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...
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2008
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Grid Vertex-Unfolding Orthogonal Polyhedra. Damian, Mirela. Flatland, Robin. O'Rourke Joseph. 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. 2008 Villanova Faculty Authorship vudl:175677 Discrete and Computational Geometry 39(1-3), March 2008, 213-238. en |
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Grid Vertex-Unfolding Orthogonal Polyhedra. |
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Damian, Mirela. Flatland, Robin. O'Rourke Joseph. |
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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 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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2008 |
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Villanova Faculty Authorship |
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vudl:175677 |
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Discrete and Computational Geometry 39(1-3), March 2008, 213-238. |
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en |
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Damian, Mirela. Flatland, Robin. O'Rourke Joseph. |
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Damian, Mirela. Flatland, Robin. O'Rourke Joseph. Grid Vertex-Unfolding Orthogonal Polyhedra. |
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Damian, Mirela. Flatland, Robin. O'Rourke Joseph. |
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Discrete and Computational Geometry 39(1-3), March 2008, 213-238. |
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Villanova Faculty Authorship |
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Damian, Mirela. |
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2008 |
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Grid Vertex-Unfolding Orthogonal Polyhedra. |
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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 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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Grid Vertex-Unfolding Orthogonal Polyhedra. |
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Grid Vertex-Unfolding Orthogonal Polyhedra. |
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Grid Vertex-Unfolding Orthogonal Polyhedra. |
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Grid Vertex-Unfolding Orthogonal Polyhedra. |
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Grid Vertex-Unfolding Orthogonal Polyhedra. |
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grid vertex-unfolding orthogonal polyhedra. |
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2008 |
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