### Abstract

We present robust and efficient algorithms for computing Voronoi diagrams of planar freeform curves. Boundaries of the Voronoi diagram consist of portions of the bisector curves between pairs of planar curves. Our scheme is based on computing critical structures of the Voronoi diagrams, such as self-intersections and junction points of bisector curves. Since the geometric objects we consider in this paper are represented as freeform NURBS curves, we were able to reformulate the solution to the problem of computing those critical structures into the zero-set solutions of a system of nonlinear piecewise rational equations in parameter space. We present a new algorithm for computing error-bounded bisector curves using a distance surface constructed from error-bounded offset approximations of planar curves. This error-bounded algorithm is fast and produces bisector curves that are correct both in topology and geometry. Once bisectors are computed, both local and global self-intersections of the bisector curves are located and trimmed away by solving a system of three piecewise rational equations in three variables. Further, our method computes junction points at which three or more trimmed bisector curves intersect by transforming them into the solutions to a system of piecewise rational equations in the merged parameter space of the planar curves. The bisectors are trimmed at those self-intersection and global junction points. The Voronoi diagram is then computed from the trimmed bisectors using a pruning algorithm. We demonstrate the effectiveness of our approach with several experimental results.

Original language | English |
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Title of host publication | Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling 2008, SPM'08 |

Pages | 67-78 |

Number of pages | 12 |

DOIs | |

Publication status | Published - 2008 Dec 17 |

Externally published | Yes |

Event | 2008 ACM Symposium on Solid and Physical Modeling 2008, SPM'08 - Stony Brook, NY, United States Duration: 2008 Jun 2 → 2008 Jun 4 |

### Other

Other | 2008 ACM Symposium on Solid and Physical Modeling 2008, SPM'08 |
---|---|

Country | United States |

City | Stony Brook, NY |

Period | 08/6/2 → 08/6/4 |

### Fingerprint

### Keywords

- Bisectors
- Freeform curves
- Junction points
- Problem reduction scheme
- Self-intersections
- Voronoi diagram
- Zero-set

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Graphics and Computer-Aided Design
- Software

### Cite this

*Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling 2008, SPM'08*(pp. 67-78) https://doi.org/10.1145/1364901.1364913

**Voronoi diagram computations for planar NURBS curves.** / Seong, Jun Kyung; Cohen, Elaine; Elber, Gershon.

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

*Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling 2008, SPM'08.*pp. 67-78, 2008 ACM Symposium on Solid and Physical Modeling 2008, SPM'08, Stony Brook, NY, United States, 08/6/2. https://doi.org/10.1145/1364901.1364913

}

TY - GEN

T1 - Voronoi diagram computations for planar NURBS curves

AU - Seong, Jun Kyung

AU - Cohen, Elaine

AU - Elber, Gershon

PY - 2008/12/17

Y1 - 2008/12/17

N2 - We present robust and efficient algorithms for computing Voronoi diagrams of planar freeform curves. Boundaries of the Voronoi diagram consist of portions of the bisector curves between pairs of planar curves. Our scheme is based on computing critical structures of the Voronoi diagrams, such as self-intersections and junction points of bisector curves. Since the geometric objects we consider in this paper are represented as freeform NURBS curves, we were able to reformulate the solution to the problem of computing those critical structures into the zero-set solutions of a system of nonlinear piecewise rational equations in parameter space. We present a new algorithm for computing error-bounded bisector curves using a distance surface constructed from error-bounded offset approximations of planar curves. This error-bounded algorithm is fast and produces bisector curves that are correct both in topology and geometry. Once bisectors are computed, both local and global self-intersections of the bisector curves are located and trimmed away by solving a system of three piecewise rational equations in three variables. Further, our method computes junction points at which three or more trimmed bisector curves intersect by transforming them into the solutions to a system of piecewise rational equations in the merged parameter space of the planar curves. The bisectors are trimmed at those self-intersection and global junction points. The Voronoi diagram is then computed from the trimmed bisectors using a pruning algorithm. We demonstrate the effectiveness of our approach with several experimental results.

AB - We present robust and efficient algorithms for computing Voronoi diagrams of planar freeform curves. Boundaries of the Voronoi diagram consist of portions of the bisector curves between pairs of planar curves. Our scheme is based on computing critical structures of the Voronoi diagrams, such as self-intersections and junction points of bisector curves. Since the geometric objects we consider in this paper are represented as freeform NURBS curves, we were able to reformulate the solution to the problem of computing those critical structures into the zero-set solutions of a system of nonlinear piecewise rational equations in parameter space. We present a new algorithm for computing error-bounded bisector curves using a distance surface constructed from error-bounded offset approximations of planar curves. This error-bounded algorithm is fast and produces bisector curves that are correct both in topology and geometry. Once bisectors are computed, both local and global self-intersections of the bisector curves are located and trimmed away by solving a system of three piecewise rational equations in three variables. Further, our method computes junction points at which three or more trimmed bisector curves intersect by transforming them into the solutions to a system of piecewise rational equations in the merged parameter space of the planar curves. The bisectors are trimmed at those self-intersection and global junction points. The Voronoi diagram is then computed from the trimmed bisectors using a pruning algorithm. We demonstrate the effectiveness of our approach with several experimental results.

KW - Bisectors

KW - Freeform curves

KW - Junction points

KW - Problem reduction scheme

KW - Self-intersections

KW - Voronoi diagram

KW - Zero-set

UR - http://www.scopus.com/inward/record.url?scp=57549104195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57549104195&partnerID=8YFLogxK

U2 - 10.1145/1364901.1364913

DO - 10.1145/1364901.1364913

M3 - Conference contribution

SP - 67

EP - 78

BT - Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling 2008, SPM'08

ER -