Grid Vertex-Unfolding Orthogonal Polyhedra

Details Grid Vertex-Unfolding Orthogonal Polyhedra Grid Vertex-Unfolding Orthogonal Polyhedra

2005-09-19

arxiv.org/abs/cs/0509054v2

Computer Science

Computational Geometry

Original: 12 pages, 8 figures, 11 references. Revised: 22 pages, 16 figures, 12 references. New version is a substantial revis

Scientific paper

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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