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

Michael Burr, Sung Woo Choi, Ben Galehouse, Chee Yap

Research output: Contribution to journalArticle

Abstract

Given a real valued function f(X, Y), a box region B0⊆R{double-struck}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{double-struck}2:f(p)=0} to B0. 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.

Original languageEnglish (US)
Pages (from-to)131-152
Number of pages22
JournalJournal of Symbolic Computation
Volume47
Issue number2
DOIs
StatePublished - Feb 2012

Fingerprint

Subdivision Algorithm
Singular Curve
Meshing
Algebraic curve
Singularity
Polygonal Approximation
Curve
Interval Arithmetic
Isolated Singularity
Evaluation Function
Numerical Algorithms
Computational Model
Branching
Function evaluation
Subroutines
Numerical Methods
Restriction
Generalise
Numerical methods
Approximation

Keywords

  • Complete numerical algorithm
  • Evaluation bound
  • Implicit algebraic curve
  • Meshing
  • Root bound
  • Singularity
  • Subdivision algorithm

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics

Cite this

Complete subdivision algorithms, II : Isotopic meshing of singular algebraic curves. / Burr, Michael; Choi, Sung Woo; Galehouse, Ben; Yap, Chee.

In: Journal of Symbolic Computation, Vol. 47, No. 2, 02.2012, p. 131-152.

Research output: Contribution to journalArticle

Burr, Michael ; Choi, Sung Woo ; Galehouse, Ben ; Yap, Chee. / Complete subdivision algorithms, II : Isotopic meshing of singular algebraic curves. In: Journal of Symbolic Computation. 2012 ; Vol. 47, No. 2. pp. 131-152.
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