Recognition of affine-equivalent polyhedra by their natural developments

TL;DR

This paper investigates whether the affine equivalence of convex polyhedra in 3D can be determined solely from their developments, extending classical rigidity results to affine transformations.

Contribution

It introduces a novel approach to identify affine equivalence of convex polyhedra using only their developments, expanding the scope of rigidity theorems.

Findings

Developments can distinguish affine-equivalent polyhedra

Extension of classical rigidity theorems to affine transformations

New criteria for affine equivalence based on developments

Abstract

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two polyhedra are isometric or not by using their developments only. In this article, we study a similar problem about whether it is possible, using only the developments of two convex polyhedra of Euclidean 3-space, to understand that these polyhedra are (or are not) affine-equivalent.