## Abstract

We present a new incremental algorithm for constructing the union of n triangles in the plane. In our experiments, the new algorithm, which we call the Disjoint-Cover (DC) algorithm, performs significantly better than the standard randomized incremental construction (RIC) of the union. Our algorithm is rather hard to analyze rigorously, but we provide an initial such analysis, which yields an upper bound on its performance that is expressed in terms of the expected cost of the RIC algorithm. Our approach and analysis generalize verbatim to the construction of the union of other objects in the plane, and, with slight modifications, to three dimensions. We present experiments with a software implementation of our algorithm using the Cgal library of geometric algorithms.

Original language | English |
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Pages (from-to) | 63-85 |

Number of pages | 23 |

Journal | Computational Geometry: Theory and Applications |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2004 |

Externally published | Yes |

### Bibliographical note

Funding Information:✩ Work reported in this paper has been supported in part by the IST Programme of the EU as Shared-cost RTD (FET Open) Projects under Contract No IST-2000-26473 (ECG—Effective Computational Geometry for Curves and Surfaces) and No IST-2001-39250 (MOVIE—Motion Planning in Virtual Environments), by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), and by the Hermann Minkowski– Minerva Center for Geometry at Tel Aviv University. Micha Sharir has also been supported by NSF Grants CCR-97-32101 and CCR-00-98246, and by a grant from the US–Israeli Binational Science Foundation. * Corresponding author.

## Keywords

- Algorithmic engineering
- Arrangements
- Exact computing
- Randomization
- Union of geometric objects