Computing in linear time a chord from which a simple polygon is weakly internally visible

  • Binay K. Bhattacharya
  • Asish Mukhopadhyay
Session 2
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1004)


A simple polygon is said to be weakly internally visible from a line segment lying inside it (chord, if the endpoints of the segment lie on the boundary of the polygon), if every point on the boundary of the polygon is visible from some point of this line segment. In this paper we describe a simple linear time algorithm for computing such a chord, when it exists. We also computed all non-redundant components of P which must be intersected by any chord of P from which P is wiv. The most interesting aspects of this algorithm are that we have been able to dispense with the complicated triangulation algorithm of Chazelle [2] as well as the algorithm in [7] for computing all shortest paths from a given vertex in a triangulated polygon. As a consequence of this, we can now use this algorithm to derive an optimal algorithm for triangulating a weakly internally visible polygon. Also these results can now be used in [3] to find a shortest line segment from which a polygon is weakly internally visible in optimal linear time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Avis and G. Toussaint. An optimal algorithm for determining the visibility of a polygon from an edge. IEEE Trans. on Computers, C-30(12):910–912, 1981.Google Scholar
  2. [2]
    B. Chazelle. Triangulating a simple polygon in linear time. Discrete and Computational Geometry, 6:485–524, 1991.Google Scholar
  3. [3]
    G. Das and G. Narasimhan OPtimal linear-time algorithm for the shortest illuninating line segment in a polygon. In Proceedings of the Tenth Annual Symposium on Computational Geometry, pages 259–266, 1994.Google Scholar
  4. [4]
    J-I Doh and K-Y Chwa. An algorithm for determining visibility of a simple polygon from an internal line segment. J. of Algorithms, 14:139–168, 1993.Google Scholar
  5. [5]
    H. El Gindy and D. Avis. A linear algorithm for computing the visbility polygon from a point. J. of Algorithms, 2:186–197, 1981.Google Scholar
  6. [6]
    S. K. Ghosh, A. Maheshwari, S. P. Pal, S. Saluja, and C. E. Veni Madhavan. Computing the shortest path tree in a weak visibility polygon. In Proceedings of the Twelvth Annual FST & TCS Conference, pages 369–389, 1991.Google Scholar
  7. [7]
    L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan. Linear time algorithms for visibility and shortest path problems inside a triangulated simple polygon. Algorithmica, 2:209–233, 1987.Google Scholar
  8. [8]
    P. Heffernan. An optimal algorithm for the two-guard problem. In Proceedings of the Ninth Annual symposium on computational geometry, pages 348–358, 1993.Google Scholar
  9. [9]
    Y. Ke. Polygon visibility algorithms for weak visibility and link distance problems. PhD thesis, The John Hopkins University, Baltimore, Maryland, 1989.Google Scholar
  10. [10]
    J. O'Rourke. Art gallery therems and algorithms. Oxford University Press, 1987.Google Scholar
  11. [11]
    F. A. Valentine. Minimal sets of visibility. Proceedings of the American Mathematical Society, 4:917–921, 1953.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Binay K. Bhattacharya
    • 1
  • Asish Mukhopadhyay
    • 2
  1. 1.Department of Computer ScienceThe University of NewcastleCallaghanAustralia
  2. 2.Department of CSEIITKanpurIndia

Personalised recommendations