Ray shooting and parametric search

Abstract

Efficient algorithms for the ray shooting problem are presented: Given a collection F of objects in d, build a data structure so that, for a query ray, the first object of F hit by the ray can be quickly determined. Using the parametric search technique, this problem is reduced to the segment emptiness problem. For various ray shooting problems, space/query-time trade-offs of the following type are achieved: For some integer b and a parameter m (n _< m < n b) the queries are answered in time O((n/m /b) log <) n), with O(m!+) space and preprocessing time (t> 0 is arbitrarily small but fixed constant), b Ld/2J is obtained for ray shooting in a convex d-polytope defined as an intersection of n half spaces, b d for an arrangement of n hyperplanes in d, and b 3 for an arrangement of n half planes in 3. This approach also yields fast procedures for finding the first k objects hit by a query ray, for searching nearest and farthest neighbors, and for the hidden surface removal. All the data structures can be maintained dynamically in amortized time O (m + / n) per insert/delete operation.