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arXiv · 1807.09355

Almost all circle polyhedra are rigid

John C. Bowers·Philip L. Bowers·Kevin Pratt
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Abstract

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane $\mathbb{E}^{2}$, as well as the infinitesimal inversive rigidity of tangency circle packings on the $2$-sphere $\mathbb{S}^{2}$. From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck's proof in~\cite{gluck75} of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and M\"obius transformations replace rigid Euclidean motions.

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John C. Bowers, Philip L. Bowers, Kevin Pratt. 2018-07-24. Almost all circle polyhedra are rigid. https://arxiv.org/abs/1807.09355

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