### Abstract

We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by km log m with m the number of segments in the convolution and k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm.

Original language | English (US) |
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Pages (from-to) | 485-509 |

Number of pages | 25 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2010 |

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### Keywords

- Minkowski sum
- kinetic framework
- robust computational geometry

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics

### Cite this

*International Journal of Computational Geometry and Applications*,

*20*(4), 485-509. https://doi.org/10.1142/S0218195910003402