Characterizing and efficiently computing quadrangulations of planar point sets

Prosenjit Bose, Godfried Toussaint

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)763-785
Number of pages23
JournalComputer Aided Geometric Design
Volume14
Issue number8
DOIs
StatePublished - Jan 1 1997

Fingerprint

Quadrangulation
Point Sets
Computing
Face
Subdivision
Odd number
Collinear
Extreme Points
Convex Hull
Set of points
Rotating
Lower bound
If and only if
Imply
Angle

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.

In: Computer Aided Geometric Design, Vol. 14, No. 8, 01.01.1997, p. 763-785.

Research output: Contribution to journalArticle

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