On intersection searching problems involving curved objects

Abstract

Efficient solutions are given for the problem of preprocessing a set of linear or curved geometric objects (e.g. lines, llne segments, circles, circular arcs, d-balls, d-spheres)such that the ones that are intersected by a curved query object can be reported (or counted) quickly. The problem is considered both in the standard setting (where one is interested in all the objects intersected) and in a generalized setting (where the input objects come aggregated in disjoint groups and one is interested in the disjoint groups that axe intersected). The solutions axe based on geometric transformations, simplex compositions, persistence, and, for the generalized problem, oa a method to progressively eliminate groups that cannot possibly be intersected.