Theory of Computing Systems

, Volume 63, Issue 3, pp 543–566 | Cite as

Capturing Points with a Rotating Polygon (and a 3D Extension)

  • Carlos Alegría-Galicia
  • David OrdenEmail author
  • Leonidas Palios
  • Carlos Seara
  • Jorge Urrutia


We study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a segment or a line. We also solve an extension to 3D where we rotate a polyhedron around a given point to contain the maximum number of elements from a set of points in the space.


Points covering Rotation Geometric optimization Polygon Polyhedron 



David Orden is supported by MINECO Projects with references MTM2014-54207 and MTM2017-83750-P, as well as by H2020-MSCA-RISE project 734922 - CONNECT. Carlos Seara is supported by projects Gen. Cat. DGR 2017SGR1640, MINECO/FEDER MTM2015-63791-R, and by H2020-MSCA-RISE project 734922 - CONNECT. Jorge Urrutia is supported in part by SEP-CONACYT of México, Proyecto 80268, and by PAPPIIT IN102117 Programa de Apoyo a la Investigación e Innovación Tecnológica, Universidad Nacional Autónoma de México.


  1. 1.
    Agarwal, P.K., Flato, E., Halperin, D.: Polygon decomposition for efficient construction of Minkowski sums. Comput. Geom. Theory Appl. 21(1–2), 39–61 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agarwal, P.K., Hagerup, T., Ray, R., Sharir, M., Smid, M., Welzl, E.: Translating a planar object to maximize point containment. In: Algorithms — ESA 2002: 10th Annual European Symposium. Rome, Italy, September 17–21, 2002. Proceedings, pp. 42–53 (2002)Google Scholar
  3. 3.
    Baran, I., Demaine, E.D., Pǎtraṡcu, M.: Subquadratic algorithms for 3SUM. Algorithmica 50(4), 584–596 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barequet, G., Dickerson, M., Pau, P.: Translating a convex polygon to contain a maximum number of points. Comput. Geom. Theory Appl. 8(4), 167–179 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barequet, G., Goryachev, A.: Offset polygon and annulus placement problems. Comput. Geom. Theory Appl. 47(3, Part A), 407–434 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barequet, G., Har-Peled, S.: Polygon containment and translation min-Hausdorff-distance between segment sets are 3SUM-hard. Int. J. Comput. Geom. Appl. 11(4), 465–474 (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chazelle, B.: Advances in Computing Research, vol. 1, Chapter The polygon containment problem, pp. 1–33. JAI Press (1983)Google Scholar
  8. 8.
    Dickerson, M., Scharstein, D.: Optimal placement of convex polygons to maximize point containment. Comput. Geom. Theory Appl. 11(1), 1–16 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gajentaan, A., Overmars, M.H.: On a class of O(n 2) problems in computational geometry. Comput. Geom. Theory nd Appl. 5(3), 165–185 (1995)CrossRefzbMATHGoogle Scholar
  10. 10.
    Grønlund, A., Pettie, S.: Threesomes, degenerates, and love triangles. J. ACM (JACM) 65(4), 22 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ishiguro, H., Yamamoto, M., Tsuji, S.: Omni-directional stereo. IEEE Trans. Pattern Anal. Mach. Intell. 14(2), 257–262 (1992)CrossRefGoogle Scholar
  12. 12.
    Kopelowitz, T., Pettie, S., Porat, E.: Higher lower bounds from the 3SUM conjecture. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1272–1287. Society for Industrial and Applied Mathematics (2016)Google Scholar
  13. 13.
    analysis, Tristan Needham.: Visual Complex Chapter 6.II.3: A Conformal Map of the Sphere, pp 283–286. Clarendon Press, Oxford (1998)Google Scholar
  14. 14.
    Pǎtraṡcu, M.: Towards polynomial lower bounds for dynamic problems. In: Proceedings of the Forty-Second ACM Symposium on Theory of Computing, pp. 603–610. ACM (2010)Google Scholar
  15. 15.
    Yap, C.K., Chang, E.-C.: Algorithms for Robot Motion Planning and Manipulation, Chapter Issues in the Metrology of Geometric Tolerancing, pp 393–400. Wellesley, A.K. Peters (1997)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Posgrado en Ciencia e Ingeniería de la ComputaciónUniversidad Nacional Autónoma de MéxicoMexico CityMéxico
  2. 2.Departamento de Física y MatemáticasUniversidad de AlcaláMadridSpain
  3. 3.Department of Computer Science and EngineeringUniversity of IoanninaIoanninaGreece
  4. 4.Departament de MatemàtiquesUniversitat Politècnica de CatalunyaBarcelonaSpain
  5. 5.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico CityMéxico

Personalised recommendations