### Abstract

Solutions of simple motion planning problems inoling collections of orthogonal rectangles in the plane, and orthogonal boxes in 3-dimensional space are described. Proofs of seeral theorems regarding collision-free translation properties of these objects are deried. A new elementary simple proof is gien that for each quadrant in the plane, eery collection of orthogonal rectangles admits precisely one ordering that is alid for collision-free translation of the rectangles in eery fixed direction contained in that quadrant. In addition, it is proed that for eery configuration of n greater than 3 orthogonal rectangles in the plane, at least four of them hae the property that each can be translated independently to infinity in some direction, without disturbing the other n-1 rectangles. The proofs are elementary, and therefore suitable for motiating undergraduate computer science students in courses on discrete mathematics. A list of more challenging adanced problems is also proided.

Original language | English (US) |
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Title of host publication | 2018 International Conference on Control and Robots, ICCR 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 73-77 |

Number of pages | 5 |

ISBN (Electronic) | 9781538682432 |

DOIs | |

State | Published - Nov 13 2018 |

Event | 2018 International Conference on Control and Robots, ICCR 2018 - Hong Kong, Hong Kong Duration: Sep 15 2018 → Sep 17 2018 |

### Other

Other | 2018 International Conference on Control and Robots, ICCR 2018 |
---|---|

Country | Hong Kong |

City | Hong Kong |

Period | 9/15/18 → 9/17/18 |

### Fingerprint

### Keywords

- algorithms
- axis-parallell rectangles
- collision aoidance
- combinatorial geometry
- computational geometry
- discrete mathematics
- robotics
- spatial motion planning

### ASJC Scopus subject areas

- Mechanical Engineering
- Control and Optimization
- Artificial Intelligence

### Cite this

*2018 International Conference on Control and Robots, ICCR 2018*(pp. 73-77). [8534492] Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ICCR.2018.8534492

**Motion Planning Problems with Boxes : An Introduction for Undergraduate Courses in Discrete Mathematics.** / Toussaint, Godfried.

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

*2018 International Conference on Control and Robots, ICCR 2018.*, 8534492, Institute of Electrical and Electronics Engineers Inc., pp. 73-77, 2018 International Conference on Control and Robots, ICCR 2018, Hong Kong, Hong Kong, 9/15/18. https://doi.org/10.1109/ICCR.2018.8534492

}

TY - GEN

T1 - Motion Planning Problems with Boxes

T2 - An Introduction for Undergraduate Courses in Discrete Mathematics

AU - Toussaint, Godfried

PY - 2018/11/13

Y1 - 2018/11/13

N2 - Solutions of simple motion planning problems inoling collections of orthogonal rectangles in the plane, and orthogonal boxes in 3-dimensional space are described. Proofs of seeral theorems regarding collision-free translation properties of these objects are deried. A new elementary simple proof is gien that for each quadrant in the plane, eery collection of orthogonal rectangles admits precisely one ordering that is alid for collision-free translation of the rectangles in eery fixed direction contained in that quadrant. In addition, it is proed that for eery configuration of n greater than 3 orthogonal rectangles in the plane, at least four of them hae the property that each can be translated independently to infinity in some direction, without disturbing the other n-1 rectangles. The proofs are elementary, and therefore suitable for motiating undergraduate computer science students in courses on discrete mathematics. A list of more challenging adanced problems is also proided.

AB - Solutions of simple motion planning problems inoling collections of orthogonal rectangles in the plane, and orthogonal boxes in 3-dimensional space are described. Proofs of seeral theorems regarding collision-free translation properties of these objects are deried. A new elementary simple proof is gien that for each quadrant in the plane, eery collection of orthogonal rectangles admits precisely one ordering that is alid for collision-free translation of the rectangles in eery fixed direction contained in that quadrant. In addition, it is proed that for eery configuration of n greater than 3 orthogonal rectangles in the plane, at least four of them hae the property that each can be translated independently to infinity in some direction, without disturbing the other n-1 rectangles. The proofs are elementary, and therefore suitable for motiating undergraduate computer science students in courses on discrete mathematics. A list of more challenging adanced problems is also proided.

KW - algorithms

KW - axis-parallell rectangles

KW - collision aoidance

KW - combinatorial geometry

KW - computational geometry

KW - discrete mathematics

KW - robotics

KW - spatial motion planning

UR - http://www.scopus.com/inward/record.url?scp=85056483764&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056483764&partnerID=8YFLogxK

U2 - 10.1109/ICCR.2018.8534492

DO - 10.1109/ICCR.2018.8534492

M3 - Conference contribution

SP - 73

EP - 77

BT - 2018 International Conference on Control and Robots, ICCR 2018

PB - Institute of Electrical and Electronics Engineers Inc.

ER -