### Abstract

We study the space of free translations of a box amidst polyhedral obstacles with n vertices. We show that the combinatorial complexity of this space is O(n
^{2}α(n)), where α(n) is the inverse Ackermann function. Our bound is within an α(n) factor off the lower bound, and it constitutes an improvement of almost an order of magnitude over the best previously known (and naive) bound for this problem, O(n
^{3}). For the case of a convex polygon of fixed (constant) size translating in the same setting (namely, a two-dimensional polygon translating in three-dimensional space), we show a tight bound Θ(n
^{2}α(n)) on the complexity of the free space.

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

Pages (from-to) | 181-196 |

Number of pages | 16 |

Journal | Computational Geometry: Theory and Applications |

Volume | 9 |

Issue number | 3 |

State | Published - Feb 1998 |

### Fingerprint

### Keywords

- Computational geometry
- Minkowski sums
- Motion planning

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*9*(3), 181-196.

**Combinatorial complexity of translating a box in polyhedral 3-space.** / Halperin, Dan; Yap, Chee.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 9, no. 3, pp. 181-196.

}

TY - JOUR

T1 - Combinatorial complexity of translating a box in polyhedral 3-space

AU - Halperin, Dan

AU - Yap, Chee

PY - 1998/2

Y1 - 1998/2

N2 - We study the space of free translations of a box amidst polyhedral obstacles with n vertices. We show that the combinatorial complexity of this space is O(n 2α(n)), where α(n) is the inverse Ackermann function. Our bound is within an α(n) factor off the lower bound, and it constitutes an improvement of almost an order of magnitude over the best previously known (and naive) bound for this problem, O(n 3). For the case of a convex polygon of fixed (constant) size translating in the same setting (namely, a two-dimensional polygon translating in three-dimensional space), we show a tight bound Θ(n 2α(n)) on the complexity of the free space.

AB - We study the space of free translations of a box amidst polyhedral obstacles with n vertices. We show that the combinatorial complexity of this space is O(n 2α(n)), where α(n) is the inverse Ackermann function. Our bound is within an α(n) factor off the lower bound, and it constitutes an improvement of almost an order of magnitude over the best previously known (and naive) bound for this problem, O(n 3). For the case of a convex polygon of fixed (constant) size translating in the same setting (namely, a two-dimensional polygon translating in three-dimensional space), we show a tight bound Θ(n 2α(n)) on the complexity of the free space.

KW - Computational geometry

KW - Minkowski sums

KW - Motion planning

UR - http://www.scopus.com/inward/record.url?scp=0037764554&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037764554&partnerID=8YFLogxK

M3 - Article

VL - 9

SP - 181

EP - 196

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 3

ER -