### 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 |

ISBN (Print) | 9783662441985 |

DOIs | |

State | Published - Jan 1 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) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### 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

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings*(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