### Abstract

Voronoi diagrams are extremely versatile Abstract: a data structure for many geometric applications. Computing this diagram "exactly" for a polyhedral set in 3-D has been a quest of computational geometers for over two decades; this quest is still unrealized. We will locate the difficulty in this quest, thanks to a recent result of Everett et al (2009). More generally, it points to the need for alternative computational models, and other notions of exactness. In this paper, we consider an alternative approach based on the well-known Subdivision Paradigm. A brief review of such algorithms for Voronoi diagrams is given. Our unique emphasis is the use of purely numerical primitives. We avoid exact (algebraic) primitives because (1) they are hard to implement correctly, and (2) they fail to take full advantage of the resolution-limited properties of subdivision.We encapsulate our numerical approach using the concept of soft primitives that conservatively converge to the exact ones in the limit. We illustrate our approach by designing the first purely numerical algorithm for the Voronoi complex of a non-degenerate polygonal set. We also discuss the critical role of filters in such algorithms. A preliminary version of our algorithm has been implemented.

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

Title of host publication | Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |

Pages | 2-16 |

Number of pages | 15 |

DOIs | |

State | Published - 2012 |

Event | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 - Piscataway, NJ, United States Duration: Jun 27 2012 → Jun 29 2012 |

### Other

Other | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
---|---|

Country | United States |

City | Piscataway, NJ |

Period | 6/27/12 → 6/29/12 |

### Fingerprint

### Keywords

- Filters
- Soft predicates
- Subdivision

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012*(pp. 2-16). [6257651] https://doi.org/10.1109/ISVD.2012.31

**Towards exact numerical Voronoi diagrams (invited talk).** / Yap, Chee K.; Sharma, Vikram; Lien, Jyh Ming.

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

*Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012.*, 6257651, pp. 2-16, 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012, Piscataway, NJ, United States, 6/27/12. https://doi.org/10.1109/ISVD.2012.31

}

TY - GEN

T1 - Towards exact numerical Voronoi diagrams (invited talk)

AU - Yap, Chee K.

AU - Sharma, Vikram

AU - Lien, Jyh Ming

PY - 2012

Y1 - 2012

N2 - Voronoi diagrams are extremely versatile Abstract: a data structure for many geometric applications. Computing this diagram "exactly" for a polyhedral set in 3-D has been a quest of computational geometers for over two decades; this quest is still unrealized. We will locate the difficulty in this quest, thanks to a recent result of Everett et al (2009). More generally, it points to the need for alternative computational models, and other notions of exactness. In this paper, we consider an alternative approach based on the well-known Subdivision Paradigm. A brief review of such algorithms for Voronoi diagrams is given. Our unique emphasis is the use of purely numerical primitives. We avoid exact (algebraic) primitives because (1) they are hard to implement correctly, and (2) they fail to take full advantage of the resolution-limited properties of subdivision.We encapsulate our numerical approach using the concept of soft primitives that conservatively converge to the exact ones in the limit. We illustrate our approach by designing the first purely numerical algorithm for the Voronoi complex of a non-degenerate polygonal set. We also discuss the critical role of filters in such algorithms. A preliminary version of our algorithm has been implemented.

AB - Voronoi diagrams are extremely versatile Abstract: a data structure for many geometric applications. Computing this diagram "exactly" for a polyhedral set in 3-D has been a quest of computational geometers for over two decades; this quest is still unrealized. We will locate the difficulty in this quest, thanks to a recent result of Everett et al (2009). More generally, it points to the need for alternative computational models, and other notions of exactness. In this paper, we consider an alternative approach based on the well-known Subdivision Paradigm. A brief review of such algorithms for Voronoi diagrams is given. Our unique emphasis is the use of purely numerical primitives. We avoid exact (algebraic) primitives because (1) they are hard to implement correctly, and (2) they fail to take full advantage of the resolution-limited properties of subdivision.We encapsulate our numerical approach using the concept of soft primitives that conservatively converge to the exact ones in the limit. We illustrate our approach by designing the first purely numerical algorithm for the Voronoi complex of a non-degenerate polygonal set. We also discuss the critical role of filters in such algorithms. A preliminary version of our algorithm has been implemented.

KW - Filters

KW - Soft predicates

KW - Subdivision

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U2 - 10.1109/ISVD.2012.31

DO - 10.1109/ISVD.2012.31

M3 - Conference contribution

SN - 9780769547244

SP - 2

EP - 16

BT - Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012

ER -