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ACM Computing Surveys (CSUR)
On lines avoiding unit balls in three dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
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Journal of the ACM (JACM)
Ray shooting and stone throwing with near-linear storage
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SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Ray shooting and intersection searching amidst fat convex polyhedra in 3-space
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Ray shooting and intersection searching amidst fat convex polyhedra in 3-space
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Journal of Scientific Computing
Ray shooting and stone throwing with near-linear storage
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SIAM Journal on Computing
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Journal of Mathematical Imaging and Vision
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We consider the problem of ray shooting amidst spheres in 3-space: given n arbitrary (possibly intersecting) spheres in 3-space and any $\epsilon$ 0, we show how to preprocess the spheres in time $O(n^{3+\epsilon})$ into a data structure of size $O(n^{3+\epsilon})$ so that any ray-shooting query can be answered in time $O(n^\epsilon)$. Our result improves previous techniques (see [P. K. Aggarwal, L. Guibas, M. Pellegrini, and M. Sharir, "Ray shooting amidst spheres," unpublished note] and [P. K. Aggarwal and J. Matousek, Discrete Comput. Geom., 11 (1994), pp. 393-418]), where roughly $O(n^4)$ storage was required to support fast queries. Our result shows that ray shooting amidst spheres has complexity comparable with that of ray shooting amidst planes in 3-space. Our technique applies to more general (convex) objects in 3-space, and we also discuss those extensions.