Abstract
We show that the region lit by a point light source inside a simple n-gon after at most k reflections off the boundary has combinatorial complexity O(n 2k), for any k ≥ 1. A lower bound of Ω((n/k)2k) is also established which matches the upper bound for any fixed k. A simple near-optimal algorithm for computing the illuminated region is presented, which runs in O(n 2k log n) time and O(n 2k) space for any k > 1.
Work on this paper by Boris Aronov has been supported by NSF grant CCR-92-11541 and Sloan Research Fellowship.
Tamal K. Dey acknowledges the support of NSF grant CCR-93-21799, USA and DST grant SR/OY/E-06/95, India.
S. P. Pal acknowledges the support of a research grant from the Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
B. Aronov, A. R. Davis, T. K. Dey, S. P. Pal, and D. C. Prasad. Visibility with reflection. Proc. 11th ACM Symp. Comput. Geom., 1995, pp. 316–325.
J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Trans. Comput., C-28:643–647, 1979.
H. N. Djidjev, A. Lingas, and J. Sack. An O(n log n) algorithm for computing the link center of a simple polygon. Discr. Comput. Geom., 8(2):131–152, 1992.
J. Foley, A. van Dam, S. Feiner, J. Hughes, and R. Phillips. Introduction to Computer Graphics. Addison-Wesley, Reading, MA 1994.
H. Edelsbrunner, L. J. Guibas, and M. Sharir. The complexity and construction of many faces in arrangements of lines and of segments. Discr. Comput. Geom., 5:161–196, 1990.
H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186–197, 1981.
E. Gutkin. Billiards in polygons. Physica D, 19:311–333, 1986.
L. J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2:209–233, 1987.
A. Horn and F. Valentine. Some properties of l-sets in the plane. Duke Math. J., 16:131–140, 1949.
Y. Ke. An efficient algorithm for link-distance problems. Proc. 5th Annu. ACM Sympos. Comput. Geom., 1989, pp. 69–78.
V. Klee. Is every polygonal region illuminable from some point? Amer. Math. Monthly, 76:180, 1969.
V. Klee and S. Wagon. Old and new unsolved problems in plane geometry and number theory. Mathematical Association of America, 1991.
J. O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, New York, NY 1987.
J. O'Rourke. Visibility. In The CRC Handbook of Discrete & Computational Geometry, Eds., J. E. Goodman and J. O'Rourke, CRC Press, to appear, 1997.
M. Pellegrini. Ray shooting and lines in space. In The CRC Handbook of Discrete & Computational Geometry, Eds., J. E. Goodman and J. O'Rourke, CRC Press, to appear, 1997.
T. C. Shermer. Recent results in art galleries. Proc. IEEE, 80(9):1384–1399, 1992.
S. Suri. On some link distance problems in a simple polygon. IEEE Trans. Robot. Autom., 6:108–113, 1990.
G. W. Tokarsky. Polygonal rooms not illuminable from every point. Amer. Math. Monthly, 102:867–879, 1995.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aronov, B., Davis, A.R., Dey, T.K., Pal, S.P., Prasad, D.C. (1996). Visibility with multiple reflections. In: Karlsson, R., Lingas, A. (eds) Algorithm Theory — SWAT'96. SWAT 1996. Lecture Notes in Computer Science, vol 1097. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61422-2_139
Download citation
DOI: https://doi.org/10.1007/3-540-61422-2_139
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61422-7
Online ISBN: 978-3-540-68529-6
eBook Packages: Springer Book Archive