Abstract
LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.
Similar content being viewed by others
References
P. Agarwal, Ray shooting and other applications of spanning trees with low stabbing number,Proc. 5th ACM Symp. on Computational Geometry, 1989, pp. 315–325.
B. Chazelle, A theorem on polygon cutting with applications,Proc. 23rd IEEE Symp. on Foundations of Computer Science, 1982, pp. 339–349.
B. Chazelle, Triangulating a simple polygon in linear time,Discrete Comput. Geom.,6 (1991), 485–524.
B. Chazelle, H. Edelsbrunner, and L. Guibas, The complexity of cutting complexes,Discrete Comput. Geom.,4 (1989), 139–181.
B. Chazelle and L. Guibas, Fractional cascading: I. A data structuring technique,Algorithmica,1 (1986), 133–162.
B. Chazelle and L. Guibas, Fractional cascading: II. Applications,Algorithmica,1 (1986), 163–191.
B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry.Discrete Comput. Geom.,4 (1989), 551–581.
D. Dobkin and D. Kirkpatrick, Fast detection of polyhedral intersection,Theoret. Comput. Sci.,27 (1983), 241–253.
H. Edelsbrunner, L. Guibas, and J. Stolfi, Optimal point location in a monotone subdivision,SIAM J. Comput.,15 (1986), 317–340.
H. ElGindy and D. Avis, A linear algorithm for computing the visibility polygon from a point,J. Algorithms,2 (1981), 186–197.
L. Guibas and J. Hershberger, Optimal shortest path queries in a simple polygon,J. Comput. System Sci.,39 (1989), 126–152.
L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. Tarjan, Linear time algorithms for visibility and shortest path problems inside triangulated simple polygons,Algorithmica,2 (1987), 209–233.
L. Guibas, J. Hershberger, and J. Snoeyink, Compact interval trees: a data structure for convex hulls,Internat. J. Comput. Geom. Appl.,1 (1991), 1–22.
D. Harel and R. E. Tarjan, Fast algorithms for finding nearest common ancestors,SIAM J. Comput.,13 (1984), 338–355.
J. Hershberger, A new data structure for shortest path queries in a simple polygon,Inform. Process. Lett,38 (1991), 231–235.
D. Kirkpatrick, Optimal search in planar subdivisions,SIAM J. Comput.,12 (1983), 28–35.
D. T. Lee and F. P. Preparata, Euclidean shortest paths in the presence of rectilinear barriers,Networks,14 (1984), 393–410.
J. Matoušek, More on cutting arrangements and spanning trees with low stabbing number, Technical Report B-90-2, Freie Universität Berlin, February 1990.
K. Mehlhorn,Data Structures and Algorithms, I: Sorting and Searching, Springer-Verlag, Heidelberg, 1984.
M. Overmars and J. van Leeuwen, Maintenance of configurations in the plane,J. Comput. System Sci.,23 (1981), 166–204.
F. Preparata and M. Shamos,Computational Geometry: An Introduction, Springer-Verlag, Heidelberg, 1985.
B. Schieber and U. Vishkin. On finding lowest common ancestors: simplification and parallelization,Proc. Third Aegean Workshop on Computing, pp. 111–123, Lecture Notes in Computer Science, Vol. 319, Springer-Verlag, Berlin, 1988.
Author information
Authors and Affiliations
Additional information
Communicated by Kurt Mehlhorn.
Work by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Micha Sharir has been supported by ONR Grants N00014-89-J-3042 and N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.
Rights and permissions
About this article
Cite this article
Chazelle, B., Edelsbrunner, H., Grigni, M. et al. Ray shooting in polygons using geodesic triangulations. Algorithmica 12, 54–68 (1994). https://doi.org/10.1007/BF01377183
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01377183