2005/09/19 by Mirela Damian, Damian, Mirela, Robin Flatland +3
Computer Science · #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #cs.CG #cs.DM
paper · pdf · doi:10.48550/arxiv.cs/0509054
Original: 12 pages, 8 figures, 11 references. Revised: 22 pages, 16 figures, 12 references. New version is a substantial revision superceding the preliminary extended abstract that appeared in Lecture Notes in Computer Science, Volume 3884, Springer, Berlin/Heidelberg, Feb. 2006, pp. 264-276
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(n2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3 x 1 refinement of the vertex grid.