### 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 131-152 |

Number of pages | 22 |

Journal | Journal of Symbolic Computation |

Volume | 47 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2012 |

### Fingerprint

### 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

*Journal of Symbolic Computation*,

*47*(2), 131-152. https://doi.org/10.1016/j.jsc.2011.08.021

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

Research output: Contribution to journal › Article

*Journal of Symbolic Computation*, vol. 47, no. 2, pp. 131-152. https://doi.org/10.1016/j.jsc.2011.08.021

}

TY - JOUR

T1 - Complete subdivision algorithms, II

T2 - Isotopic meshing of singular algebraic curves

AU - Burr, Michael

AU - Choi, Sung Woo

AU - Galehouse, Ben

AU - Yap, Chee

PY - 2012/2

Y1 - 2012/2

N2 - 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.

AB - 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.

KW - Complete numerical algorithm

KW - Evaluation bound

KW - Implicit algebraic curve

KW - Meshing

KW - Root bound

KW - Singularity

KW - Subdivision algorithm

UR - http://www.scopus.com/inward/record.url?scp=81555214084&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81555214084&partnerID=8YFLogxK

U2 - 10.1016/j.jsc.2011.08.021

DO - 10.1016/j.jsc.2011.08.021

M3 - Article

AN - SCOPUS:81555214084

VL - 47

SP - 131

EP - 152

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

IS - 2

ER -