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Grid Vertex-Unfolding Orthogonal Polyhedra

Mirela Damian·Robin Flatland·Joseph O'Rourke
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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.

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Mirela Damian, Robin Flatland, Joseph O'Rourke. 2006-09-27. Grid Vertex-Unfolding Orthogonal Polyhedra. https://arxiv.org/abs/cs/0509054

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