TY - GEN

T1 - Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space

AU - Mayer, Naama

AU - Fogel, Efi

AU - Halperin, Dan

PY - 2010

Y1 - 2010

N2 - We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in ℝ3, where one of the polytopes frequently 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 collisiondetection 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. We use Cgal arrangements and in particular the support for spherical Gaussian-maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments to verify the correctness of the algorithm and its implementation, and to compare its efficiency with an alternative (static) exact method. The results are reported.

AB - We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in ℝ3, where one of the polytopes frequently 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 collisiondetection 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. We use Cgal arrangements and in particular the support for spherical Gaussian-maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments to verify the correctness of the algorithm and its implementation, and to compare its efficiency with an alternative (static) exact method. The results are reported.

UR - http://www.scopus.com/inward/record.url?scp=77958037078&partnerID=8YFLogxK

U2 - 10.1145/1839778.1839780

DO - 10.1145/1839778.1839780

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AN - SCOPUS:77958037078

SN - 9781605589848

T3 - Proceedings - 14th ACM Symposium on Solid and Physical Modeling, SPM'10

SP - 1

EP - 10

BT - Proceedings - 14th ACM Symposium on Solid and Physical Modeling, SPM'10

T2 - 14th ACM Symposium on Solid and Physical Modeling, SPM'10

Y2 - 1 September 2010 through 3 September 2010

ER -