球面测地弧排列的Minkowski和构造及其它应用
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
摘要：
我们提出了两个精确的实现高效的输出敏感的算法，计算Minkowski和两个凸多面体在3D。我们不采取一般立场。也就是说，我们处理退化输入，并产生精确的结果。我们提供了一个严格的上限上的确切的最大复杂性的Minkowski总和的多面体在3D的被加数多面体的方面。该算法采用的数据结构，表示嵌入在三维的二维参数表面上的安排的变体，他们利用许多操作应用到这些表示的安排。我们已经开发了支持排列数据结构变体和应用于它们的操作的软件组件。这些软件组件是通用的，因为它们可以用任何数字类型实例化。然而，我们的算法只需要（精确的）理性算术。这些软件组件与精确的有理数运算一起实现了Minkowski和构造及其相关应用的强大，高效和优雅的实现。这些软件组件是通过一个名为“表面上的计算”的计算几何算法库（CGAL）包提供的。我们还提出了其他应用程序，利用嵌入在球体上的圆弧的安排的确切实现。我们使用它们作为基本块在一个有效的算法，分区组装多面体在3D用两只手使用无限的翻译的精确实现。这个应用程序清楚地表明了精确计算的重要性，因为不精确的计算可能会导致有效的分割运动的解雇。
英文摘要：
We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.
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