### Abstract

Given a real function f(X, Y), a box regionB and ε 0, we want to compute an ε-isotopic polygonal approximation to the curve C: f(X, Y) = 0 within B. We focus on subdivision algorithms because of their adaptive complexity. Plantinga & Vegter (2004) gave a numerical subdivision algorithm that is exact when the curve C is non-singular. They used a computational model that relies only on function evaluation and interval arithmetic. We generalize their algorithm to any (possibly non-simply connected) region B that does not contain singularities of C. With this generalization as subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete numerical method to treat implicit algebraic curves with isolated singularities.

Original language | English (US) |
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Title of host publication | ISSAC'08: Proceedings of the 21st International Symposium on Symbolic and Algebraic Computation 2008 |

Pages | 87 |

Number of pages | 1 |

DOIs | |

State | Published - 2008 |

Event | 21st Annual Meeting of the International Symposium on Symbolic Computation, ISSAC 2008 - Linz, Hagenberg, Austria Duration: Jul 20 2008 → Jul 23 2008 |

### Other

Other | 21st Annual Meeting of the International Symposium on Symbolic Computation, ISSAC 2008 |
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Country | Austria |

City | Linz, Hagenberg |

Period | 7/20/08 → 7/23/08 |

### Fingerprint

### Keywords

- Evaluation bound
- Im
- Meshing
- Root bound
- Singularity

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*ISSAC'08: Proceedings of the 21st International Symposium on Symbolic and Algebraic Computation 2008*(pp. 87) https://doi.org/10.1145/1390768.1390783

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*ISSAC'08: Proceedings of the 21st International Symposium on Symbolic and Algebraic Computation 2008.*pp. 87, 21st Annual Meeting of the International Symposium on Symbolic Computation, ISSAC 2008, Linz, Hagenberg, Austria, 7/20/08. https://doi.org/10.1145/1390768.1390783

}

TY - GEN

T1 - Complete subdivision algorithms, II

T2 - Isotopic meshing of singular algebraic curves

AU - Burr, Michael

AU - Choi, Sung Woo

AU - Galehouse, Benjamin

AU - Yap, Chee

PY - 2008

Y1 - 2008

N2 - Given a real function f(X, Y), a box regionB and ε 0, we want to compute an ε-isotopic polygonal approximation to the curve C: f(X, Y) = 0 within B. We focus on subdivision algorithms because of their adaptive complexity. Plantinga & Vegter (2004) gave a numerical subdivision algorithm that is exact when the curve C is non-singular. They used a computational model that relies only on function evaluation and interval arithmetic. We generalize their algorithm to any (possibly non-simply connected) region B that does not contain singularities of C. With this generalization as subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete numerical method to treat implicit algebraic curves with isolated singularities.

AB - Given a real function f(X, Y), a box regionB and ε 0, we want to compute an ε-isotopic polygonal approximation to the curve C: f(X, Y) = 0 within B. We focus on subdivision algorithms because of their adaptive complexity. Plantinga & Vegter (2004) gave a numerical subdivision algorithm that is exact when the curve C is non-singular. They used a computational model that relies only on function evaluation and interval arithmetic. We generalize their algorithm to any (possibly non-simply connected) region B that does not contain singularities of C. With this generalization as subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete numerical method to treat implicit algebraic curves with isolated singularities.

KW - Evaluation bound

KW - Im

KW - Meshing

KW - Root bound

KW - Singularity

UR - http://www.scopus.com/inward/record.url?scp=57449099508&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57449099508&partnerID=8YFLogxK

U2 - 10.1145/1390768.1390783

DO - 10.1145/1390768.1390783

M3 - Conference contribution

AN - SCOPUS:57449099508

SN - 9781595939043

SP - 87

BT - ISSAC'08: Proceedings of the 21st International Symposium on Symbolic and Algebraic Computation 2008

ER -