### Abstract

The Voronoi diagram of a set of geometric entities on a plane, such as points, line segments, or arcs, is a collection of Voronoi polygons associated with each entity, where the Voronoi polygon of an entity is a set of points which are closer to the associated entity than any other entity. A Voronoi diagram is one of the most fundamental geometrical constructs, and it is well known for its theoretical elegance and the wealth of applications. Various geometric problems can be solved with the aid of Voronoi diagrams. The paper discusses an algorithm to construct the Voronoi diagram of the interior of a simple polygon which consists of simple curves such as line segments as well as arcs in a plane with O(N log N) time complexity by the use of a divide-and-conquer scheme. Particular emphasis is placed on the parameterization of bisectors using a rational quadratic Bézier curve representation which unifies four different bisector cases.

Original language | English |
---|---|

Pages (from-to) | 605-614 |

Number of pages | 10 |

Journal | Computer-Aided Design |

Volume | 27 |

Issue number | 8 |

DOIs | |

State | Published - 1995 Jan 1 |

### Fingerprint

### Keywords

- curves
- rational polynomials
- Voronoi diagrams

### Cite this

*Computer-Aided Design*,

*27*(8), 605-614. https://doi.org/10.1016/0010-4485(95)99797-C

}

*Computer-Aided Design*, vol. 27, no. 8, pp. 605-614. https://doi.org/10.1016/0010-4485(95)99797-C

**Representing the Voronoi diagram of a simple polygon using rational quadratic Bézier curves.** / Kim, Deok-Soo; Hwang, Il Kyu; Park, Bum Joo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Representing the Voronoi diagram of a simple polygon using rational quadratic Bézier curves

AU - Kim, Deok-Soo

AU - Hwang, Il Kyu

AU - Park, Bum Joo

PY - 1995/1/1

Y1 - 1995/1/1

N2 - The Voronoi diagram of a set of geometric entities on a plane, such as points, line segments, or arcs, is a collection of Voronoi polygons associated with each entity, where the Voronoi polygon of an entity is a set of points which are closer to the associated entity than any other entity. A Voronoi diagram is one of the most fundamental geometrical constructs, and it is well known for its theoretical elegance and the wealth of applications. Various geometric problems can be solved with the aid of Voronoi diagrams. The paper discusses an algorithm to construct the Voronoi diagram of the interior of a simple polygon which consists of simple curves such as line segments as well as arcs in a plane with O(N log N) time complexity by the use of a divide-and-conquer scheme. Particular emphasis is placed on the parameterization of bisectors using a rational quadratic Bézier curve representation which unifies four different bisector cases.

AB - The Voronoi diagram of a set of geometric entities on a plane, such as points, line segments, or arcs, is a collection of Voronoi polygons associated with each entity, where the Voronoi polygon of an entity is a set of points which are closer to the associated entity than any other entity. A Voronoi diagram is one of the most fundamental geometrical constructs, and it is well known for its theoretical elegance and the wealth of applications. Various geometric problems can be solved with the aid of Voronoi diagrams. The paper discusses an algorithm to construct the Voronoi diagram of the interior of a simple polygon which consists of simple curves such as line segments as well as arcs in a plane with O(N log N) time complexity by the use of a divide-and-conquer scheme. Particular emphasis is placed on the parameterization of bisectors using a rational quadratic Bézier curve representation which unifies four different bisector cases.

KW - curves

KW - rational polynomials

KW - Voronoi diagrams

UR - http://www.scopus.com/inward/record.url?scp=0029349763&partnerID=8YFLogxK

U2 - 10.1016/0010-4485(95)99797-C

DO - 10.1016/0010-4485(95)99797-C

M3 - Article

AN - SCOPUS:0029349763

VL - 27

SP - 605

EP - 614

JO - CAD Computer Aided Design

JF - CAD Computer Aided Design

SN - 0010-4485

IS - 8

ER -