### Abstract

We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the arrangement of the convolution, as is the case for the kinetic convolution. The monotonic crossing number is bounded by the kinetic crossing number, and also by the maximum number of intersecting pairs of monotone boundary chains, which is typically much smaller. We give a Minkowski sum algorithm based on the monotonic convolution. The running time is O(s + nα(n) log(n) + m^{2}), versus O(s + n^{2}) for the kinetic algorithm, with s the input size and with n and m the number of segments in the kinetic and monotonic convolutions. For inputs with a bounded number of turning points and inflection points, the running time is O(sα(s) log s), versus Ω(s^{2}) for the kinetic algorithm. The monotonic convolution is 37% smaller than the kinetic convolution and its arrangement is 62% smaller based on 21 test pairs.

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

Pages (from-to) | 383-396 |

Number of pages | 14 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2007 |

### Fingerprint

### Keywords

- Convolution
- Kinetic framework
- Minkowski sum

### ASJC Scopus subject areas

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

### Cite this

*International Journal of Computational Geometry and Applications*,

*17*(4), 383-396. https://doi.org/10.1142/S0218195907002392

**A monotonic convolution for Minkowski sums.** / Milenkovic, Victor; Sacks, Elisha.

Research output: Contribution to journal › Article

*International Journal of Computational Geometry and Applications*, vol. 17, no. 4, pp. 383-396. https://doi.org/10.1142/S0218195907002392

}

TY - JOUR

T1 - A monotonic convolution for Minkowski sums

AU - Milenkovic, Victor

AU - Sacks, Elisha

PY - 2007/8

Y1 - 2007/8

N2 - We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the arrangement of the convolution, as is the case for the kinetic convolution. The monotonic crossing number is bounded by the kinetic crossing number, and also by the maximum number of intersecting pairs of monotone boundary chains, which is typically much smaller. We give a Minkowski sum algorithm based on the monotonic convolution. The running time is O(s + nα(n) log(n) + m2), versus O(s + n2) for the kinetic algorithm, with s the input size and with n and m the number of segments in the kinetic and monotonic convolutions. For inputs with a bounded number of turning points and inflection points, the running time is O(sα(s) log s), versus Ω(s2) for the kinetic algorithm. The monotonic convolution is 37% smaller than the kinetic convolution and its arrangement is 62% smaller based on 21 test pairs.

AB - We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the arrangement of the convolution, as is the case for the kinetic convolution. The monotonic crossing number is bounded by the kinetic crossing number, and also by the maximum number of intersecting pairs of monotone boundary chains, which is typically much smaller. We give a Minkowski sum algorithm based on the monotonic convolution. The running time is O(s + nα(n) log(n) + m2), versus O(s + n2) for the kinetic algorithm, with s the input size and with n and m the number of segments in the kinetic and monotonic convolutions. For inputs with a bounded number of turning points and inflection points, the running time is O(sα(s) log s), versus Ω(s2) for the kinetic algorithm. The monotonic convolution is 37% smaller than the kinetic convolution and its arrangement is 62% smaller based on 21 test pairs.

KW - Convolution

KW - Kinetic framework

KW - Minkowski sum

UR - http://www.scopus.com/inward/record.url?scp=34548103270&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548103270&partnerID=8YFLogxK

U2 - 10.1142/S0218195907002392

DO - 10.1142/S0218195907002392

M3 - Article

AN - SCOPUS:34548103270

VL - 17

SP - 383

EP - 396

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

SN - 0218-1959

IS - 4

ER -