Document Open Access Logo

Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems

Authors Pankaj K. Agarwal , Boris Aronov , Esther Ezra , Matthew J. Katz , Micha Sharir



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2022.4.pdf
  • Filesize: 0.73 MB
  • 14 pages

Document Identifiers

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Esther Ezra
  • School of Computer Science, Bar Ilan University, Ramat Gan, Israel
Matthew J. Katz
  • Department of Computer Science, Ben Gurion University, Beer Sheva, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel

Acknowledgements

We thank Peyman Afshani for sharing with us problems that have motivated our study of segment-intersection searching amid spherical caps, Ovidiu Daescu for suggesting the collision-detection application that motivated some aspects of our work, and the reviewers of this work for their helpful comments.

Cite AsGet BibTex

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 4:1-4:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.4

Abstract

Let 𝒯 be a set of n planar semi-algebraic regions in ℝ³ 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 γ, which is also a plate, we can quickly answer various intersection queries, such as detecting whether γ intersects any plate of 𝒯, reporting all the plates intersected by γ, 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 ℝ³ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in ℝ³. 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^{ρ}), where ρ = (2t-3)/3(t-1) < 2/3 and t ≥ 3 is the number of parameters needed to represent the query arcs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Intersection searching
  • Semi-algebraic range searching
  • Point-enclosure queries
  • Ray-shooting queries
  • Polynomial partitions
  • Cylindrical algebraic decomposition
  • Multi-level partition trees
  • Collision detection

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. P. K. Agarwal. Ray shooting and other applications of spanning trees with low stabbing number. SIAM J. Comput., 21(3):540-570, 1992. Google Scholar
  2. P. K. Agarwal. Simplex range searching and its variants: A review. In Journey through Discrete Mathematics: A Tribute to Jiří Matoušek, pages 1-30. Springer Verlag, Berlin-Heidelberg, 2017. Google Scholar
  3. P. K. Agarwal, B. Aronov, E. Ezra, M. J. Katz, and M. Sharir. Intersection queries for flat semi-algebraic objects in three dimensions and related problems. URL: http://arxiv.org/abs/2203.10241.
  4. P. K. Agarwal, B. Aronov, E. Ezra, and J. Zahl. An efficient algorithm for generalized polynomial partitioning and its applications. SIAM J. Comput., 50:760-787, 2021. Also in Proc. Sympos. on Computational Geometry (SoCG), 2019, 5:1-5:14. Also in URL: https://arxiv.org/abs/1812.10269.
  5. P. K. Agarwal, B. Aronov, and S. Suri. Stabbing triangulations by lines in 3D. In Proc. 11th Annu. Sympos. on Computational Geometry, pages 267-276, 1995. Google Scholar
  6. P. K. Agarwal and J. Matousek. On range searching with semialgebraic sets. Discret. Comput. Geom., 11:393-418, 1994. Google Scholar
  7. P. K. Agarwal and J. Matoušek. Ray shooting and parameric search. SIAM J. Comput., 22:794-806, 1993. Google Scholar
  8. P. K. Agarwal, J. Matoušek, and M. Sharir. On range searching with semialgebraic sets II. SIAM J. Comput., 42:2039-2062, 2013. Also in URL: https://arxiv.org/abs/1208.3384.
  9. P. K. Agarwal and Micha Sharir. Ray shooting amidst convex polyhedra and polyhedral terrains in three dimensions. SIAM J. Comput., 25(1):100-116, 1996. Google Scholar
  10. B. Aronov and S. Fortune. Approximating minimum-weight triangulations in three dimensions. Discrete Comput. Geom., 21(4):527-549, 1999. Google Scholar
  11. S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics 10. Springer-Verlag, Berlin, 2003. Google Scholar
  12. B. Chazelle. Fast searching in a real algebraic manifold with applications to geometric complexity. In Colloquium on Trees in Algebra and Programming, pages 145-156, 1985. Google Scholar
  13. G. E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In Proc. 2nd GI Conf. Automata Theory and Formal Languages, volume 33 of LNCS, pages 134-183. Springer, 1975. Google Scholar
  14. M. de Berg, D. Halperin, M. H. Overmars, J. Snoeyink, and M. J. van Kreveld. Efficient ray shooting and hidden surface removal. Algorithmica, 12(1):30-53, 1994. Google Scholar
  15. D. P. Dobkin and D. G. Kirkpatrick. Fast detection of polyhedral intersection. Theor. Comput. Sci., 27:241-253, 1983. Google Scholar
  16. E. Ezra and M. Sharir. On ray shooting for triangles in 3-space and related problems. SIAM J. Comput., in press. Also in Proc. 37th Sympos. on Computational Geometry (2021), 34:1-34:15. Also in URL: https://arxiv.org/abs/2102.07310.
  17. L. Guth. Polynomial partitioning for a set of varieties. Math. Proc. Camb. Phil. Soc., 159:459-469, 2015. Also in URL: https://arxiv.org/abs/1410.8871.
  18. L. Guth and N. H. Katz. On the Erdőrig os distinct distances problem in the plane. Annals Math., 181:155-190, 2015. Also in URL: https://arxiv.org/abs/1011.4105.
  19. J. Hershberger and S. Suri. A pedestrian approach to ray shooting: Shoot a ray, take a walk. J. Algorithms, 18(3):403-431, 1995. Google Scholar
  20. J. Matoušek and Z. Patáková. Multilevel polynomial partitions and simplified range searching. Discrete Comput. Geom., 54:22-41, 2015. Google Scholar
  21. S. Mohaban and M. Sharir. Ray shooting amidst spheres in 3 dimensions and related problems. SIAM J. Comput., 26:654-674, 1997. Google Scholar
  22. M. Pellegrini. Ray shooting on triangles in 3-space. Algorithmica, 9:471-494, 1993. Google Scholar
  23. M. Pellegrini. Ray shooting and lines in space. In Handbook on Discrete and Computational Geometry, chapter 41, pages 1093-1112. CRC Press, Boca Raton, Florida, 3rd edition, 2017. Google Scholar
  24. J. T. Schwartz and M. Sharir. On the Piano Movers' problem: II. General techniques for computing topological properties of real algebraic manifolds. Advances in Appl. Math., 4:298-351, 1983. Google Scholar
  25. M. Sharir and H. Shaul. Ray shooting and stone throwing with near-linear storage. Comput. Geom. Theory Appls., 30:239-252, 2005. Google Scholar
  26. A. C. Yao and F. F. Yao. A general approach to d-dimensional geometric queries (extended abstract). In Proc. 17th Annu. ACM Symp. Theory of Computing, pages 163-168, 1985. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail