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Efficient retrieval of Minkowski sum of rotating polytopes

A subset of the Criticality map of two

identical tetrahedra for two axes rotation

[gaussian-map] [critical-map]
The Gaussian map A subset of the Criticality map

Abstract

We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R3, where one of the polytopes rotates. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph-structure of the Minkowski sum, while one of the polytopes rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis.

Our work targets applications that require exact collision detection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra of exact rotations about an arbitrary axis in R3, and it well balances between preprocessing time and space on the one hand, and query time on the other.

 

Links

  • Naama Mayer, Efi Fogel, and Dan Halperin
    Fast and Robust Retrieval of Minkowski Sums of Rotating Convex Polyhedra in 3-Space
    Computer Aided Design, 43(10): 1258-1269, 2011 [link] [bibtex]
    A preliminary version appeared in Proceedings of the 14th ACM Symposium on Solid and Physical Modeling (SPM), Pages 1-10, Haifa, Israel 2010 [link] [bibtex]
  • Contacts

    Naama Mayer
    contact
    Efi Fogel http://www.cs.tau.ac.il/~efif efif@post.tau.ac.il
    Dan Halperin http://acg.cs.tau.ac.il/danhalperin danha@post.tau.ac.il
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