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Linear Data Structures for Fast Ray-Shooting amidst Convex Polyhedra

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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).

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Authors and Affiliations

  1. School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Haim Kaplan, Natan Rubin & Micha Sharir

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Micha Sharir

Authors
  1. Haim Kaplan
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  2. Natan Rubin
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  3. Micha Sharir
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Correspondence to Haim Kaplan.

Additional 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.

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Kaplan, H., Rubin, N. & Sharir, M. Linear Data Structures for Fast Ray-Shooting amidst Convex Polyhedra. Algorithmica 55, 283–310 (2009). https://doi.org/10.1007/s00453-008-9220-0

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  • DOI: https://doi.org/10.1007/s00453-008-9220-0

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