### Abstract

An algorithm and implementation is given for rotational polygon containment: given polygons P_{1}, P^{2}, P_{3},..., P_{k} and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C∈C of minimum area for which containment has a solution. Minimum enclosure algorithms are given for the following classes: 1) rectangles of fixed width, 2) scaled copies of a fixed convex polygon, 3) arbitrary rectangles. Containment and minimum enclosure are NP-hard (even in the purely translational case). The minimum enclosure is approximate: it bounds the minimum area between (1 - ε)A and A. Experiments are done to determine the largest practical value of k for both containment and minimum enclosure. Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles.

Original language | English (US) |
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Title of host publication | Proceedings of the Annual Symposium on Computational Geometry |

Publisher | ACM |

Pages | 1-8 |

Number of pages | 8 |

State | Published - 1998 |

Event | Proceedings of the 1998 14th Annual Symposium on Computational Geometry - Minneapolis, MN, USA Duration: Jun 7 1998 → Jun 10 1998 |

### Other

Other | Proceedings of the 1998 14th Annual Symposium on Computational Geometry |
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City | Minneapolis, MN, USA |

Period | 6/7/98 → 6/10/98 |

### Fingerprint

### ASJC Scopus subject areas

- Chemical Health and Safety
- Software
- Safety, Risk, Reliability and Quality
- Geometry and Topology

### Cite this

*Proceedings of the Annual Symposium on Computational Geometry*(pp. 1-8). ACM.

**Rotational polygon containment and minimum enclosure.** / Milenkovic, Victor.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Annual Symposium on Computational Geometry.*ACM, pp. 1-8, Proceedings of the 1998 14th Annual Symposium on Computational Geometry, Minneapolis, MN, USA, 6/7/98.

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TY - GEN

T1 - Rotational polygon containment and minimum enclosure

AU - Milenkovic, Victor

PY - 1998

Y1 - 1998

N2 - An algorithm and implementation is given for rotational polygon containment: given polygons P1, P2, P3,..., Pk and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C∈C of minimum area for which containment has a solution. Minimum enclosure algorithms are given for the following classes: 1) rectangles of fixed width, 2) scaled copies of a fixed convex polygon, 3) arbitrary rectangles. Containment and minimum enclosure are NP-hard (even in the purely translational case). The minimum enclosure is approximate: it bounds the minimum area between (1 - ε)A and A. Experiments are done to determine the largest practical value of k for both containment and minimum enclosure. Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles.

AB - An algorithm and implementation is given for rotational polygon containment: given polygons P1, P2, P3,..., Pk and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C∈C of minimum area for which containment has a solution. Minimum enclosure algorithms are given for the following classes: 1) rectangles of fixed width, 2) scaled copies of a fixed convex polygon, 3) arbitrary rectangles. Containment and minimum enclosure are NP-hard (even in the purely translational case). The minimum enclosure is approximate: it bounds the minimum area between (1 - ε)A and A. Experiments are done to determine the largest practical value of k for both containment and minimum enclosure. Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles.

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M3 - Conference contribution

SP - 1

EP - 8

BT - Proceedings of the Annual Symposium on Computational Geometry

PB - ACM

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