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 \(\tilde{O}(n{\rm poly}(k))\) preprocessing time and storage, 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 ℓ0, but the directions of the rays are arbitrary, the more general case when the supporting lines of the rays pass through ℓ0, and the case of rays orthogonal to z-axis 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).
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, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.
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Kaplan, H., Rubin, N., Sharir, M. (2007). Linear Data Structures for Fast Ray-Shooting Amidst Convex Polyhedra. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_27
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DOI: https://doi.org/10.1007/978-3-540-75520-3_27
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