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 " > 0, we show how to preprocess the spheres in time O(n 3+"), into a data structure of size O(n 3+"), so that any ray-shooting query can be answered in time O(n "). Our result improves previous techniques (see 3, 5]), 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 these extensions.