Advertisement

Computational Geometry

  • Michal Forišek
  • Monika Steinová
Chapter
Part of the SpringerBriefs in Computer Science book series (BRIEFSCOMPUTER)

Abstract

The second main chapter of this book is dedicated to problems from the area of computational geometry. We consider four different problems. First, we discuss the well-known use of the rubber band metaphor to find the Euclidean shortest path in a plane with obstacles. In the second problem, we present our original use of the rubber band metaphor to significantly simplify simple distance calculations, in particular the distance between two line segments in two, three, and even more dimensions. The third problem considered is concerned with testing whether a point is contained in a polygon. For this problem we develop an original metaphor based on the definition of the winding number, and we use it to design an algorithm that is easy to implement. Finally, we show another physical metaphor that can be used to triangulate a polygon easily.

Keywords

Short Path Line Segment Computational Geometry Query Point Rubber Band 
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.

References

  1. 1.
    Asano, T., Asano, T., Guibas, L.J., Hershberger, J., Imai, H.: Visibility of disjoint polygons. Algorithmica 1(1), 49–63 (1986)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)MATHGoogle Scholar
  3. 3.
    Canny, J.F.: The Complexity of Robot Motion Planning. MIT Press, Cambridge (1988)Google Scholar
  4. 4.
    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6(5), 485–524 (1991)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Delaunay, B.N.: Sur la sphère vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793–800 (1934)Google Scholar
  6. 6.
    Eberly, D.H.: 3D Game Engine Design: A Practical Approach to Real-Time Computer Graphics. CRC Press, Boca Raton (2000)Google Scholar
  7. 7.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)MATHCrossRefGoogle Scholar
  8. 8.
    Guibas, L.J., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2, 209–233 (1987)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Haines, E.: Point in polygon strategies. In: Heckbert, P. (ed.) Graphics Gems IV, pp. 24–46. Academic Press, San Diego (1994)Google Scholar
  10. 10.
    Hershberger, J., Suri, S.: Efficient computation of Euclidean shortest paths in the plane. In: Proceedings of the 34th Annual Symposium on Foundations of Computer Science (FOCS 1993), pp. 508–517. IEEE Computer Society (1993)Google Scholar
  11. 11.
    Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28(6), 2215–2256 (1999)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Li, F., Klette, R.: Rubberband algorithms for solving various 2D or 3D shortest path problems. In: Computing: Theory and Applications, 2007. ICCTA ’07, pp. 9–19. doi: 10.1109/ICCTA.2007.113 (2007)
  13. 13.
    O’Rourke, J.: Computational Geometry in C. Cambridge University Press, Cambridge (1998)MATHCrossRefGoogle Scholar
  14. 14.
    Shimrat, M.: Algorithm 112: position of point relative to polygon. Commun. ACM 5(8), 434 (1962)CrossRefGoogle Scholar
  15. 15.
    Sunday, D.: Inclusion of a Point in a Polygon. http://geomalgorithms.com/a03_inclusion.html (2012). Accessed 8 Dec 2012
  16. 16.
    Tarjan, R.E., Wyk, C.J.V.: An \(O(n \log \log n)\)-time algorithm for triangulating a simple polygon. SIAM J. Comput. 17(1), 143–178 (1988)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Welzl, E.: Constructing the visibility graph for \(n\)-line segments in \(O(n^2)\) time. Inf. Process. Lett. 20(4), 167–171 (1985)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceComenius UniversityBratislavaSlovakia
  2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland

Personalised recommendations