# Intersecting line segments, ray shooting, and other applications of geometric partitioning techniques

## Abstract

We present a variety of applications of certain techniques, based on partition trees, that were originally developed for range searching problems. Our results are obtained by enhancing and extending these techniques, and include: (i) An *O*(*n*^{4/3+δ}+*k*)-time (for any δ>0), *O*(*n*)-space randomized algorithm for finding all *k* intersections of *n* line segments in the plane (we can count the number of these intersections in *O*(*n*^{4/3+δ}) time and linear space). (ii) Preprocessing a collection of *n* (possibly intersecting) segments in the plane so that, given any query ray, we can find quickly the first segment it hits. Other applications concern “implicit” point location, hidden surface removal in three dimensions, polygon placement queries, and problems involving overlapping planar maps. We also present several efficient algorithms involving the analysis of the connectivity and other useful properties of arrangements of line segments.

## Keywords

Line Segment Computational Geometry Query Time Simple Polygon Partition Tree## Preview

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