Planar pseudo-triangulations, spherical pseudo-tilings and hyperbolic virtual polytopes

TL;DR

This paper explores the connection between pseudo-triangulations and hyperbolic virtual polytopes, introducing spherical pseudo-tilings to facilitate the construction of hyperbolic polytopes and advancing graph embedding techniques.

Contribution

It introduces a novel approach linking pseudo-triangulations with hyperbolic virtual polytopes through spherical pseudo-tilings, and presents a theorem on spherically embedded Laman-plus-one graphs.

Findings

Avoidance of non-pointed vertices using pseudo-di-gons

Reduction of hyperbolic polytope construction to spherical graph embedding

Announcement of a theorem on spherically embedded Laman-plus-one graphs

Abstract

We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for convex bodies (A. D. Alexandrov's problem): suppose that a constant $C$ separates (non-strictly) everywhere the principal curvature radii of a smooth 3-dimensional convex body $K$. Then $K$ is necessarily a ball of radius $C$. The two key ideas are: Passing from planar pseudo-triangulations to spherical pseudo-tilings, we avoid non-poited vertices. Instead, we use pseudo-di-gons. A theorem on spherically embedded Laman-plus-one graphs is announced. The difficult problem of hyperbolic polytopes constructing can be reduced to finding spherically embedded graphs.