On translational motion planning of a convex polyhedron in 3-space

Boris Aronov*, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

47 Scopus citations


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(nk log k), and that it can be Ω(nkα(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).

Original languageEnglish
Pages (from-to)1785-1803
Number of pages19
JournalSIAM Journal on Computing
Issue number6
StatePublished - Dec 1997


  • Algorithmic motion planning
  • Combinatorial complexity
  • Combinatorial geometry
  • Computational geometry
  • Convex polyhedra
  • Geometric algorithms
  • Randomized algorithms


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