An Attack on Flexibility and Stoker's Problem

TL;DR

This paper explores the geometric realization space of triangulated Euclidean polyhedra in three dimensions, focusing on face and dihedral angles, and analyzes its structure and dimension.

Contribution

It provides a new description of the realization space using face and dihedral angles, and computes its dimension at smooth points.

Findings

Characterization of the realization space in terms of angles

Dimension computation at smooth points

Connection between face angles and polyhedral cones

Abstract

In view of solving problems of geometric realizability of polyhedra with given geometric constraints, we describe the space of geometric realizations of a simply-connected triangulated euclidean polyhedron in $\mathbb{R}^3$ up to similarity in terms of the angles of its faces and the angles between its faces. To do so we describe it as the set of its triangular faces glued together correspondingly and as the set of the polyhedral cones that it defines around its vertices. We recompute its dimension at smooth points modulo a combinatorial lemma.