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 th...
| Main Authors: | , , |
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| Format: | |
| Language: | English |
| Published: |
2008
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| Online Access: | http://ezproxy.villanova.edu/login?url=https://digital.library.villanova.edu/Item/vudl:175677 |
| Summary: | 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. |
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