### Abstract

It is shown that a set of n spheres in 3-space, unlike convex polyhedra, exhibits the translation ordering property. Furthermore, such an ordering can be obtained in O(n log n) time. It is also proven that there always exist at least H spheres that can be translated to infinity without disturbing the others, where H is the number of spheres whose centers lie on the convex hull of the centers of the entire collection of spheres. In the worst case H equals minn, 4 and thus this strengthens the result of R. Dawson (1984). It also implies that at least H spheres can be identified with this property in O(n log n) time. Finally, for the two-dimensional problem, it is shown that for a given circle C//i in a collection of circles, all directions of allowable translations for C//i can be computed in O(n log n) time and, therefore, all circles that can be so translated individually without disturbing the others can be identified in O(n**2 log n) time. Some open problems are also discussed.

Original language | English (US) |
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Pages | 291-295 |

Number of pages | 5 |

State | Published - Dec 1 1985 |

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### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*SOME COLLISION AVOIDANCE PROBLEMS BETWEEN SPHERES.*. 291-295.

**SOME COLLISION AVOIDANCE PROBLEMS BETWEEN SPHERES.** / Toussaint, Godfried.

Research output: Contribution to conference › Paper

}

TY - CONF

T1 - SOME COLLISION AVOIDANCE PROBLEMS BETWEEN SPHERES.

AU - Toussaint, Godfried

PY - 1985/12/1

Y1 - 1985/12/1

N2 - It is shown that a set of n spheres in 3-space, unlike convex polyhedra, exhibits the translation ordering property. Furthermore, such an ordering can be obtained in O(n log n) time. It is also proven that there always exist at least H spheres that can be translated to infinity without disturbing the others, where H is the number of spheres whose centers lie on the convex hull of the centers of the entire collection of spheres. In the worst case H equals minn, 4 and thus this strengthens the result of R. Dawson (1984). It also implies that at least H spheres can be identified with this property in O(n log n) time. Finally, for the two-dimensional problem, it is shown that for a given circle C//i in a collection of circles, all directions of allowable translations for C//i can be computed in O(n log n) time and, therefore, all circles that can be so translated individually without disturbing the others can be identified in O(n**2 log n) time. Some open problems are also discussed.

AB - It is shown that a set of n spheres in 3-space, unlike convex polyhedra, exhibits the translation ordering property. Furthermore, such an ordering can be obtained in O(n log n) time. It is also proven that there always exist at least H spheres that can be translated to infinity without disturbing the others, where H is the number of spheres whose centers lie on the convex hull of the centers of the entire collection of spheres. In the worst case H equals minn, 4 and thus this strengthens the result of R. Dawson (1984). It also implies that at least H spheres can be identified with this property in O(n log n) time. Finally, for the two-dimensional problem, it is shown that for a given circle C//i in a collection of circles, all directions of allowable translations for C//i can be computed in O(n log n) time and, therefore, all circles that can be so translated individually without disturbing the others can be identified in O(n**2 log n) time. Some open problems are also discussed.

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M3 - Paper

SP - 291

EP - 295

ER -