### Abstract

The star unfolding of a convex polytope with respect to a point x on its surface 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: it 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 conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: • The construction of the ridge tree (in preparation for shortest-path queries, for instance) can be achieved by an especially simple O(n^{2}) algorithm. This is no worst-case complexity improvement, but a considerable simplification nonetheless. • The exact set of all shortest-path "edge sequences" on a polytope can be found by an algorithm considerably simpler than was known previously, with a time improvement of roughly a factor of n over the old bound of O(n^{7} log n). • The geodesic diameter of a polygon can be found in O(n^{9} log n) time, an improvement of the previous best O(n^{10}) algorithm. Our results suggest conjectures on "unfoldings" of general convex surfaces.

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

Pages (from-to) | 219-250 |

Number of pages | 32 |

Journal | Discrete and Computational Geometry |

Volume | 8 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1992 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Discrete and Computational Geometry*,

*8*(1), 219-250. https://doi.org/10.1007/BF02293047

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

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 8, no. 1, pp. 219-250. https://doi.org/10.1007/BF02293047

}

TY - JOUR

T1 - Nonoverlap of the star unfolding

AU - Aronov, Boris

AU - O'Rourke, Joseph

PY - 1992/12

Y1 - 1992/12

N2 - The star unfolding of a convex polytope with respect to a point x on its surface 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: it 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 conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: • The construction of the ridge tree (in preparation for shortest-path queries, for instance) can be achieved by an especially simple O(n2) algorithm. This is no worst-case complexity improvement, but a considerable simplification nonetheless. • The exact set of all shortest-path "edge sequences" on a polytope can be found by an algorithm considerably simpler than was known previously, with a time improvement of roughly a factor of n over the old bound of O(n7 log n). • The geodesic diameter of a polygon can be found in O(n9 log n) time, an improvement of the previous best O(n10) algorithm. Our results suggest conjectures on "unfoldings" of general convex surfaces.

AB - The star unfolding of a convex polytope with respect to a point x on its surface 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: it 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 conceptual simplification of several algorithms concerned with shortest paths on polytopes, and sometimes a worst-case complexity improvement as well: • The construction of the ridge tree (in preparation for shortest-path queries, for instance) can be achieved by an especially simple O(n2) algorithm. This is no worst-case complexity improvement, but a considerable simplification nonetheless. • The exact set of all shortest-path "edge sequences" on a polytope can be found by an algorithm considerably simpler than was known previously, with a time improvement of roughly a factor of n over the old bound of O(n7 log n). • The geodesic diameter of a polygon can be found in O(n9 log n) time, an improvement of the previous best O(n10) algorithm. Our results suggest conjectures on "unfoldings" of general convex surfaces.

UR - http://www.scopus.com/inward/record.url?scp=21144464875&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21144464875&partnerID=8YFLogxK

U2 - 10.1007/BF02293047

DO - 10.1007/BF02293047

M3 - Article

VL - 8

SP - 219

EP - 250

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -