### Abstract

The star unfolding of a convex polytope with respect to a point x is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: (1) It does not self-overlap: its boundary is a simple polygon. (2) The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit the conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: for constructing the ridge tree, for finding the exact set of all shortest-path "edge sequences," and for computing the geodesic diameter of a polytope. Our results suggest conjectures on "unfoldings" of general convex surfaces.

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

Title of host publication | Proceedings of the Annual Symposium on Computational Geometry |

Publisher | Association for Computing Machinery |

Pages | 105-114 |

Number of pages | 10 |

Volume | Part F129851 |

ISBN (Print) | 0897914260 |

DOIs | |

State | Published - Jun 1 1991 |

Event | 7th Annual Symposium on Computational Geometry, SCG 1991 - North Conway, United States Duration: Jun 10 1991 → Jun 12 1991 |

### Other

Other | 7th Annual Symposium on Computational Geometry, SCG 1991 |
---|---|

Country | United States |

City | North Conway |

Period | 6/10/91 → 6/12/91 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the Annual Symposium on Computational Geometry*(Vol. Part F129851, pp. 105-114). Association for Computing Machinery. https://doi.org/10.1145/109648.109660

**Nonoverlap of the star unfolding.** / Aronov, Boris; O'Rourke, Joseph.

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

*Proceedings of the Annual Symposium on Computational Geometry.*vol. Part F129851, Association for Computing Machinery, pp. 105-114, 7th Annual Symposium on Computational Geometry, SCG 1991, North Conway, United States, 6/10/91. https://doi.org/10.1145/109648.109660

}

TY - GEN

T1 - Nonoverlap of the star unfolding

AU - Aronov, Boris

AU - O'Rourke, Joseph

PY - 1991/6/1

Y1 - 1991/6/1

N2 - The star unfolding of a convex polytope with respect to a point x is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: (1) It does not self-overlap: its boundary is a simple polygon. (2) The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit the conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: for constructing the ridge tree, for finding the exact set of all shortest-path "edge sequences," and for computing the geodesic diameter of a polytope. Our results suggest conjectures on "unfoldings" of general convex surfaces.

AB - The star unfolding of a convex polytope with respect to a point x is obtained by cutting the surface along the shortest paths from x to every vertex, and flattening the surface on the plane. We establish two main properties of the star unfolding: (1) It does not self-overlap: its boundary is a simple polygon. (2) The ridge tree in the unfolding, which is the locus of points with more than one shortest path from x, is precisely the Voronoi diagram of the images of x, restricted to the unfolding. These two properties permit the conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: for constructing the ridge tree, for finding the exact set of all shortest-path "edge sequences," and for computing the geodesic diameter of a polytope. Our results suggest conjectures on "unfoldings" of general convex surfaces.

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U2 - 10.1145/109648.109660

DO - 10.1145/109648.109660

M3 - Conference contribution

SN - 0897914260

VL - Part F129851

SP - 105

EP - 114

BT - Proceedings of the Annual Symposium on Computational Geometry

PB - Association for Computing Machinery

ER -