## Abstract

In a generalized intersection searching problem, a set S of colored geometric objects is to be preprocessed so that, given a query object q, the distinct colors of the objects of S that are intersected by q can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and have many applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the color classes satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving objects that are not necessarily iso-oriented. These problems include: generalized halfspace range searching in ℝ^{d}, for any fixed d ≥ 2, and segment intersection searching, triangle stabbing, and triangle range searching in ℝ2 for certain classes of line segments and triangles. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.

Original language | English (US) |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Computational Geometry: Theory and Applications |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1996 |

### Bibliographical note

Funding Information:Keywords: Computational geometry; Data structures; Filtering search; Geometric duality; Intersection searching; Persistence ~A preliminary version appears in Proc. 10th ACM Symp. on Computational Geometry (1994) under the title "Efficient algorithms for generalized intersection searching on non-iso-oriented objects". * Corresponding author. E-mail: janardan@cs.umn.edu. Research supported in part by NSF grant CCR-92-00270. Portions of this work were done while RJ was visiting MS at the Max-Planck-Institut fur Informatik, in Saarb~cken, Germany, in July 1993. RJ would like to thank the MPI for its generous support. E-mail: pgupta@cs.umn.edu. Research supported in part by NSF grant CCR-92-00270. 2E-mail: michiel@mpi-sb.mpg.de. This author was supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II).

## Keywords

- Computational geometry
- Data structures
- Filtering search
- Geometric duality
- Intersection searching
- Persistence