### Abstract

We present a purely numerical (i.e., non-algebraic) subdivision algorithm for computing an isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function on the plane, along with effective interval forms of the function and its partial derivatives. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A preliminary implementation is available in Core Library.

Original language | English (US) |
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Title of host publication | Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings |

Publisher | Springer Verlag |

Pages | 277-282 |

Number of pages | 6 |

Volume | 8592 LNCS |

ISBN (Print) | 9783662441985 |

DOIs | |

State | Published - 2014 |

Event | 4th International Congress on Mathematical Software, ICMS 2014 - Seoul, Korea, Republic of Duration: Aug 5 2014 → Aug 9 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8592 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 4th International Congress on Mathematical Software, ICMS 2014 |
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Country | Korea, Republic of |

City | Seoul |

Period | 8/5/14 → 8/9/14 |

### Fingerprint

### Keywords

- arrangement of curves
- interval arithmetic
- Isotopy
- marching-cube
- subdivision algorithms

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings*(Vol. 8592 LNCS, pp. 277-282). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8592 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-662-44199-2_43

**Isotopic arrangement of simple curves : An exact numerical approach based on subdivision.** / Lien, Jyh Ming; Sharma, Vikram; Vegter, Gert; Yap, Chee.

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

*Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings.*vol. 8592 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8592 LNCS, Springer Verlag, pp. 277-282, 4th International Congress on Mathematical Software, ICMS 2014, Seoul, Korea, Republic of, 8/5/14. https://doi.org/10.1007/978-3-662-44199-2_43

}

TY - GEN

T1 - Isotopic arrangement of simple curves

T2 - An exact numerical approach based on subdivision

AU - Lien, Jyh Ming

AU - Sharma, Vikram

AU - Vegter, Gert

AU - Yap, Chee

PY - 2014

Y1 - 2014

N2 - We present a purely numerical (i.e., non-algebraic) subdivision algorithm for computing an isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function on the plane, along with effective interval forms of the function and its partial derivatives. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A preliminary implementation is available in Core Library.

AB - We present a purely numerical (i.e., non-algebraic) subdivision algorithm for computing an isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function on the plane, along with effective interval forms of the function and its partial derivatives. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A preliminary implementation is available in Core Library.

KW - arrangement of curves

KW - interval arithmetic

KW - Isotopy

KW - marching-cube

KW - subdivision algorithms

UR - http://www.scopus.com/inward/record.url?scp=84905845331&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84905845331&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-44199-2_43

DO - 10.1007/978-3-662-44199-2_43

M3 - Conference contribution

AN - SCOPUS:84905845331

SN - 9783662441985

VL - 8592 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 277

EP - 282

BT - Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings

PB - Springer Verlag

ER -