Almost all circle polyhedra are rigid

TL;DR

This paper proves that nearly all triangulated circle polyhedra and tangency circle packings are infinitesimally inversively rigid, extending classical Euclidean rigidity results to the setting of circle polyhedra using inversive distances and Möbius transformations.

Contribution

It establishes the infinitesimal inversive rigidity of almost all circle polyhedra and packings, adapting Gluck's Euclidean rigidity proof to the circle setting.

Findings

Almost all triangulated circle polyhedra are infinitesimally inversively rigid.

Tangency circle packings on the 2-sphere are also infinitesimally inversively rigid.

The proof adapts classical Euclidean rigidity methods to the circle polyhedra context.

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.