Abstract
In this paper we explore some novel aspects of visibility for stationary and moving points inside a simple polygon P. We provide a mechanism for expressing the visibility polygon from a point as the disjoint union of logarithmically many canonical pieces using a quadraticspace data structure. This allows us to report visibility polygons in time proportional to their size, but without the cubic space overhead of earlier methods. The same canonical decomposition can be used to determine visibility within a frustum, or to compute various attributes of the visibility polygon efficiently. By exploring the connection between visibility polygons and shortest path trees, we obtain a kinetic algorithm that can track the visibility polygon as the viewpoint moves along polygonal paths inside P, at a polylogarithmic cost per combinatorial change in the visibility. The combination of the static and kinetic algorithms leads to a space query-time tradeoff for the visibility from a point problem and an output-sensitive algorithm for the weak visibility from a segment problem.
Boris Aronov has been partially supported by NSF Grant CCR-92-11541 and a Sloan Research Fellowship. Leonidas Guibas and Li Zhang were supported in part by Army Research O?ce MURI grant 5-23542-A and NSF grant CCR-9623851. Marek Teichmann is supported by the National Science and Engineering Council of Canada.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J. Basch, L. Guibas, and J. Hershberger. Data structures for mobile data. In Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pages 747–756, 1997.
P. Bose, A. Lubiw, and J. I. Munro. E.cient visibility queries in simple polygons. In Proc. 4th Canad. Conf. Comput. Geom., pages 23–28, 1992.
B. Chazelle. A theorem on polygon cutting with applications. In Proc. 23rd Annu. IEEE Sympos. Found. Comput. Sci., pages 339–349, 1982.
D. Z. Chen and O. Daescu. Maintaining visibility of a polygon with a moving point of view. In Proc. 8th Canad. Conf. Comput. Geom., pages 240–245, 1996.
L. J. Guibas and J. Hershberger. Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci., 39:126–152, 1989.
L. J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2:209–233, 1987.
L. J. Guibas, R. Motwani, and P. Raghavan. The robot localization problem in two dimensions. In Proc. 3rd ACM-SIAM Sympos. Discrete Algorithms, pages 259–268, 1992.
J. Hershberger and S. Suri. A pedestrian approach to ray shooting: Shoot a ray, take a walk. In Proc. 4th ACM-SIAM Sympos. Discrete Algorithms, pages 54–63, 1993.
M. H. Overmars. The Design of Dynamic Data Structures, volume 156 of Lecture Notes Comput. Sci. Springer-Verlag, Heidelberg, West Germany, 1983.
N. Sarnak and R. E. Tarjan. Planar point location using persistent search trees. Commun. ACM, 29:669–679, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aronov, B., Guibas, L.J., Teichmann, M., Zhang, L. (1998). Visibility Queries in Simple Polygons and Applications. In: Chwa, KY., Ibarra, O.H. (eds) Algorithms and Computation. ISAAC 1998. Lecture Notes in Computer Science, vol 1533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49381-6_38
Download citation
DOI: https://doi.org/10.1007/3-540-49381-6_38
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65385-1
Online ISBN: 978-3-540-49381-5
eBook Packages: Springer Book Archive