## Abstract

Let T be a set of n planar semi-algebraic regions in R^{3} of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess T into a data structure so that for a query object ?, which is also a plate, we can quickly answer various intersection queries, such as detecting whether ? intersects any plate of T, reporting all the plates intersected by ?, or counting them. We focus on two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in R^{3} (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in R^{3}. These interesting special cases form the building blocks for the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if T is a set of plates and the query objects are arcs, we obtain a data structure that uses O^{*}(n^{4}/^{3}) storage (where the O^{*}(·) notation hides subpolynomial factors) and answers an intersection query in O^{*}(n^{2}/^{3}) time. Alternatively, by increasing the storage to O^{*}(n^{3}/^{2}), the query time can be decreased to O^{*}(n^{?}), where ? = (2t - 3)/3(t - 1) < 2/3 and t = 3 is the number of parameters needed to represent the query arcs.

Original language | English |
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Title of host publication | 38th International Symposium on Computational Geometry, SoCG 2022 |

Editors | Xavier Goaoc, Michael Kerber |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772273 |

DOIs | |

State | Published - 1 Jun 2022 |

Event | 38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany Duration: 7 Jun 2022 → 10 Jun 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 224 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 38th International Symposium on Computational Geometry, SoCG 2022 |
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Country/Territory | Germany |

City | Berlin |

Period | 7/06/22 → 10/06/22 |

### Bibliographical note

Funding Information:Funding Pankaj K. Agarwal: Work partially supported by NSF grants IIS-18-14493 and CCF-20-07556. Boris Aronov: Work partially supported by NSF Grants CCF-15-40656 and CCF-20-08551, and by Grant 2014/170 from the US-Israel Binational Science Foundation. Esther Ezra: Work partially supported by NSF CAREER under Grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation. Matthew J. Katz: Work partially supported by Grant 1884/16 from the Israel Science Foundation, and by Grant 2019715/CCF-20-08551 from the US-Israel Binational Science Foundation/US National Science Foundation. Micha Sharir: Work partially supported by Grant 260/18 from the Israel Science Foundation.

Publisher Copyright:

© Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir; licensed under Creative Commons License CC-BY 4.0

## Keywords

- Collision detection
- Cylindrical algebraic decomposition
- Intersection searching
- Multilevel partition trees
- Point-enclosure queries
- Polynomial partitions
- Ray-shooting queries
- Semi-algebraic range searching