Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Visibility and intersection problems in plane geometry


We develop new data structures for solving various visibility and intersection problems about a simple polygonP onn vertices. Among our results are a simpleO(n logn)-time algorithm for computing the illuminated subpolygon ofP from a luminous side, and anO(logn)-time algorithm for determining which side ofP is first hit by a bullet fired from a point in a certain direction. The latter method requires preprocessing onP which takes timeO(n logn) and spaceO(n). The two main tools in attacking these problems are geometric duality on the two-sided plane and fractional cascading.


  1. 1.

    Chazelle, B. A theorem on polygon cutting with applications,Proc. 23rd Ann. IEEE Symp. Found. Comput. Sci. (1982), 339–349.

  2. 2.

    Chazelle, B. Filtering search: a new approach to query-answering,SIAM J. Comput. 15 (1986), 703–724.

  3. 3.

    Chazelle, B. Computing on a free tree via complexity-preserving mappings,Algorithmica 2 (1987), 337–361.

  4. 4.

    Chazelle, B., Guibas, L. J. Fractional cascading: I. A data structuring technique,Algorithmica 1 (1986), 133–162.

  5. 5.

    Chazelle, B., Guibas, L. J. Fractional cascading: II. Applications,Algorithmica 1 (1986), 163–191.

  6. 6.

    Edelsbrunner, H., Guibas, L. J., Stolfi, J. Optimal point location in a monotone subdivision,SIAM J. Comput. 15 (1986), 317–340.

  7. 7.

    El Gindy, H. A. An efficient algorithm for computing the weak visibility polygon from an edge in simply polygons, unpublished manuscript, McGill University, 1984.

  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 (1987), 209–233.

  9. 9.

    Guibas, L. J., Hershberger, J. Optimal shortest-path queries in a simple polygon,Proc. 3rd Ann. ACM Symp. Comput Geom. (1987), 50–63.

  10. 10.

    Guibas, L. J., Ramshaw, L., Stolfi, J. A kinetic framework for computational geometry,Proc. 24th Ann. IEEE Symp. Found. Comput. Sci. (1983), 100–111.

  11. 11.

    Lee, D. T., Lin, A. Computing the visibility polygon from an edge, unpublished manuscript, Northwestern University, 1984.

  12. 12.

    Tarjan, R. E., Van Wyk, C. AnO(n log logn)-time algorithm for triangulating simple polygons,SIAM J. Comput.,17 (1988), 143–178.

  13. 13.

    van Emde Boas, P., Kaas, R., Zijlstra, E. Design and implementation of an efficient priority queue,Math. Systems Theory 10 (1977), 99–127.

Download references

Author information

Additional information

Bernard Chazelle wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8700917. A preliminary version of this paper was presented at the First Annual ACM Symposium on Computational Geometry, June 1985.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chazelle, B., Guibas, L.J. Visibility and intersection problems in plane geometry. Discrete Comput Geom 4, 551–581 (1989).

Download citation


  • Convex Polygon
  • Plane Geometry
  • Query Time
  • Intersection Problem
  • Simple Polygon