Grid Vertex-Unfolding Orthogonal Polyhedra.

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 vertexunfolding 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.

Main Author: Damian, Mirela.
Other Authors: Flatland, Robin., O'Rourke Joseph.
Language: English
Published: 2008
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