### Abstract

Given a set S of n ≥ 3 points in the plane (not all on a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the given points in S. For example, the shortest circuit through S is a simple polygon. In 1994, Grünbaum showed that an analogous theorem holds in R^{3}. More precisely, if S is a set of n ≥ 4 points in R^{3} (not all of which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (sphere-like) polyhedron P such that the vertices of P are precisely the given points in S. Grünbaum's constructive proof may yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are star-shaped, have Hamiltonian skeletons, and admit efficient point-location queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in O (n log n) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O (n^{1 + ε{lunate}}), for any ε{lunate} > 0.

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

Pages (from-to) | 42-54 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 156 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2008 |

### Fingerprint

### Keywords

- Computational geometry
- Polyhedra
- Triangulation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*156*(1), 42-54. https://doi.org/10.1016/j.dam.2007.08.033

**On polyhedra induced by point sets in space.** / Agarwal, Pankaj K.; Hurtado, Ferran; Toussaint, Godfried; Trias, Joan.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 156, no. 1, pp. 42-54. https://doi.org/10.1016/j.dam.2007.08.033

}

TY - JOUR

T1 - On polyhedra induced by point sets in space

AU - Agarwal, Pankaj K.

AU - Hurtado, Ferran

AU - Toussaint, Godfried

AU - Trias, Joan

PY - 2008/1/1

Y1 - 2008/1/1

N2 - Given a set S of n ≥ 3 points in the plane (not all on a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the given points in S. For example, the shortest circuit through S is a simple polygon. In 1994, Grünbaum showed that an analogous theorem holds in R3. More precisely, if S is a set of n ≥ 4 points in R3 (not all of which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (sphere-like) polyhedron P such that the vertices of P are precisely the given points in S. Grünbaum's constructive proof may yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are star-shaped, have Hamiltonian skeletons, and admit efficient point-location queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in O (n log n) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O (n1 + ε{lunate}), for any ε{lunate} > 0.

AB - Given a set S of n ≥ 3 points in the plane (not all on a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the given points in S. For example, the shortest circuit through S is a simple polygon. In 1994, Grünbaum showed that an analogous theorem holds in R3. More precisely, if S is a set of n ≥ 4 points in R3 (not all of which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (sphere-like) polyhedron P such that the vertices of P are precisely the given points in S. Grünbaum's constructive proof may yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are star-shaped, have Hamiltonian skeletons, and admit efficient point-location queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in O (n log n) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O (n1 + ε{lunate}), for any ε{lunate} > 0.

KW - Computational geometry

KW - Polyhedra

KW - Triangulation

UR - http://www.scopus.com/inward/record.url?scp=35648997008&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35648997008&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2007.08.033

DO - 10.1016/j.dam.2007.08.033

M3 - Article

VL - 156

SP - 42

EP - 54

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 1

ER -