关于区域图和双区域图的计算
On the computation of zone and double zone diagrams
摘要：
计算几何中的经典对象是由显式关系定义的。几年前，T. Asano，J. Matousek和T.德山提出了“隐式计算几何“，其中几何对象由涉及集合的隐式关系定义。在这个家庭中的一个重要成员被称为“区图”。正如在原始作品中已经观察到的那样，区域图的隐含性质意味着它们的计算是一项具有挑战性的任务。在一个连续的设置这一任务已解决（简要）只有这些作者在欧几里德平面与点的网站。我们讨论的可能性，计算区域图在广泛的一类空间，也揭示了新的光在原来的设置计算。类的空间，这是介绍这里，特别是包括欧几里德领域和有限维严格凸赋范空间。网站的一般形式是允许的，它表明，概括的迭代法建议浅野，Matousek和Tokuyama收敛到一个双区图，另一个隐含的几何对象，其存在是已知的一般。有时可以从这个过程中得到一个区域图。迭代的实际（近似）计算基于一个简单的算法，该算法能够在一般设置中近似计算沃罗诺伊图。我们的分析还产生了一些独立感兴趣的副产品，例如Voronoi细胞的某些拓扑性质（例如，在所考虑的设置中，它们的边界不能是“胖”）。
英文摘要：
Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").
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。
