### Abstract

Given a bounded open subset Ω of the plane whose boundary is the union of finitely many polygons, and a real number d > 0, a manifold FP (the [free placements]) may be defined as the set of placements of a closed oriented line‐segment B (a [ladder]) of length d inside Ω. FP is a three‐dimensional manifold. A [Voronoi complex] in this manifold, a two‐dimensional cell complex, is defined by analogy with the classical geometric construction in the plane; within this complex a one‐dimensional subcomplex N, called the skeleton, is defined. It is shown that every component of FP contains a unique component of N, and canonical motions are given to move the ladder to placements within N. In this way, general motion planning is reduced to searching in a suitable representation of N as a (combinatorial) graph. Efficient construction of N is described in a companion paper.

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

Pages (from-to) | 423-483 |

Number of pages | 61 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 39 |

Issue number | 4 |

DOIs | |

State | Published - 1986 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*39*(4), 423-483. https://doi.org/10.1002/cpa.3160390402

**Generalized voronoi diagrams for moving a ladder. I : Topological analysis.** / Ó'Dunlaing, Colm; Sharir, Micha; Yap, Chee K.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 39, no. 4, pp. 423-483. https://doi.org/10.1002/cpa.3160390402

}

TY - JOUR

T1 - Generalized voronoi diagrams for moving a ladder. I

T2 - Topological analysis

AU - Ó'Dunlaing, Colm

AU - Sharir, Micha

AU - Yap, Chee K.

PY - 1986

Y1 - 1986

N2 - Given a bounded open subset Ω of the plane whose boundary is the union of finitely many polygons, and a real number d > 0, a manifold FP (the [free placements]) may be defined as the set of placements of a closed oriented line‐segment B (a [ladder]) of length d inside Ω. FP is a three‐dimensional manifold. A [Voronoi complex] in this manifold, a two‐dimensional cell complex, is defined by analogy with the classical geometric construction in the plane; within this complex a one‐dimensional subcomplex N, called the skeleton, is defined. It is shown that every component of FP contains a unique component of N, and canonical motions are given to move the ladder to placements within N. In this way, general motion planning is reduced to searching in a suitable representation of N as a (combinatorial) graph. Efficient construction of N is described in a companion paper.

AB - Given a bounded open subset Ω of the plane whose boundary is the union of finitely many polygons, and a real number d > 0, a manifold FP (the [free placements]) may be defined as the set of placements of a closed oriented line‐segment B (a [ladder]) of length d inside Ω. FP is a three‐dimensional manifold. A [Voronoi complex] in this manifold, a two‐dimensional cell complex, is defined by analogy with the classical geometric construction in the plane; within this complex a one‐dimensional subcomplex N, called the skeleton, is defined. It is shown that every component of FP contains a unique component of N, and canonical motions are given to move the ladder to placements within N. In this way, general motion planning is reduced to searching in a suitable representation of N as a (combinatorial) graph. Efficient construction of N is described in a companion paper.

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UR - http://www.scopus.com/inward/citedby.url?scp=84990619101&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160390402

DO - 10.1002/cpa.3160390402

M3 - Article

AN - SCOPUS:84990619101

VL - 39

SP - 423

EP - 483

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 4

ER -