We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in $R^3$, (ii) reporting intersections between query lines (segments, or rays) and input triangles, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra, (iv) detecting, counting, or reporting intersections in a set of lines in $R^3$, and (v) output-sensitive construction of an arrangement of triangles in three dimensions. 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 given triangles, which uses $O(n^{3/2+\varepsilon})$ storage and preprocessing, and answers a query in $O(n^{1/2+\varepsilon})$ 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 algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer $m$ queries on $n$ objects in \[ \max \left\{ O(m^{2/3}n^{5/6+\varepsilon} + n^{1+\varepsilon}),\; O(m^{5/6+\varepsilon}n^{2/3} + m^{1+\varepsilon}) \right\} \] time, for any $\varepsilon>0$, again an improvement over the earlier bounds.

翻译：这些问题包括 (一) 在三角形中射线,以R33美元计算, (二) 报告查询线(部分,或射线)和输入三角形之间的交叉点, 并大致计算这些交叉点的数量, (三) 计算两个非convex 多元赫德拉的交叉点, (四) 在一个行中检测、计数或报告交叉点, 以3美元计算, 以及 (五) 在三个维度中构建一个对产出敏感的三角形安排。 我们的方法以多盘分布技术为基础。 例如, 我们的射线算法进程(部分,或射线) 以$3美元处理, 并大致计算在给定三角形中对射线询问的交叉点, (四) 使用美元(n) 3/2 ⁇ creepsial) 的存储和预处理, 并用 $O(n) (n)\\\\\\ 1/ {qu) 解算出一个对三角形的查询。 (n) 在任何时间里, 5 这样的存储结果中, 这个数量是相当的。