Discrete & Computational Geometry

, Volume 12, Issue 1, pp 91–104 | Cite as

Camera placement in integer lattices

  • E. Kranakis
  • M. Pocchiola


The camera placement problem concerns the placement of a fixed number of point-cameras on thed-dimensional integer lattice in order to maximize their visibility. We reduce the problem to a finite discrete optimization problem and give a characterization of optimal configurations of size at most 3 d .


Lattice Point Prime Number Discrete Comput Geom Optimal Configuration Integer Lattice 


  1. 1.
    H. L. Abbott, Some results in combinatorial geometry.Discrete Mathematics, 9: 199–204, 1974.MathSciNetCrossRefGoogle Scholar
  2. 2.
    V. Boltjansky and I. Gohberg.Results and Problems in Combinatorial Geometry. Cambridge University Press, Cambridge, 1985.CrossRefMATHGoogle Scholar
  3. 3.
    P. Erdős, P.M. Gruber, and J. Hammer.Lattice Points. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 39. Longman Scientific and Technical, Harlow, 1989.Google Scholar
  4. 4.
    J. Hammer,Unsolved Problems concerning Lattice Points. Research Notes in Mathematics. Pitman, London, 1977.MATHGoogle Scholar
  5. 5.
    C. Jordan.Calculus of Finite Differences. Chelsea, New York, 1965.MATHGoogle Scholar
  6. 6.
    M. Kac.Statistical Independence in Probability, Analysis and Number Theory. The Carus Mathematical Monographys, Vol. 12. Mathematical Association of America, Washington, DC, 1959.MATHGoogle Scholar
  7. 7.
    D. Knuth.The Art of Computer Programming: Seminumerical Algorithms, 2nd edn. Computer Science and Information Processing, Addison-Wesley, Reading, MA, 1981.MATHGoogle Scholar
  8. 8.
    E. Kranakis and M. Pocchiola. Enumeration and visibility problems in integer lattices.Proceedings of the 6th Annual ACM Symposium on Computational Geometry, 1990, pp. 261–270.Google Scholar
  9. 9.
    E. Kranakis and M. Pocchiola. A brief survey of art gallery problems in integer lattice systems.CWI Quartely, 4 (4): 269–282, 1991.MathSciNetMATHGoogle Scholar
  10. 10.
    E. Kranakis and M. Pocchiola, Camera Placement in Integer Lattices. Technical Report 92-20, Lab. Inform., Ecole Normale Supérieure, Paris, 1992. Also available as TR 214, Carleton University, School of Computer Science.Google Scholar
  11. 11.
    W. O. J. Moser. Problems on extremal properties of a finite set of points. InDiscrete Geometry and Convexity, Goodmanet al., eds. New York Academy of Sciences, Washington, DC, 1985, pp. 52–64.Google Scholar
  12. 12.
    J. O’Rourke.Art Gallery Theorems and Algorithms. International Series of Monographs on Computer Science. Oxford University press, Oxford, 1987.Google Scholar
  13. 13.
    M. Pocchiola. Trois thèmes sur la visibilité: énumération, optimisation et graphique 2D. Technical Report 90-23, Lab. Inform., Ecole Normale Supérieure, Paris, 1990. Ph.D. thesis.Google Scholar
  14. 14.
    D. F. Rearick. Mutually visible lattice points.Norske Vid. Selsk. Forh. (Trondheim), 39: 41–45, 1966.MathSciNetMATHGoogle Scholar
  15. 15.
    H. Rumsey, Jr. Sets of visible points.Duke Mathematical Journal, 33: 263–274, 1966.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • E. Kranakis
    • 1
    • 2
  • M. Pocchiola
    • 3
  1. 1.Centrum voor Wiskunde en InformaticaAmsterdamThe Netherlands
  2. 2.School of compute ScienceCarleton UniversityOttawa, OntarioCanada
  3. 3.Département de Mathématiques et d’InformatiqueEcole Normale Supérieure, URA 1327 CNRSParis Cédex 05France

Personalised recommendations