Pipes, cigars, and kreplach: the union of Minkowski sums in three dimensions
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Motion planning of a ball amid segments in three dimensions
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
The union of congruent cubes in three dimensions
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Polyhedral Voronoi diagrams of polyhedra in three dimensions
Proceedings of the eighteenth annual symposium on Computational geometry
On the Complexity of the Union of Geometric Objects
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
Recent Developments in the Theory of Arrangements of Surfaces
Proceedings of the 19th Conference on Foundations of Software Technology and Theoretical Computer Science
Translating a convex polyhedron over monotone polyhedra
Computational Geometry: Theory and Applications
A New Cell-subdivision Approach to Plan Free Translations in Cluttered Environments
Journal of Intelligent and Robotic Systems
On lines avoiding unit balls in three dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On the union of κ-round objects
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Almost tight bound for a single cell in an arrangement of convex polyhedra in R3
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Pianos are not flat: rigid motion planning in three dimensions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Decompositions and Boundary Coverings of Non-convex Fat Polyhedra
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Decompositions and boundary coverings of non-convex fat polyhedra
Computational Geometry: Theory and Applications
On the union of fat tetrahedra in three dimensions
Journal of the ACM (JACM)
Motion planning via manifold samples
ESA'11 Proceedings of the 19th European conference on Algorithms
A step towards automated design of side actions in injection molding of complex parts
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
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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 $P_i=A_i\oplus (-B)$, for i= 1,...,k. We show that the combinatorial complexity of the free configuration space of B is O(nk log k), and that it can be $\Omega(nk\alpha(k))$ in the worst case, where n is the total complexity of the individual Minkowski sums P1,...,Pk. We also derive an efficient randomized algorithm that constructs this configuration space in expected time O(nk log k log n).