### Abstract

Given a set of points P = { p1, p2, · · ·, pn} in three dimensions, the width of P, W(P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in 0(n log n + I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and in the worst case I = Ω(n2). For convex polyhedra, the time complexity becomes 0(n + I). If P is a set of points in the plane, the complexity can be reduced to 0(n log n). Finally, for simple polygons linear time suffices.

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

Pages (from-to) | 761-765 |

Number of pages | 5 |

Journal | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Volume | 10 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 1988 |

### Fingerprint

### Keywords

- Algorithms
- antipodal pairs
- artificial intelligence
- computational geometry
- convex hull
- geometric complexity
- geometric transforms
- image processing
- minimax approximating line
- minimax approximating plane
- pattern recognition
- rotating calipers
- width

### ASJC Scopus subject areas

- Software
- Computer Vision and Pattern Recognition
- Computational Theory and Mathematics
- Artificial Intelligence
- Applied Mathematics

### Cite this

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,

*10*(5), 761-765. https://doi.org/10.1109/34.6790

**Computing the Width of a Set.** / Houle, Michael E.; Toussaint, Godfried.

Research output: Contribution to journal › Article

*IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 10, no. 5, pp. 761-765. https://doi.org/10.1109/34.6790

}

TY - JOUR

T1 - Computing the Width of a Set

AU - Houle, Michael E.

AU - Toussaint, Godfried

PY - 1988/1/1

Y1 - 1988/1/1

N2 - Given a set of points P = { p1, p2, · · ·, pn} in three dimensions, the width of P, W(P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in 0(n log n + I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and in the worst case I = Ω(n2). For convex polyhedra, the time complexity becomes 0(n + I). If P is a set of points in the plane, the complexity can be reduced to 0(n log n). Finally, for simple polygons linear time suffices.

AB - Given a set of points P = { p1, p2, · · ·, pn} in three dimensions, the width of P, W(P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in 0(n log n + I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and in the worst case I = Ω(n2). For convex polyhedra, the time complexity becomes 0(n + I). If P is a set of points in the plane, the complexity can be reduced to 0(n log n). Finally, for simple polygons linear time suffices.

KW - Algorithms

KW - antipodal pairs

KW - artificial intelligence

KW - computational geometry

KW - convex hull

KW - geometric complexity

KW - geometric transforms

KW - image processing

KW - minimax approximating line

KW - minimax approximating plane

KW - pattern recognition

KW - rotating calipers

KW - width

UR - http://www.scopus.com/inward/record.url?scp=0024069812&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024069812&partnerID=8YFLogxK

U2 - 10.1109/34.6790

DO - 10.1109/34.6790

M3 - Article

VL - 10

SP - 761

EP - 765

JO - IEEE Transactions on Pattern Analysis and Machine Intelligence

JF - IEEE Transactions on Pattern Analysis and Machine Intelligence

SN - 0162-8828

IS - 5

ER -