Epsilon-Unfolding Orthogonal Polyhedra

Abstract

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as ε = 1/2 Ω(n) .

Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

“…We begin with a review the epsilon-unfolding algorithm [6], starting in this section with a high-level overview, and then in Section 3 detailing those aspects of the algorithm that we modify to achieve quadratic refinement.…”

“…Therefore, we follow [6] in describing the algorithm for this simple shape class, before extending the ideas to all orthogonal polyhedra. All modifications needed for delta-unfolding are also present in unfolding orthogonal extrusions, and so we describe them in terms of this simple shape class.…”

“…But for completeness, we briefly summarize the remainder of the epsilon-unfolding algorithm for extrusions, and refer the reader to [6] for additional details. To complete the unfolding of P , ξ is thicken in the +y and −y direction (as viewed in the 3D coordinate system of Figure 2a) so that it completely covers each band.…”

“…The breakthrough was the discovery that arbitrary orthogonal polyhedra (homeomorphic to a sphere) unfold with finite grid refinement [6]. Unfortunately, the amount of grid refinement is exponential in the worst case (though polynomial for "well-balanced" polyhedra).…”

“…We show how to modify the epsilon-unfolding algorithm of [6] to reduce the refinement from worst-case exponential (2 Θ(n) ) to worst-case quadratic (Θ(n 2 )), while still unfolding any orthogonal polyhedron (with n vertices) homeomorphic to a sphere. We call our algorithm the delta-unfolding algorithm, to suggest that the resulting surface strips are still narrow but wider than those produced by epsilon-unfolding.…”