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

TL;DR

This paper introduces a purely numerical subdivision algorithm for accurately approximating simple arrangements of curves, ensuring isotopic correctness using interval methods, and generalizing previous algorithms.

Contribution

It presents the first non-algebraic, numerical subdivision approach for isotopic curve arrangements, extending prior algebraic algorithms with certified interval methods.

Findings

Algorithm is practical and easy to implement.

Ensures topological correctness of curve approximations.

Integrates geometric and topological computations effectively.

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).