三维空间中三角形的射线打靶及相关问题
On Ray Shooting for Triangles in 3-Space and Related Problems
摘要：
我们考虑几个问题，涉及线在三维空间，并提出改进的算法来解决这些问题。这些问题包括：（i）在$R^3$中的三角形之间的射线射击，（ii）查询线之间的相交报告（iii）计算两个非凸多面体的交点，（iv）检测、计数或报告$R^3$中的一组线中的交点，以及（v）三维三角形排列的输出敏感构造。我们的方法是基于多项式分区技术。例如，我们的射线射击算法处理$R^3$中的一组$n$三角形到一个数据结构中，用于回答给定三角形中的射线射击查询，该数据结构使用$O(n^{3/2+\varepsilon})$存储和预处理，并且对于任何$\varepsilon>0$，在$O(n^{1/2+\varepsilon})$时间内回答查询。这是对25年前获得的已知结果的显著改进，在已知结果中，利用该存储量，查询时间界限大致为$n^{5/8}$。其他问题的算法有类似的性能界限，与以前的结果有类似的改进。我们还得到了一个非平凡的改进存储和查询时间之间的权衡。使用它，我们得到的算法，回答$m$查询$n$对象在\[ \max \left\{ O(m^{2/3}n^{5/6+\varepsilon} + n^{1+\varepsilon}),\; O(m^{5/6+\varepsilon}n^{2/3} + m^{1+\varepsilon}) \right\} \]的时间，任何$\varepsilon>0$，再次改进了早期的界限。
英文摘要：
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.
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