Optimal binary space partitions for orthogonal objects
Journal of Algorithms
Vertex-unfoldings of simplicial manifolds
Proceedings of the eighteenth annual symposium on Computational geometry
Grid vertex-unfolding orthostacks
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
Common unfoldings of polyominoes and polycubes
CGGA'10 Proceedings of the 9th international conference on Computational Geometry, Graphs and Applications
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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 polyhedra 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(n2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3 × 1 refinement of the vertex grid.