Adaptive isotopic approximation of nonsingular curves

Abstract

We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y ) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Vegter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R0 with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities in the region, and intersects the boundary of R0 transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.

Isotopic Arrangement of Simple Curves: An Exact Numerical Approach Based on Subdivision

“…The approach of this paper has previously been successfully applied to the isotopic approximation of a single non-singular curve or surface by Plantinga and Vegter [18,17] and Lin and Yap [11,10]. The current paper is a non-trivial extension of these previous works.…”

“…Let x be a point on the boundary of the box B. From the definition of G and from the mean value theorem (10) we know that…”