562 research outputs found
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 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(n^2) time.
Enroute to explaining this algorithm, we present a simpler vertex-unfolding
algorithm that requires a 3 x 1 refinement of the vertex grid.Comment: 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. 264-27
A New Algorithm in Geometry of Numbers
A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the
only ellipsoid circumscribed about P. We present a new algorithm for finding
perfect Delaunay polytopes. Our method overcomes the major shortcomings of the
previously used method. We have implemented and used our algorithm for finding
perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new
conjecture that sheds light on the structure of lattice Delaunay tilings.Comment: 7 pages, 3 figures; Proceedings of ISVD-07, International Symposium
on Voronoi diagrams in Science and Engineering held in July of 2007 in Wales,
U
Vertex-Unfoldings of Simplicial Polyhedra
We present two algorithms for unfolding the surface of any polyhedron, all of
whose faces are triangles, to a nonoverlapping, connected planar layout. The
surface is cut only along polyhedron edges. The layout is connected, but it may
have a disconnected interior: the triangles are connected at vertices, but not
necessarily joined along edges.Comment: 10 pages; 7 figures; 8 reference
On the dominant of the `s`-`t`-cut polytope
The natural linear programming formulation of the maximum s-t-flow problem in path-variables has a dual linear program whose underlying polyhedron is the dominant of the s-t-cut polytope. We present a complete characterization of this polyhedron with respect to vertices, facets, and adjacency
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