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Throwing Stones Inside Simple Polygons

  • Otfried Cheong
  • Hazel Everett
  • Hyo-Sil Kim
  • Sylvain Lazard
  • René Schott
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)

Abstract

Given two sets A and B of m non-intersecting line segments in the plane, we show how to compute in O(m logm) time a data structure that uses O(m) space and allows to answer the following query in O(logm) time: Given a parabola γ: y = ax 2 + bx + c, does γ separate A and B? This structure can be used to build a data structure that stores a simple polygon and allows ray-shooting queries along parabolic trajectories with vertical main axis. For a polygon with complexity n, we can answer such “stone throwing” queries in O(log2 n) time, using O(n logn) space and O(n log2 n) preprocessing time. This matches the best known bound for circular ray shooting in simple polygons.

Keywords

Line Segment Voronoi Diagram Computational Geometry Query Point Simple Polygon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, P., Sharir, M.: Circle shooting in a simple polygon. J. Algorithms 14, 69–87 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ahn, H.-K., Cheong, O., van Oostrum, R.: Casting a polyhedron with directional uncertainty. Computational Geometry: Theory and Applications 26, 129–141 (2003)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bae, S.-W., Chwa, K.-Y.: Voronoi diagrams with a transportation network on the Euclidean plane. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 101–112. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Cheng, S.-W., Cheong, O., Everett, H., van Oostrum, R.: Hierarchical decompositions and circular ray shooting in simple polygons. Discrete Comput. Geom. 32, 401–415 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Berlin (2000)zbMATHGoogle Scholar
  6. 6.
    Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: Shoot a ray, take a walk. J. Algorithms 18, 403–431 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Klein, R.: Concrete and Abstract Voronoi Diagrams. LNCS, vol. 400. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  8. 8.
    Klein, R., Mehlhorn, K., Meiser, S.: Randomized incremental construction of abstract Voronoi diagrams. Computational Geometry 3, 157–184 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Sharir, M., Shaul, H.: Ray shooting and stone throwing with near-linear storage. Computational Geometry 30, 239–252 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Otfried Cheong
    • 1
  • Hazel Everett
    • 2
  • Hyo-Sil Kim
    • 1
  • Sylvain Lazard
    • 2
  • René Schott
    • 2
  1. 1.Division of Computer ScienceKAISTDaejeonSouth Korea
  2. 2.LORIA & IECN – INRIA Lorraine, Universities Nancy 1 & 2NancyFrance

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