On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
An optimal algorithm for intersecting three-dimensional convex polyhedra
SIAM Journal on Computing
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Combinatorial complexity of translating a box in polyhedral 3-space
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Computing a face in an arrangement of line segments and related problems
SIAM Journal on Computing
Piecewise linear paths among convex obstacles
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
A new technique for analyzing substructures in arrangements
Proceedings of the eleventh annual symposium on Computational geometry
Proceedings of the twelfth annual symposium on Computational geometry
Voronoi diagrams of lines in 3-space under polyhedral convex distance functions
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
A Simple Algorithm for Complete Motion Planning of Translating Polyhedral Robots
International Journal of Robotics Research
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Let B be a convex polyhedron translating in 3-space amidst k convex polyhedral obstacles A1,…,Ak with pairwise disjoint interiors. The free configuration space (space of all collision-free placements) of B can be represented as the complement of the union of the Minkowski sums Pi=Ai ⊕ (−B), for i=1,…,k. We show that the combinatorial complexity of the free configuration space of B is O(nklog2k), where n is the total complexity of the individual Minkowski sums P1,…,Pk. The bound is almost tight in the worst case. We also derive an efficient randomized algorithm that constructs this configuration space in expected time O(nklog3k).