Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves

TL;DR

This paper presents a generalized subdivision algorithm for computing epsilon-isotopic polygonal approximations of algebraic curves, capable of handling singularities and complex regions using purely numerical methods.

Contribution

It extends previous algorithms to non-simply connected regions and introduces a method to detect and analyze algebraic singularities numerically.

Findings

First complete purely numerical method for isotopic approximation of algebraic curves with singularities.

Algorithm can detect isolated singularities and their branching degrees.

Applicable to complex, bounded regions without singularities.

Abstract

Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive epsilon, we want to compute an epsilon-isotopic polygonal approximation to the restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga and Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities.