Almost all circle polyhedra are rigid

2018/07/24 by Bowers, John C., Bowers, Philip L., Pratt, Kevin
#52C26 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1807.09355

Abstract

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane 𝔼², as well as the infinitesimal inversive rigidity of tangency circle packings on the 2-sphere \mathbbS². From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck's proof in~\citegluck75 of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and Möbius transformations replace rigid Euclidean motions.

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• Rigidity of Polyhedral Surfaces, III
• Rigidity of infinite inversive distance circle packings in the plane
• Infinite rigidity of inversive distance circle packings in the Poincaré disk
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