Polyhedral line transversals in space
Discrete & Computational Geometry - ACM Symposium on Computational Geometry, Waterloo
Visibility problems for polyhedral terrains
Journal of Symbolic Computation
Arrangements of lines in 3-space: a data structure with applications
SCG '88 Proceedings of the fourth annual symposium on Computational 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
The maximum number of ways to stab n convex nonintersecting sets in the plane is 2n - 2
Discrete & Computational Geometry
Discrete & Computational Geometry
The different ways of stabbing disjoint convex sets
Discrete & Computational Geometry
Lower bounds on stabbing lines in 3-space
Computational Geometry: Theory and Applications
Finding stabbing lines in 3-space
Discrete & Computational Geometry
On stabbing lines for convex polyhedra in 3D
Computational Geometry: Theory and Applications
Ray Shooting Amidst Convex Polyhedra and PolyhedralTerrains in Three Dimensions
SIAM Journal on Computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
The Union of Convex Polyhedra in Three Dimensions
SIAM Journal on Computing
Helly-type theorems and geometric transversals
Handbook of discrete and computational geometry
Ray shooting and lines in space
Handbook of discrete and computational geometry
Geometric Permutations Induced by Line Transversals through a Fixed Point
Discrete & Computational Geometry
Transversals to Line Segments in Three-Dimensional Space
Discrete & Computational Geometry
On incremental rendering of silhouette maps of a polyhedral scene
Computational Geometry: Theory and Applications
Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra
SIAM Journal on Computing
Linear data structures for fast ray-shooting amidst convex polyhedra
ESA'07 Proceedings of the 15th annual European conference on Algorithms
On the complexity of sets of free lines and line segments among balls in three dimensions
Proceedings of the twenty-sixth annual symposium on Computational geometry
Lines avoiding balls in three dimensions revisited
Proceedings of the twenty-sixth annual symposium on Computational geometry
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
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We establish a bound of O(n2k1+ε), for any ε 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R3 with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus, when k ≪ n, the new bounds on the complexity (and construction cost) of T improve upon the previously best known bounds, which are nearly cubic in n. To obtain the above result, we study the set Tl0 of line transversals which emanate from a fixed line l0, establish an almost tight bound of O(nk1+ε) on the complexity of Tl0, and provide a randomized algorithm which computes Tl0 in comparable expected time. Slightly improved combinatorial bounds for the complexity of Tl0, and comparable improvements in the cost of constructing this set, are established for two special cases, both assuming that the polyhedra of P are pairwise disjoint: the case where l0 is disjoint from the polyhedra of P, and the case where the polyhedra of P are unbounded in a direction parallel to l0.