Diffuse Reflection Radius in a Simple Polygon

  • Eli Fox-Epstein
  • Csaba D. Tóth
  • Andrew Winslow
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8591)


Light reflecting diffusely off of a surface leaves in all directions. It is shown that every simple polygon with n vertices can be illuminated from a single point light source s after at most \(\lfloor (n-2)/4\rfloor\)diffuse reflections, and this bound is the best possible. A point s with this property can be computed in O(nlogn) time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aanjaneya, M., Bishnu, A., Pal, S.P.: Directly visible pairs and illumination by reflections in orthogonal polygons. In: Abstracts of 24th European Workshop on Comput. Geom., pp. 241–244 (2008)Google Scholar
  2. 2.
    Aronov, B., Davis, A.R., Iacono, J., Yu, A.S.C.: The complexity of diffuse reflections in a simple polygon. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 93–104. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Aronov, B., Guibas, L.J., Teichmann, M., Zhang, L.: Visibility queries and maintenance in simple polygons. Discrete Comput. Geom. 27(4), 461–483 (2002)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bose, P., Lubiw, A., Munro, J.I.: Efficient visibility queries in simple polygons. Comput. Geometry Theory Appl. 23(3), 313–335 (2002)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bae, S.W., Korman, M., Okamoto, Y., Wang, H.: Computing the L 1 Geodesic diameter and center of a simple polygon in linear time. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 120–131. Springer, Heidelberg (2014)Google Scholar
  6. 6.
    Barequet, G., Cannon, S.M., Fox-Epstein, E., Hescott, B., Souvaine, D.L., Tóth, C.D., Winslow, A.: Diffuse reflections in simple polygons. Electronic Notes in Discrete Math. 44(5), 345–350 (2013)CrossRefGoogle Scholar
  7. 7.
    Brahma, S., Pal, S.P., Sarkar, D.: A linear worst-case lower bound on the number of holes in regions visible due to multiple diffuse reflections. J. of Geometry 81(1-2), 5–14 (2004)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Breen, M.: A Helly-type theorem for simple polygons. Geometriae Dedicata 60(3), 283–288 (1996)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chen, D.Z., Daescu, O.: Maintaining visibility of a polygon with a moving point of view. Information Processing Letters 65(5), 269–275 (1998)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, D.Z., Wang, H.: Weak visibility queries of line segments in simple polygons. In: Chao, K.-M., Hsu, T.-s., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 609–618. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Demaine, E.D., Erickson, J., Hurtado, F., Iacono, J., Langerman, S., Meijer, H., Overmars, M., Whitesides, S.: Separating point sets in polygonal environments. Intl. J. Comput. Geom. Appl. 15(4), 403–419 (2005)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Djidjev, H.N., Lingas, A., Sack, J.-R.: An O(n logn) algorithm for computing the link center of a simple polygon. Discrete. Comput. Geom. 8, 131–152 (1992)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Ghosh, S.K.: Visibility algorithms in the plane, ch. 7. Cambridge Univ. Press (2007)Google Scholar
  14. 14.
    Guibas, L., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2(1-4), 209–233 (1987)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Hershberger, J., Suri, S.: Matrix searching with the shortest path metric. SIAM J. Comput. 26(6), 1612–1634 (1997)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Lee, D.T., Preparata, F.: An optimal algorithm for finding the kernel of a polygon. J. ACM 26, 415–421 (1979)MATHMathSciNetGoogle Scholar
  17. 17.
    Maheshwari, A., Sack, J., Djidjev, H.N.: Link distance problems. In: Handbook of Computational Geometry, ch.12. Elsevier (2000)Google Scholar
  18. 18.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4, 611–626 (1989)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Schuierer, S.: Computing the L 1-diameter and center of a simple rectilinear polygon. In: Proc. Intl. Conf. on Computing and Information (ICCI), pp. 214–229 (1994)Google Scholar
  20. 20.
    Suri, S.: A linear time algorithm for minimum link paths inside a simple polygon. Comput. Vision. Graph. Image Process. 35, 99–110 (1986)CrossRefMATHGoogle Scholar
  21. 21.
    Suri, S.: On some link distance problems in a simple polygon. IEEE Trans. Robot. Autom. 6, 108–113 (1990)CrossRefGoogle Scholar
  22. 22.
    Toussaint, G.T.: An optimal algorithm for computing the relative convex hull of a set of points in a polygon. In: Signal Processing III: Theories and Applications (EURASIP 1986), Part 2, pp. 853–856 (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Eli Fox-Epstein
    • 1
  • Csaba D. Tóth
    • 2
  • Andrew Winslow
    • 3
  1. 1.Department of Computer ScienceBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsCalifornia State UniversityLos AngelesUSA
  3. 3.Department of Computer ScienceTufts UniversityMedfordUSA

Personalised recommendations