Ray Shooting Amidst Spheres in Three Dimensions and Related Problems

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previous techniques (see [P. K. Aggarwal, L. Guibas, M. Pellegrini, and M. Sharir, "Ray shooting amidst spheres," unpublished note] and [P. K. Aggarwal and J. Matousek, Discrete Comput.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.

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