Minimizing Visible Edges in Polyhedra

TL;DR

The paper establishes tight bounds on the minimum number of edges visible from points outside or inside a polyhedron, revealing fundamental geometric visibility properties in three-dimensional space.

Contribution

It proves tight bounds on the number of edges visible from any point relative to a polyhedron or polygon arrangement in 3D space, extending understanding of geometric visibility.

Findings

Points outside a polyhedron see at least 8 edges, and this is tight.

Points inside a polyhedron see at least 6 edges, and this is tight.

Interior points see a positive portion of at least 6 edges.

Abstract

We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}^3$, every point in $\mathbb R^3$ that does not see any vertex of $\mathcal P$ must see eight or more edges of $\mathcal P$, and this bound is tight. More generally, this remains true if $\mathcal P$ is any finite arrangement of internally disjoint polygons in $\mathbb{R}^3$. We also prove that every point in $\mathbb{R}^3$ can see six or more edges of $\mathcal{P}$ (possibly only the endpoints of some these edges) and every point in the interior of $\mathcal{P}$ can see a positive portion of at least six edges of $\mathcal{P}$. These bounds are also tight.