Isotopic Arrangement of Simple Curves: an Exact Numerical Approach based on Subdivision

Abstract

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).