Grid Vertex-Unfolding Orthogonal Polyhedra

TL;DR

This paper proves that all genus-zero orthogonal polyhedra can be unfolded into a flat net using grid vertex-unfolding, and provides an efficient algorithm for doing so.

Contribution

It introduces the first algorithm for grid vertex-unfolding of genus-zero orthogonal polyhedra, expanding unfolding techniques beyond previous limitations.

Findings

All genus-zero orthogonal polyhedra have a grid vertex-unfolding.

An O(n^2) time algorithm for vertex-unfolding is provided.

A simpler vertex-unfolding algorithm with a 3x1 grid refinement is presented.

Abstract

An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a *net*, a connected planar piece with no overlaps. A *grid unfolding* allows additional cuts along grid edges induced by coordinate planes passing through every vertex. A vertex-unfolding permits faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedron of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of "gridding" of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertex-unfolds P in O(n^2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3 x 1 refinement of the vertex grid.