### Abstract

In 1994 Grünbaum showed that, given a point set S in R ^{3}, it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n ^{log6}n) expected time which constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al.

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

Pages (from-to) | 148-153 |

Number of pages | 6 |

Journal | Computational Geometry: Theory and Applications |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2013 |

### Fingerprint

### Keywords

- Convex hull
- Gift wrapping
- Serpentine
- Tetrahedralization

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Science Applications
- Computational Mathematics
- Control and Optimization
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*46*(2), 148-153. https://doi.org/10.1016/j.comgeo.2012.02.008

**Bounded-degree polyhedronization of point sets.** / Barequet, Gill; Benbernou, Nadia; Charlton, David; Demaine, Erik D.; Demaine, Martin L.; Ishaque, Mashhood; Lubiw, Anna; Schulz, André; Souvaine, Diane L.; Toussaint, Godfried; Winslow, Andrew.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 46, no. 2, pp. 148-153. https://doi.org/10.1016/j.comgeo.2012.02.008

}

TY - JOUR

T1 - Bounded-degree polyhedronization of point sets

AU - Barequet, Gill

AU - Benbernou, Nadia

AU - Charlton, David

AU - Demaine, Erik D.

AU - Demaine, Martin L.

AU - Ishaque, Mashhood

AU - Lubiw, Anna

AU - Schulz, André

AU - Souvaine, Diane L.

AU - Toussaint, Godfried

AU - Winslow, Andrew

PY - 2013/2/1

Y1 - 2013/2/1

N2 - In 1994 Grünbaum showed that, given a point set S in R 3, it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n log6n) expected time which constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al.

AB - In 1994 Grünbaum showed that, given a point set S in R 3, it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n log6n) expected time which constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al.

KW - Convex hull

KW - Gift wrapping

KW - Serpentine

KW - Tetrahedralization

UR - http://www.scopus.com/inward/record.url?scp=84867580919&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867580919&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2012.02.008

DO - 10.1016/j.comgeo.2012.02.008

M3 - Article

VL - 46

SP - 148

EP - 153

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 2

ER -