### Abstract

We consider domain subdivision algorithms for computing isotopic approximations of a nonsingular algebraic curve. The curve is given by a polynomial equation f(X, Y)= 0. Two algorithms in this area are from Snyder (1992) SIGGRAPH Comput. Graphics, 26(2), 121 and Plantinga and Vegter (2004) In Proc. Eurographics Symposium on Geometry Processing, pp. 245-254. We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parameterizability criterion for subdivision, and like Plantinga and Vegter, we exploit nonlocal 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
_{0} transversally. Our algorithm is practical and easy to implement exactly. We report some very encouraging experimental results, showing that our algorithms can be much more efficient than the algorithms of Plantinga-Vegter and Snyder.

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

Pages (from-to) | 760-795 |

Number of pages | 36 |

Journal | Discrete and Computational Geometry |

Volume | 45 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2011 |

### Fingerprint

### Keywords

- Curve approximation
- Exact algorithms
- Isotopy
- Meshing
- Parameterizability
- Subdivision algorithms
- Topological correctness

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

**Adaptive Isotopic Approximation of Nonsingular Curves : The Parameterizability and Nonlocal Isotopy Approach.** / Lin, Long; Yap, Chee.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 45, no. 4, pp. 760-795. https://doi.org/10.1007/s00454-011-9345-9

}

TY - JOUR

T1 - Adaptive Isotopic Approximation of Nonsingular Curves

T2 - The Parameterizability and Nonlocal Isotopy Approach

AU - Lin, Long

AU - Yap, Chee

PY - 2011/6

Y1 - 2011/6

N2 - We consider domain subdivision algorithms for computing isotopic approximations of a nonsingular algebraic curve. The curve is given by a polynomial equation f(X, Y)= 0. Two algorithms in this area are from Snyder (1992) SIGGRAPH Comput. Graphics, 26(2), 121 and Plantinga and Vegter (2004) In Proc. Eurographics Symposium on Geometry Processing, pp. 245-254. We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parameterizability criterion for subdivision, and like Plantinga and Vegter, we exploit nonlocal 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 0 transversally. Our algorithm is practical and easy to implement exactly. We report some very encouraging experimental results, showing that our algorithms can be much more efficient than the algorithms of Plantinga-Vegter and Snyder.

AB - We consider domain subdivision algorithms for computing isotopic approximations of a nonsingular algebraic curve. The curve is given by a polynomial equation f(X, Y)= 0. Two algorithms in this area are from Snyder (1992) SIGGRAPH Comput. Graphics, 26(2), 121 and Plantinga and Vegter (2004) In Proc. Eurographics Symposium on Geometry Processing, pp. 245-254. We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parameterizability criterion for subdivision, and like Plantinga and Vegter, we exploit nonlocal 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 0 transversally. Our algorithm is practical and easy to implement exactly. We report some very encouraging experimental results, showing that our algorithms can be much more efficient than the algorithms of Plantinga-Vegter and Snyder.

KW - Curve approximation

KW - Exact algorithms

KW - Isotopy

KW - Meshing

KW - Parameterizability

KW - Subdivision algorithms

KW - Topological correctness

UR - http://www.scopus.com/inward/record.url?scp=79954634201&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79954634201&partnerID=8YFLogxK

U2 - 10.1007/s00454-011-9345-9

DO - 10.1007/s00454-011-9345-9

M3 - Article

VL - 45

SP - 760

EP - 795

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -