### Abstract

Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an Ω(n log n) time lower bound for the problem. Our results imply that a k-angulation of a set of points can be achieved with the addition of at most k - 3 extra points within the same time bound. Finally, we present an experimental comparison between three quadrangulation algorithms which shows that the Spiraling Rotating Calipers (SRC) algorithm produces quadrangulations with the greatest number of convex quadrilaterals as well as those with the smallest difference between the average minimum and maximum angle over all quadrangles.

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

Pages (from-to) | 763-785 |

Number of pages | 23 |

Journal | Computer Aided Geometric Design |

Volume | 14 |

Issue number | 8 |

DOIs | |

State | Published - Jan 1 1997 |

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### ASJC Scopus subject areas

- Modeling and Simulation
- Automotive Engineering
- Aerospace Engineering
- Computer Graphics and Computer-Aided Design

### Cite this

**Characterizing and efficiently computing quadrangulations of planar point sets.** / Bose, Prosenjit; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Computer Aided Geometric Design*, vol. 14, no. 8, pp. 763-785. https://doi.org/10.1016/S0167-8396(97)00013-7

}

TY - JOUR

T1 - Characterizing and efficiently computing quadrangulations of planar point sets

AU - Bose, Prosenjit

AU - Toussaint, Godfried

PY - 1997/1/1

Y1 - 1997/1/1

N2 - Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an Ω(n log n) time lower bound for the problem. Our results imply that a k-angulation of a set of points can be achieved with the addition of at most k - 3 extra points within the same time bound. Finally, we present an experimental comparison between three quadrangulation algorithms which shows that the Spiraling Rotating Calipers (SRC) algorithm produces quadrangulations with the greatest number of convex quadrilaterals as well as those with the smallest difference between the average minimum and maximum angle over all quadrangles.

AB - Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an Ω(n log n) time lower bound for the problem. Our results imply that a k-angulation of a set of points can be achieved with the addition of at most k - 3 extra points within the same time bound. Finally, we present an experimental comparison between three quadrangulation algorithms which shows that the Spiraling Rotating Calipers (SRC) algorithm produces quadrangulations with the greatest number of convex quadrilaterals as well as those with the smallest difference between the average minimum and maximum angle over all quadrangles.

UR - http://www.scopus.com/inward/record.url?scp=0031245851&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031245851&partnerID=8YFLogxK

U2 - 10.1016/S0167-8396(97)00013-7

DO - 10.1016/S0167-8396(97)00013-7

M3 - Article

AN - SCOPUS:0031245851

VL - 14

SP - 763

EP - 785

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

SN - 0167-8396

IS - 8

ER -