### Abstract

We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Veg- ter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R
_{0} with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities in the region, and intersects the boundary of R
_{o} transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.

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

Title of host publication | Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09 |

Pages | 351-360 |

Number of pages | 10 |

DOIs | |

State | Published - 2009 |

Event | 25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark Duration: Jun 8 2009 → Jun 10 2009 |

### Other

Other | 25th Annual Symposium on Computational Geometry, SCG'09 |
---|---|

Country | Denmark |

City | Aarhus |

Period | 6/8/09 → 6/10/09 |

### Fingerprint

### Keywords

- Curve approximation
- Exact numerical algorithms
- Isotopy
- Meshing
- Parametrizability
- Subdivision algorithms
- Topological correctness

### ASJC Scopus subject areas

- Computational Mathematics
- Geometry and Topology
- Theoretical Computer Science

### Cite this

*Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09*(pp. 351-360) https://doi.org/10.1145/1542362.1542423

**Adaptive isotopic approximation of nonsingular curves : The parametrizability and nonlocal isotopy approach.** / Lin, Long; Yap, Chee.

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

*Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09.*pp. 351-360, 25th Annual Symposium on Computational Geometry, SCG'09, Aarhus, Denmark, 6/8/09. https://doi.org/10.1145/1542362.1542423

}

TY - GEN

T1 - Adaptive isotopic approximation of nonsingular curves

T2 - The parametrizability and nonlocal isotopy approach

AU - Lin, Long

AU - Yap, Chee

PY - 2009

Y1 - 2009

N2 - We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Veg- ter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R 0 with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities in the region, and intersects the boundary of R o transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.

AB - We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Veg- ter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R 0 with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities in the region, and intersects the boundary of R o transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.

KW - Curve approximation

KW - Exact numerical algorithms

KW - Isotopy

KW - Meshing

KW - Parametrizability

KW - Subdivision algorithms

KW - Topological correctness

UR - http://www.scopus.com/inward/record.url?scp=70849116923&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70849116923&partnerID=8YFLogxK

U2 - 10.1145/1542362.1542423

DO - 10.1145/1542362.1542423

M3 - Conference contribution

SN - 9781605585017

SP - 351

EP - 360

BT - Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09

ER -