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Isotopic Arrangement of Simple Curves: an Exact Numerical Approach based on Subdivision
arXiv - CS - Computational Geometry Pub Date : 2020-09-02 , DOI: arxiv-2009.00811
Jyh-Ming Lien and Vikram Sharma and Gert Vegter and Chee Yap

This paper presents the first purely numerical (i.e., non-algebraic) subdivision algorithm for the isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$, and effective interval forms of $f, \frac{\partial{f}}{\partial{x}}, \frac{\partial{f}}{\partial{y}}$ are available. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A version of this paper without the appendices appeared in Lien et al. (2014).

中文翻译:

简单曲线的同位素排列:一种基于细分的精确数值方法

本文介绍了第一个用于简单曲线排列的同位素近似的纯数值(即非代数)细分算法。这种排列是“简单的”,即任意三条曲线没有公共交点,任意两条曲线横向相交,并且每条曲线都是非奇异的。曲线由解析函数 $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ 的零集和 $f, \frac{\partial{f}} 的有效区间形式给出{\partial{x}}、\frac{\partial{f}}{\partial{y}}$ 可用。我们的解决方案概括了 Plantinga-Vegter (2004) 和 Lin-Yap (2009) 的同位素曲线近似算法。我们使用基于区间方法的经过认证的数字基元。此类算法具有许多有利的特性:它们实用、易于实现、没有实现漏洞、将拓扑与几何计算相结合,具有自适应和局部复杂性。这篇论文的一个版本没有附录出现在 Lien 等人。(2014)。
更新日期:2020-09-03
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