Abstract
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 3d.
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H. L. Abbott, Some results in combinatorial geometry.Discrete Mathematics, 9: 199–204, 1974.
V. Boltjansky and I. Gohberg.Results and Problems in Combinatorial Geometry. Cambridge University Press, Cambridge, 1985.
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.
J. Hammer,Unsolved Problems concerning Lattice Points. Research Notes in Mathematics. Pitman, London, 1977.
C. Jordan.Calculus of Finite Differences. Chelsea, New York, 1965.
M. Kac.Statistical Independence in Probability, Analysis and Number Theory. The Carus Mathematical Monographys, Vol. 12. Mathematical Association of America, Washington, DC, 1959.
D. Knuth.The Art of Computer Programming: Seminumerical Algorithms, 2nd edn. Computer Science and Information Processing, Addison-Wesley, Reading, MA, 1981.
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.
E. Kranakis and M. Pocchiola. A brief survey of art gallery problems in integer lattice systems.CWI Quartely, 4 (4): 269–282, 1991.
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.
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.
J. O’Rourke.Art Gallery Theorems and Algorithms. International Series of Monographs on Computer Science. Oxford University press, Oxford, 1987.
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.
D. F. Rearick. Mutually visible lattice points.Norske Vid. Selsk. Forh. (Trondheim), 39: 41–45, 1966.
H. Rumsey, Jr. Sets of visible points.Duke Mathematical Journal, 33: 263–274, 1966.
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A preliminary version of this work appears in [8]. The research of E. Kranakis was supported by NSERC Grant No. 907002.
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Kranakis, E., Pocchiola, M. Camera placement in integer lattices. Discrete Comput Geom 12, 91–104 (1994). https://doi.org/10.1007/BF02574368
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DOI: https://doi.org/10.1007/BF02574368