503 research outputs found

    Overlap of convex polytopes under rigid motion

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    We present an algorithm to compute a rigid motion that approximately maximizes the volume of the intersection of two convex polytopes P-1 and P-2 in R-3. For all epsilon is an element of (0, 1/2] and for all n >= 1/epsilon, our algorithm runs in O(epsilon(-3) n log(3.5) n) time with probability 1 - n(-O(1)). The volume of the intersection guaranteed by the output rigid motion is a (1 - epsilon)-approximation of the optimum, provided that the optimum is at least lambda . max{vertical bar P-1 vertical bar . vertical bar P-2 vertical bar} for some given constant lambda is an element of (0, 1]. (C) 2013 Elsevier B.V. All rights reserved.X1155Ysciescopu

    Approximating the Maximum Overlap of Polygons under Translation

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    Let PP and QQ be two simple polygons in the plane of total complexity nn, each of which can be decomposed into at most kk convex parts. We present an (1ε)(1-\varepsilon)-approximation algorithm, for finding the translation of QQ, which maximizes its area of overlap with PP. Our algorithm runs in O(cn)O(c n) time, where cc is a constant that depends only on kk and ε\varepsilon. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

    A method for dense packing discovery

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    The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density ϕ=128/2190.5845\phi=128/219\approx0.5845 and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure

    Probabilistic Matching of Planar Regions

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    We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes AA and BB, the algorithm computes a transformation tt such that with high probability the area of overlap of t(A)t(A) and BB is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices

    Maximum Overlap Area of a Convex Polyhedron and a Convex Polygon Under Translation

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