### Abstract

One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. 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 e{open}-strongly convex O(e{open} + 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) |
---|---|

Pages (from-to) | 345-364 |

Number of pages | 20 |

Journal | Algorithmica (New York) |

Volume | 8 |

Issue number | 1-6 |

DOIs | |

State | Published - Dec 1992 |

Externally published | Yes |

### Fingerprint

### Keywords

- Approximate arithmetic
- Convex hulls
- Exact rational arithmetic
- Numerical analysis
- Robust geometry
- Strong and weak convexity

### ASJC Scopus subject areas

- Applied Mathematics
- Safety, Risk, Reliability and Quality
- Software
- Computer Graphics and Computer-Aided Design

### Cite this

*Algorithmica (New York)*,

*8*(1-6), 345-364. https://doi.org/10.1007/BF01758851

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

Research output: Contribution to journal › Article

*Algorithmica (New York)*, vol. 8, no. 1-6, pp. 345-364. https://doi.org/10.1007/BF01758851

}

TY - JOUR

T1 - Constructing strongly convex hulls using exact or rounded arithmetic

AU - Li, Zhenyu

AU - Milenkovic, Victor

PY - 1992/12

Y1 - 1992/12

N2 - One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. 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 e{open}-strongly convex O(e{open} + 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 n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. 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 e{open}-strongly convex O(e{open} + 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.

KW - Approximate arithmetic

KW - Convex hulls

KW - Exact rational arithmetic

KW - Numerical analysis

KW - Robust geometry

KW - Strong and weak convexity

UR - http://www.scopus.com/inward/record.url?scp=52449145571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52449145571&partnerID=8YFLogxK

U2 - 10.1007/BF01758851

DO - 10.1007/BF01758851

M3 - Article

VL - 8

SP - 345

EP - 364

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1-6

ER -