Storing the subdivision of a polyhedral surface
SCG '86 Proceedings of the second annual symposium on Computational geometry
On shortest paths in polyhedral spaces
SIAM Journal on Computing
On shortest paths amidst convex polyhedra
SIAM Journal on Computing
SCG '87 Proceedings of the third annual symposium on Computational geometry
Convex hulls of finite sets of points in two and three dimensions
Communications of the ACM
A Bibliography on Digital and Computational Convexity (1961-1988)
IEEE Transactions on Pattern Analysis and Machine Intelligence
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
A single-exponential upper bound for finding shortest paths in three dimensions
Journal of the ACM (JACM)
Approximating shortest paths on a convex polytope in three dimensions
Proceedings of the twelfth annual symposium on Computational geometry
Approximating shortest paths on a convex polytope in three dimensions
Journal of the ACM (JACM)
Parallel implementation of geometric shortest path algorithms
Parallel Computing - Special issue: High performance computing with geographical data
Automatic generation of computeranimation: using AI for movie animation
Automatic generation of computeranimation: using AI for movie animation
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The problem of computing the Euclidean shortest path between two points in three-dimensional space bounded by a collection of convex and disjoint polyhedral obstacles having n faces altogether is considered. This problem is known to be NP-hard and in exponential time for arbitrarily many obstacles; it can be solved in O(n2log n) time for a single convex polyhedral obstacle and in polynomial time for any fixed number of convex obstacles. In this paper Mount's technique is extended to the case of two convex polyhedral obstacles and an algorithm that solves this problem in time O(n3 · 2O{&agr;(n)4}log n) (where &agr;(n) is the functional inverse of Ackermann's function, and is thus extremely slowly growing) is presented, thus improving significantly Sharir's previous results for this special case. This result is achieved by constructing a new kind of Voronoi diagram, called peeper's Voronoi diagram, which is introduced and analyzed in this paper, and which may be of interest in its own right.