Polyhedra inscribed in a quadric

TL;DR

This paper characterizes convex polyhedra inscribed in quadric surfaces (sphere, hyperboloid, cylinder) using geometric and combinatorial methods, revealing conditions for their 1-skeletons and unifying their study through various geometries.

Contribution

It provides a complete characterization of polyhedra inscribed in hyperboloids and cylinders, extending Rivin's sphere results via a unified geometric framework.

Findings

Polyhedra inscribed in hyperboloids or cylinders correspond to those inscribed in spheres with Hamiltonian cycles.

Parameterization of convex ideal polyhedra in anti-de Sitter and half-pipe geometries.

Unified geometric approach links the three quadric surfaces through degeneration and limits.

Abstract

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $\Gamma$ is realized as the $1$-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if $\Gamma$ is realized as the $1$-skeleton of a polyhedron inscribed in the sphere and $\Gamma$ admits a Hamiltonian cycle. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout.