Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Approximating shortest paths on a convex polytope in three dimensions
Journal of the ACM (JACM)
Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus
Proceedings of the sixteenth annual symposium on Computational geometry
Determining the Separation of Preprocessed Polyhedra - A Unified Approach
ICALP '90 Proceedings of the 17th International Colloquium on Automata, Languages and Programming
Optimal Algorithms for the Intersection and the Minimum Distance Problems Between Planar Polygons
IEEE Transactions on Computers
Computer-Aided Design
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Let A and B be two convex polytopes in R3 with m and n facets, respectively. The penetration depth of A and B, denoted as π(A, B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes π(A, B) in O(m3/4+Ɛn3/4+Ɛ + m1+Ɛ + n1+Ɛ) expected time, for any constant Ɛ 0. It also computes a vector t such that ∥t∥ = π(A, B) and int(A + t) ∩ B = θ. We show that if the Minkowski sum B⊕(-A) has K facets, then the expected running time of our algorithm is O(K1/2+Ɛm1/4n1/4+m1+Ɛ+n1+Ɛ), for any Ɛ 0. We also present an approximation algorithm for computing π(A, B). For any δ 0, we can compute, in time O(m+n+(log2(m+n))/δ), a vector t such that ∥t∥ ≤ (1 + δ)π(A, B) and int(A + t) ∩ B = θ. Our result also gives a δ-approximation algorithm for computing the width of A in time O(n + (log2 n)/δ), which is simpler and slightly faster than the recent algorithm by Chan [4].