### Abstract

One useful generalization of the convex hull of a set S of points is the ε-strongly convex δ-hull. It is defined to be a convex polygon P with vertices taken from S such that no point in S lies farther than δ outside P and such that even if the vertices of P are perturbed by as much as ε, P remains convex. It was an open question as to whether an ε-strongly convex O(ε)-hull existed for all positive ε. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an ε-strongly convex O(ε + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded arithmetic convex hull algorithm which guarantees a convex output and which has error independent of n.

Original language | English (US) |
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Title of host publication | Proc Sixth Annu Symp Comput Geom |

Publisher | Publ by ACM |

Pages | 235-243 |

Number of pages | 9 |

ISBN (Print) | 0897913620 |

State | Published - 1990 |

Externally published | Yes |

Event | Proceedings of the Sixth Annual Symposium on Computational Geometry - Berkeley, CA, USA Duration: Jun 6 1990 → Jun 8 1990 |

### Other

Other | Proceedings of the Sixth Annual Symposium on Computational Geometry |
---|---|

City | Berkeley, CA, USA |

Period | 6/6/90 → 6/8/90 |

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proc Sixth Annu Symp Comput Geom*(pp. 235-243). Publ by ACM.

**Constructing strongly convex hulls using exact or rounded arithmetic.** / Li, Zhenyu; Milenkovic, Victor.

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

*Proc Sixth Annu Symp Comput Geom.*Publ by ACM, pp. 235-243, Proceedings of the Sixth Annual Symposium on Computational Geometry, Berkeley, CA, USA, 6/6/90.

}

TY - GEN

T1 - Constructing strongly convex hulls using exact or rounded arithmetic

AU - Li, Zhenyu

AU - Milenkovic, Victor

PY - 1990

Y1 - 1990

N2 - One useful generalization of the convex hull of a set S of points is the ε-strongly convex δ-hull. It is defined to be a convex polygon P with vertices taken from S such that no point in S lies farther than δ outside P and such that even if the vertices of P are perturbed by as much as ε, P remains convex. It was an open question as to whether an ε-strongly convex O(ε)-hull existed for all positive ε. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an ε-strongly convex O(ε + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded arithmetic convex hull algorithm which guarantees a convex output and which has error independent of n.

AB - One useful generalization of the convex hull of a set S of points is the ε-strongly convex δ-hull. It is defined to be a convex polygon P with vertices taken from S such that no point in S lies farther than δ outside P and such that even if the vertices of P are perturbed by as much as ε, P remains convex. It was an open question as to whether an ε-strongly convex O(ε)-hull existed for all positive ε. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an ε-strongly convex O(ε + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded arithmetic convex hull algorithm which guarantees a convex output and which has error independent of n.

UR - http://www.scopus.com/inward/record.url?scp=0024983394&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024983394&partnerID=8YFLogxK

M3 - Conference contribution

SN - 0897913620

SP - 235

EP - 243

BT - Proc Sixth Annu Symp Comput Geom

PB - Publ by ACM

ER -