On Flat Polyhedra deriving from Alexandrov's Theorem

TL;DR

This paper presents an efficient algorithm to identify and reconstruct degenerate flat polyhedra derived from Alexandrov's theorem, enhancing understanding of polyhedral geometry and computational methods.

Contribution

It introduces a straightforward O(n^3) algorithm to determine and reconstruct flat polyhedra from gluing instructions, expanding practical applications of Alexandrov's theorem.

Findings

Algorithm runs in O(n^3) time

Can determine degeneracy of flat polyhedra

Reconstructs flat polyhedra from gluing data

Abstract

We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov's gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n^3) time for polygons whose gluings are specified by n labels.