On ray shooting in convex polytopes

Abstract

Let P be a convex polytope with n facets in the Euclidean space of a (small) fixed dimension d. We consider the membership problem for P (given a query point, decide whether it lies in P) and the ray shooting problem in P (given a query ray originating inside P, determine the first facet of P hit by it). It was shown in [AM2] that a data structure for the membership problem satisfying certain mild assumptions can also be used for the ray shooting problem, with a logarithmic overhead in query time. Here we show that some specific data structures for the membership problem can be used for ray shooting in a more direct way, reducing the overhead in the query time and eliminating the use of parametric search. We also describe an improved static solution for the membership problem, approaching the conjectured lower bounds more tightly. © 1993 Springer-Verlag New York Inc.