Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves

Abstract

Given a real valued function f (X, Y ), a box region B 0 ⊆ R 2 and ε > 0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curve S = f −1 (0) = {p ∈ R 2 : f (p) = 0} to B 0 . We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & 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.

GRASS GIS: A multi-purpose open source GIS

“…• There are many bisection algorithms where continuous amortization may be useful, see, for example, Henrici (1970), Yakoubsohn (2005), Yap andSagraloff (2011), Plantinga andVegter (2004), Plantinga (2006), Snyder (1992, Galehouse (2009), Burr et al (2012, Eigenwillig et al (2006), Sagraloff (2011), Du et al (2007 and Lin and Yap (2009). We plan on extending our techniques to these cases.…”

“…EVAL-type algorithms have been studied in the univariate case in Henrici (1970), Yakoubsohn (2005, Yap and Sagraloff (2011), Burr et al (in preparation) and Burr et al (2009), in the bivariate and trivariate cases in Lorensen and Cline (1987), Snyder (1992), Plantinga and Vegter (2004), Plantinga (2006), Lin and Yap (2009) and Burr et al (2012), and in the multivariate case in Galehouse (2009) and Dedieu and Yakoubsohn (1992). All of these algorithms are devoted to approximating algebraic (and in some cases analytic varieties) in the real or complex settings.…”

“…The two-dimensional EVAL-type algorithm (Plantinga and Vegter, 2004;Plantinga, 2006) was presented for approximating smooth and bounded varieties. It was extended to singular and unbounded varieties in Burr et al (2012); in addition, the tests performed by the algorithm were improved in Lin and Yap (2009).…”

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

“…There are also approaches [24] that achieve locally quadratic convergence towards the simple roots of polynomial systems and there very efficient in practice. Another interesting direction of subdivision algorithms, of more geometric nature, concerns the approximation of algebraic varieties [30,25,5,29,37,21], and the computation of the approximate Voronoi diagrams [39]. There are also quite important applications of these algorithms to the problem of robot motion planning [33].…”

“…Additionally, by further subdivision, the isotopy can be made sufficiently small so that the Hausdorff distance between the approximation and the variety is as small as desired. We note that it is possible to extend the PV algorithm in the plane to provide an approximation even when V R (f ) is unbounded, V R (f ) is singular, and I is not a bounding box, see [5]. In this paper, however, we focus on the original PV algorithm without the restriction of a bounded curve.…”