### Abstract

Consider n i.i.d. random lines in the plane defined by their slope and distance from the origin. The slope is uniformly distributed on [0, 2π] and independent of the distance R from the origin. These lines define a set I of n(n - 1)/2 intersection points. It was recently shown by Atallah and Ching and Lee that the cardinality of the convex hull of these intersection points is O(n), and they exhibited an O(n log n) time algorithm for computing such a convex hull. Let N_{ch} and N_{ol} be the number of points on the convex hull and outer layer of I, respectively. We show that there exist arrangements of lines in which N_{ol} = n(n - l)/2. We show that, nevertheless, both N_{ch} and N_{ol} have expected values O(1), and give bounds that are uniform over all distributions of R with 0 ≤ ER < ∞. These results lead to an algorithm for computing the convex hull of I in O(n log n) worst-case time and O(n) expected time under these conditions.

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

Pages (from-to) | 381-394 |

Number of pages | 14 |

Journal | Journal of Algorithms |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - May 1993 |

### ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Computational Theory and Mathematics

## Fingerprint Dive into the research topics of 'Convex Hulls for Random Lines'. Together they form a unique fingerprint.

## Cite this

*Journal of Algorithms*,

*14*(3), 381-394. https://doi.org/10.1006/jagm.1993.1020