatures of arcs meeting at common vertices, a proof that this characterization remains invariant under Moebius transformations, an application of Moebius invariance to prove bridgelessness, and a Moebius-invariant power diagram of circles previously developed by the author for its applications in graph drawing.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Digital Image Processing Techniques