### Abstract

We present a data structure for ray-shooting queries in a set of convex fat polyhedra of total complexity n in ℝ^{3}. The data structure uses O(n^{2+ε}) storage and preprocessing time, and queries can be answered in O(log^{2} n) time. A trade-off between storage and query time is also possible: for any m with n < m < n^{2}, we can construct a structure that uses O(m^{1+ε}) storage and preprocessing time such that queries take O((n/√m) log^{2} n) time. We also describe a data structure for simplex intersection queries in a set of n convex fat constant-complexity polyhedra in ℝ^{3}. For any m with n< m < n^{2}, we can construct a structure that uses O(m^{1+ε}) storage and preprocessing time such that all polyhedra intersecting a query simplex can be reported in O((n/m^{1/3}) log n + k) time, where k is the number of answers.

Original language | English (US) |
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Title of host publication | Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06 |

Pages | 88-94 |

Number of pages | 7 |

Volume | 2006 |

State | Published - 2006 |

Event | 22nd Annual Symposium on Computational Geometry 2006, SCG'06 - Sedona, AZ, United States Duration: Jun 5 2006 → Jun 7 2006 |

### Other

Other | 22nd Annual Symposium on Computational Geometry 2006, SCG'06 |
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Country | United States |

City | Sedona, AZ |

Period | 6/5/06 → 6/7/06 |

### Fingerprint

### Keywords

- Computational geometry
- Fat objects
- Intersection searching
- Ray shooting

### ASJC Scopus subject areas

- Chemical Health and Safety
- Software
- Safety, Risk, Reliability and Quality
- Geometry and Topology

### Cite this

*Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06*(Vol. 2006, pp. 88-94)

**Ray shooting and intersection searching amidst fat convex polyhedra in 3-space.** / Aronov, Boris; De Berg, Mark; Gray, Chris.

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

*Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06.*vol. 2006, pp. 88-94, 22nd Annual Symposium on Computational Geometry 2006, SCG'06, Sedona, AZ, United States, 6/5/06.

}

TY - GEN

T1 - Ray shooting and intersection searching amidst fat convex polyhedra in 3-space

AU - Aronov, Boris

AU - De Berg, Mark

AU - Gray, Chris

PY - 2006

Y1 - 2006

N2 - We present a data structure for ray-shooting queries in a set of convex fat polyhedra of total complexity n in ℝ3. The data structure uses O(n2+ε) storage and preprocessing time, and queries can be answered in O(log2 n) time. A trade-off between storage and query time is also possible: for any m with n < m < n2, we can construct a structure that uses O(m1+ε) storage and preprocessing time such that queries take O((n/√m) log2 n) time. We also describe a data structure for simplex intersection queries in a set of n convex fat constant-complexity polyhedra in ℝ3. For any m with n< m < n2, we can construct a structure that uses O(m1+ε) storage and preprocessing time such that all polyhedra intersecting a query simplex can be reported in O((n/m1/3) log n + k) time, where k is the number of answers.

AB - We present a data structure for ray-shooting queries in a set of convex fat polyhedra of total complexity n in ℝ3. The data structure uses O(n2+ε) storage and preprocessing time, and queries can be answered in O(log2 n) time. A trade-off between storage and query time is also possible: for any m with n < m < n2, we can construct a structure that uses O(m1+ε) storage and preprocessing time such that queries take O((n/√m) log2 n) time. We also describe a data structure for simplex intersection queries in a set of n convex fat constant-complexity polyhedra in ℝ3. For any m with n< m < n2, we can construct a structure that uses O(m1+ε) storage and preprocessing time such that all polyhedra intersecting a query simplex can be reported in O((n/m1/3) log n + k) time, where k is the number of answers.

KW - Computational geometry

KW - Fat objects

KW - Intersection searching

KW - Ray shooting

UR - http://www.scopus.com/inward/record.url?scp=33748081122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748081122&partnerID=8YFLogxK

M3 - Conference contribution

SN - 1595933409

SN - 9781595933409

VL - 2006

SP - 88

EP - 94

BT - Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06

ER -