Finding stabbing lines in 3-space

Discrete & Computational Geometry, Aug 1992

A line intersecting all polyhedra in a setℬ is called a “stabber” for the setℬ. This paper addresses some combinatorial and algorithmic questions about the setℒ(ℬ) of all lines stabbingℬ. We prove that the combinatorial complexity ofℒ(ℬ) has an\(O(n^3 2^{c\sqrt {\log n} } )\) upper bound, wheren is the total number of facets inℬ, andc is a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one.

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Finding stabbing lines in 3-space

Discrete Comput Geom Finding Stabbing Lines in 1 -Space 1 M. Pellegrini I 1 P. W. Shor 0 1 0 AT&T Bell Laboratories , 2D-149, 600 Mountain Avenue, Murray Hill, NJ 07974 , USA 1 Courant Institute, New York University , 251 Mercer Street, New York, NY 10012 , USA A line intersecting all polyhedra in a set ~ is called a "stabber" for the set ~. This paper addresses some combinatorial and algorithmic questions about the set 6a(~) of all lines stabbing ~. We prove that the combinatorial complexity of ~ ( ~ ) has a n 0(n32c ' f ~ ) upper bound, where n is the total number of facets in ~, and c is a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one. * The research of M. Pellegrini was partially supported by Eni and Enidata within the AXL project, and by NSF Grant CCR-8901484. A preliminary version appeared in the Proceedings of the Second ACM-SIAM Symposium on Discrete Al#orithms, pp. 24-31. Two lines are in the same isotopy class if it is possible to move one into the other without crossing edges of ,~. 1. Introduction The first algorithm for finding line stabbers for a set ~ of polyhedra in R 3 with total complexity n, due to Avis and Wenger [ A W l ] , [AW2], has an O(n4 log n)time bound. M c K e n n a a n d O ' R o u r k e I-MO] improve the time complexity to O(n*~(n)), where ct(n) is a functional inverse of the Ackerman function. The algorithm in [ M O ] finds all the isotopy classes 1 of lines generated by the polyhedra in ~ and it is within an 0t(n) factor from the optimal for that problem. The set ba(~) of stabbing lines coincides with the union of some of the isotopy classes and it was conjectured that the complexity of 6e(~) could be less than the complexity of all the isotopy classes. Jaromczyk and K o w a l u k [ J K ] claimed an 0(n32 ~(")) upper bound to the complexity of the set of stabbing lines, but unfortunately there are cases in which the analysis used in [ J K ] is incorrect (see [P2]). A line is characterized by four parameters, therefore a natural representation for a line in R 3 is a point m R 4. The locus of lines intersecting a given line is a quadric (hyperbolic) surface in R4. Given a set of polyhedra ~ , consider the lines spanning edges of the polyhedra, and for each such line the corresponding surface in R4. We obtain an arrangement d ( ~ ) of surfaces in R 4 that divide the space into cells (McKenna and O'Rourke [ M O ] implicitly construct this arrangement). Each cell of d ( ~ ) contains points whose stabbed set is invariant within the cell. The set 6e(~) of all stabbing lines is therefore the union of some cells in the arrangement. Our aim is to find a worst-case upper bound on the combinatorial complexity of 6P(~) which is significantly lower than the complexity of the whole arrangement d ( ~ ) . We consider the lines spanning edges in ~ to be in general position when no four lines are on the same ruled surface (planes, one sheet hyperboloids, and hyperbolic paraboloids [B]). For simplicity, we deal mostly with edges in general position and we give additional arguments to cope with degeneracies. The complexity of 6~(~) is bounded by the number of zero-dimensional faces (vertices) of S~(M). Each vertex represents an extremal stabbin# line. An extremal stabbing line l is a stabbing line for ~ which falls into one of the following three categories: t. I intersects four edges in four distinct polyhedra in ~¢ and is tangent to the same four polyhedra. 2. 1 intersects one vertex and two edges in three distinct polyhedra in ~ and is tangent to the same three polyhedra. 3. l meets two vertices in two distinct polyhedra in ~ and is tangent to those two polyhedra. For subclasses 2 and 3 an O(na) upper bound is trivially established. In this paper we concentrate our attention on subclass 1 of extremal stabbing lines. An f~(n3) lower bound for the complexity of ~ ( ~ ) is shown in [P2] and [P1] and can also be derived by results in [CEGS]. We prove in this paper an almost matching O(n32c ~ ) upper bound on the complexity of 6a(M) and on the time to answer the question of whether a stabbing line exists (stabbing problem). Initially (Section 3) we consider only triangles to show the analysis in a simplified setting. The main tools used are the Pliicker coordinates of lines, which are introduced in Section 2, and the random sampling technique of Clarkson [C]. All randomized algorithms in this paper can be turned into deterministic algorithms within the same time bounds using the methods of Matou~ek [Ma2]. The results for triangles are extended to the general case of convex polyhedra in Section 4. The query problem (given a line, is it a stabber?) is solved using O(n2÷~) preprocessing and storage, and O(log n) query time (Section 5). The query algorithm is used as a subroutine of an algorithm to find a stabbing line which uses O(na2cl'fi~-~)time (Section 5). The techniques used in [AWl, [MO] a (...truncated)


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M. Pellegrini, P. W. Shor. Finding stabbing lines in 3-space, Discrete & Computational Geometry, 1992, pp. 191-208, Volume 8, Issue 2, DOI: 10.1007/BF02293043