### Abstract

Let S be a set of (possibly degenerate) triangles in R^{3} whose interiors are disjoint. A triangulation of R^{3} with respect to S, denoted by T(S), is a simplicial complex in which each face of T(S) is either disjoint from S or contained in a higher dimensional face of S. The line stabbing number of T(S) is the maximum number of tetrahedra of T(S) intersected by a segment that does not intersect any triangle of S. We investigate the line stabbing number of triangulations in several cases-when S is a set of points, when the triangles of 5 form the boundary of a convex or a nonconvex polyhedron, or when the triangles of S form the boundaries of k disjoint convex polyhedra. We prove almost tight worst-case upper and lower bounds on line stabbing numbers for these cases. We also estimate the number of tetrahedra necessary to guarantee low stabbing number.

Original language | English (US) |
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Title of host publication | Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995 |

Publisher | Association for Computing Machinery |

Pages | 267-276 |

Number of pages | 10 |

Volume | Part F129372 |

ISBN (Electronic) | 0897917243 |

DOIs | |

State | Published - Sep 1 1995 |

Event | 11th Annual Symposium on Computational Geometry, SCG 1995 - Vancouver, Canada Duration: Jun 5 1995 → Jun 7 1995 |

### Other

Other | 11th Annual Symposium on Computational Geometry, SCG 1995 |
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Country | Canada |

City | Vancouver |

Period | 6/5/95 → 6/7/95 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995*(Vol. Part F129372, pp. 267-276). Association for Computing Machinery. https://doi.org/10.1145/220279.220308

**Stabbing triangulations by lines in 3D.** / Agarwal, Pankaj K.; Aronov, Boris; Suri, Subhash.

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

*Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995.*vol. Part F129372, Association for Computing Machinery, pp. 267-276, 11th Annual Symposium on Computational Geometry, SCG 1995, Vancouver, Canada, 6/5/95. https://doi.org/10.1145/220279.220308

}

TY - GEN

T1 - Stabbing triangulations by lines in 3D

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - Suri, Subhash

PY - 1995/9/1

Y1 - 1995/9/1

N2 - Let S be a set of (possibly degenerate) triangles in R3 whose interiors are disjoint. A triangulation of R3 with respect to S, denoted by T(S), is a simplicial complex in which each face of T(S) is either disjoint from S or contained in a higher dimensional face of S. The line stabbing number of T(S) is the maximum number of tetrahedra of T(S) intersected by a segment that does not intersect any triangle of S. We investigate the line stabbing number of triangulations in several cases-when S is a set of points, when the triangles of 5 form the boundary of a convex or a nonconvex polyhedron, or when the triangles of S form the boundaries of k disjoint convex polyhedra. We prove almost tight worst-case upper and lower bounds on line stabbing numbers for these cases. We also estimate the number of tetrahedra necessary to guarantee low stabbing number.

AB - Let S be a set of (possibly degenerate) triangles in R3 whose interiors are disjoint. A triangulation of R3 with respect to S, denoted by T(S), is a simplicial complex in which each face of T(S) is either disjoint from S or contained in a higher dimensional face of S. The line stabbing number of T(S) is the maximum number of tetrahedra of T(S) intersected by a segment that does not intersect any triangle of S. We investigate the line stabbing number of triangulations in several cases-when S is a set of points, when the triangles of 5 form the boundary of a convex or a nonconvex polyhedron, or when the triangles of S form the boundaries of k disjoint convex polyhedra. We prove almost tight worst-case upper and lower bounds on line stabbing numbers for these cases. We also estimate the number of tetrahedra necessary to guarantee low stabbing number.

UR - http://www.scopus.com/inward/record.url?scp=0040172907&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040172907&partnerID=8YFLogxK

U2 - 10.1145/220279.220308

DO - 10.1145/220279.220308

M3 - Conference contribution

VL - Part F129372

SP - 267

EP - 276

BT - Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995

PB - Association for Computing Machinery

ER -