On recognizing shapes of polytopes from their shadows

TL;DR

This paper proves that convex polytopes can be uniquely identified by their shadow shapes from any point on a surrounding sphere, extending shape recognition principles.

Contribution

It establishes that congruent sight cones from all points on a sphere uniquely determine a convex polytope, with an analogous result for spherical projections.

Findings

Polytopes are uniquely determined by their sight cones from the sphere.

The result applies to both Euclidean and spherical projections.

Provides a new method for shape recognition of convex polytopes.

Abstract

Let $P$ and $Q$ be two convex polytopes both contained in the interior of an Euclidean ball $r\textbf{B}^{d}$. We prove that $P=Q$ provided that their sight cones from any point on the sphere $rS^{d-1}$ are congruent. We also prove an analogous result for spherical projections.