arXiv subjects

John C. Bowers

Publications and source records attributed to John C. Bowers.

6 recordsLinked to original sources

arXiv2026

Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings

Hyperbolic inversive distance circle packings on the $2$-sphere correspond to Koebe polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic $3$-space. Koebe polyhedra are triangulated convex hyperbolic polyhedra with hyperideal vertices whose faces meet $\mathbb{B}^{3}$. We prove the global rigidity of these circle packings or, equivalently, of these Koebe polyhedra under mild assumptions on the links of their vertices. Previous rigidity results apply only when all edges of the Koebe polyhedron are tangent or, alternatively, when no edge is tangent to the ideal boundary of hyperbolic space. We remove these restrictions. This generalizes the global rigidity results of both Bao-Bonahon and Bowers-Bowers-Pratt (arXiv:1703.09338), as well as the uniqueness part of the celebrated Koebe-Andre'ev-Thurston Theorem to the case where adjacent circles need not touch.

math.MG↗

arXiv2020

A proof of the Koebe-Andre'ev-Thurston theorem via flow from tangency packings

Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles. In this paper, we extend the Connelly-Gortler method to allow circles to overlap by angles up to $π/2$. As a result, we obtain a new proof of the general Koebe-Andre'ev-Thurston theorem for disk packings on $\mathbb{S}^2$ with overlaps and a numerical algorithm for computing them. Our development makes use of the correspondence between circles and disks on $\mathbb{S}^2$ and hyperplanes and half-spaces in the 4-dimensional Minkowski spacetime $\mathbb{R}^{1,3}$, which we illuminate in a preliminary section. Using this view we generalize a notion of convexity of circle polyhedra that has recently been used to prove the global rigidity of certain circle packings. Finally, we use this view to show that all convex circle polyhedra are infinitesimally rigid, generalizing a recent related result.

math.MG↗

arXiv2018

Almost all circle polyhedra are rigid

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 Möbius transformations replace rigid Euclidean motions.

math.MG↗

arXiv2017

Rigidity of Circle Polyhedra in the 2-Sphere and of Hyperideal Polyhedra in Hyperbolic 3-Space

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper non-unitary c-polyhedra with Möbius-congruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$.

math.MG↗

arXiv2016

Ma-Schlenker c-Octahedra in the 2-Sphere

We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary constructions use only the inversive geometry of the $2$-sphere.

math.DG↗

arXiv2014

Faster Reductions for Straight Skeletons to Motorcycle Graphs

We give an algorithm that reduces the straight skeleton to the motorcycle graph in $O(n\log n)$ time for simple polygons and $O(n(\log n)\log m)$ time for a planar straight line graph (PSLG) with $m$ connected components. This improves on the previous best of $O(n(\log n)\log r)$ for polygons with $r$ reflex vertices (possibly with holes) and $O(n^2\log n)$ for general planar straight line graphs. This allows us to speed up the straight skeleton algorithm for polygons and PSLGs. For a polygon with $h$ holes and $r$ reflex vertices we achieve a speedup from $O(n(\log n)\log r + r^{4/3+ε})$ time to $O(n(\log n)\log h + r^{4/3 + ε})$ time in the non-degenerate case and from $O(n(\log n)\log r + r^{17/11 + ε})$ to $O(n(\log n)\log h + r^{17/11 + ε})$ in degenerate cases. For a PSLG with $m$ connected components and $r$ reflex vertices, we gain a speed up from $O(n^{1 + ε} + n^{8/11 + ε}r^{9/11+ε})$ to $O(n(\log n)\log m + r^{4/3 + ε})$ in the non-degenerate case and from $O(n^{1 + ε} + n^{8/11 + ε}r^{9/11+ε})$ to $O(n(\log n)\log m + r^{17/11 + ε})$ in the degenerate case.

cs.CG↗