We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them . Our approach is based on the polynomial partitioning technique . For example, our ray-shooting algorithm processes a set of $n$ triangles in $R^3$ into a data structure for answering ray shooting queries amid the giventriangles . The algorithms for the other problems have similar performancebounds, with similar performance bounds . We also derive a nontrivial improved tradeoff between storage and query time, and we obtain algorithms that answer $m$ queries in O(m^{2/3) time, for any$\varepsilon>0$ This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly \$n^{5/8 . The algorithm for other problems has similar performanceBounds, but similar improvements over previous results. We also derived a nontribial tradeoff of storage and

Author(s) : Esther Ezra, Micha Sharir