@article{f882f853c40b4f1db2eaa924e9091788,
title = "Linear data structures for fast ray-shooting amidst convex polyhedra",
abstract = "We consider the problem of ray shooting in a three-dimensional scene consisting of k (possibly intersecting) convex polyhedra with a total of n facets. That is, we want to preprocess them into a data structure, so that the first intersection point of a query ray and the given polyhedra can be determined quickly. We describe data structures that require tilde{O}(n\cdot \mathrm{poly}(k)) preprocessing time and storage (where the tilde{O}(\cdot) notation hides polylogarithmic factors), and have polylogarithmic query time, for several special instances of the problem. These include the case when the ray origins are restricted to lie on a fixed line ℓ 0, but the directions of the rays are arbitrary, the more general case when the supporting lines of the rays pass through ℓ 0, and the case of rays orthogonal to some fixed line with arbitrary origins and orientations. We also present a simpler solution for the case of vertical ray-shooting with arbitrary origins. In all cases, this is a significant improvement over previously known techniques (which require Ω(n 2) storage, even when k ∼ n).",
keywords = "Computational geometry, Convex polyhedra, Lines in space, Ray-shooting",
author = "Haim Kaplan and Natan Rubin and Micha Sharir",
note = "Funding Information: Work by Haim Kaplan and Natan Rubin has been supported by Grant 975/06 from the Israel Science Fund. Work by Micha Sharir and Natan Rubin was partially supported by NSF Grant CCF-05-14079, by a grant from the U.S.–Israeli Binational Science Foundation, by grant 155/05 from the Israel Science Fund, Israeli Academy of Sciences, by a grant from the AFIRST French–Israeli program, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper appeared in Proc. 15th Annu. Europ. Sympos. Alg. (2007), 287–298.",
year = "2009",
month = oct,
doi = "10.1007/s00453-008-9220-0",
language = "אנגלית",
volume = "55",
pages = "283--310",
journal = "Algorithmica",
issn = "0178-4617",
publisher = "Springer New York",
number = "2",
}