基于Voronoi网格的凸多边形点包含判定
A Novel Point Inclusion Test for Convex Polygons Based on Voronoi Tessellations
摘要：
多边形的点包含检验，即多边形中点包含检验（PIP）算法，是计算几何相关领域的基本工具，已被研究了很长时间。PIP算法直接或间接地得到多边形实体的几何定义，并验证其对给定点的包容性。PIP算法直接作用于几何实体，导出多边形边缘的线性边界定义。此外，几乎所有的直接检验方法都依赖于线方程的两点形式来将空间划分为半空间。Voronoi镶嵌使用另一种方法进行半空间分割。代替直线方程，生成器点之间的距离比较用于完成相同的任务。Voronoi镶嵌由凸多边形组成，这些凸多边形在生成点之间定义。因此，Voronoi曲面细分方法的提出为我们开发一种新的凸多边形PIP测试方法提供了灵感。导出了凸多边形到Voronoi多边形的转换方程。作为参考，选择了非常标准的凸PIP测试算法\textit{the sign of offset}进行比较。为了推广比较，使用\textit{the ray crossing}算法作为另一个参考。所有的算法都被实现为向量和矩阵操作，没有任何分支。这使我们能够受益于底层线性代数库的CPU优化。实验结果表明，我们提出的算法可以有相当的性能特性与参考算法。此外，它具有简单性，无论是从几何表示还是心理模型。
英文摘要：
The point inclusion tests for polygons, in other words the point-in-polygon (PIP) algorithms, are fundamental tools for many scientific fields related to computational geometry, and they have been studied for a long time. The PIP algorithms get direct or indirect geometric definition of a polygonal entity, and validate its containment of a given point. The PIP algorithms, which are working directly on the geometric entities, derive linear boundary definitions for the edges of the polygons. Moreover, almost all direct test methods rely on the two-point form of the line equation to partition the space into half-spaces. Voronoi tessellations use an alternate approach for half-space partitioning. Instead of line equation, distance comparison between generator points is used to accomplish the same task. Voronoi tessellations consist of convex polygons, which are defined between generator points. Therefore, Voronoi tessellations have become an inspiration for us to develop a new approach of the PIP testing, specialized for convex polygons. The equations, essential to the conversion of a convex polygon to a Voronoi polygon, are derived. As a reference, a very standard convex PIP testing algorithm, \textit{the sign of offset}, is selected for comparison. For generalization of the comparisons, \textit{the ray crossing} algorithm is used as another reference. All algorithms are implemented as vector and matrix operations without any branching. This enabled us to benefit from the CPU optimizations of the underlying linear algebra libraries. Experimentation showed that, our proposed algorithm can have comparable performance characteristics with the reference algorithms. Moreover, it has simplicity, both from a geometric representation and the mental model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
