### Abstract

We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by Cgal. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning.
The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small amount of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations.
The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r 2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron.

Original language | English |
---|---|

Title of host publication | Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007) 8-10 October 2007, Eilat, Israel |

Editors | L. Arge, M. Hoffmann, E. Welzl |

Place of Publication | Berlin |

Publisher | Springer |

Pages | 669-680 |

ISBN (Print) | 978-3-540-75519-7 |

DOIs | |

Publication status | Published - 2007 |

Event | conference; ESA 2007, Eilat, Israel; 2007-10-08; 2007-10-10 - Duration: 8 Oct 2007 → 10 Oct 2007 |

### Publication series

Name | Lecture Notes in Computer Science |
---|---|

Volume | 4698 |

ISSN (Print) | 0302-9743 |

### Conference

Conference | conference; ESA 2007, Eilat, Israel; 2007-10-08; 2007-10-10 |
---|---|

Period | 8/10/07 → 10/10/07 |

Other | ESA 2007, Eilat, Israel |

## Fingerprint Dive into the research topics of 'Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces'. Together they form a unique fingerprint.

## Cite this

Hachenberger, P. (2007). Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces. In L. Arge, M. Hoffmann, & E. Welzl (Eds.),

*Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007) 8-10 October 2007, Eilat, Israel*(pp. 669-680). (Lecture Notes in Computer Science; Vol. 4698). Springer. https://doi.org/10.1007/978-3-540-75520-3_59