带洞多边形的精确Minkowski和
Exact Minkowski Sums of Polygons With Holes
摘要：
我们提出了一个有效的算法，计算Minkowski和两个多边形，这可能有洞。新算法是基于卷积方法。它的效率部分源于带洞多边形的闵可夫斯基和的一个性质，这实际上在任何维度上都成立：给定两个带洞的多边形，对于每个输入多边形，我们可以填充与另一个多边形相比相对较小的洞。具体地说，我们总是可以填满至少一个多边形的所有洞，将其转换为一个简单的多边形，并且仍然获得完全相同的Minkowski和。消除输入和数中的洞可以加快Minkowski和的计算速度。我们引入了一个强大的实现新算法，它遵循精确几何计算范式，从而保证准确的结果。我们还提出了一个实证比较的性能的Minkowski和建设的各种输入的例子，我们表明，在许多情况下，新算法的实现表现出更好的性能比其他几个实现。特别是，我们比较了新算法的实现，标准卷积算法的实现，以及使用各种凸分解方法的分解方法的实现，包括两种处理带孔多边形的新方法-一种是基于垂直分解，另一种是基于三角剖分。该软件是作为CGAL（计算几何算法库）的“二维Minkowski和”软件包的扩展而开发的。其他信息和补充材料可在我们的项目页面http://acg.cs.tau.ac.il/projects/rc上获得
英文摘要：
We present an efficient algorithm that computes the Minkowski sum of two polygons, which may have holes. The new algorithm is based on the convolution approach. Its efficiency stems in part from a property for Minkowski sums of polygons with holes, which in fact holds in any dimension: Given two polygons with holes, for each input polygon we can fill up the holes that are relatively small compared to the other polygon. Specifically, we can always fill up all the holes of at least one polygon, transforming it into a simple polygon, and still obtain exactly the same Minkowski sum. Obliterating holes in the input summands speeds up the computation of Minkowski sums. We introduce a robust implementation of the new algorithm, which follows the Exact Geometric Computation paradigm and thus guarantees exact results. We also present an empirical comparison of the performance of Minkowski sum construction of various input examples, where we show that the implementation of the new algorithm exhibits better performance than several other implementations in many cases. In particular, we compared the implementation of the new algorithm, an implementation of the standard convolution algorithm, and an implementation of the decomposition approach using various convex decomposition methods, including two new methods that handle polygons with holes - one is based on vertical decomposition and the other is based on triangulation. The software has been developed as an extension of the "2D Minkowski Sums" package of CGAL (Computational Geometry Algorithms Library). Additional information and supplementary material is available at our project page http://acg.cs.tau.ac.il/projects/rc
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