Abstract
The paper presents two algorithms involving shooting in three dimensions. We first present a new algorithm for performing ray shooting amid several special classes of n triangles in three dimensions. We show how to implement this technique to obtain improved query time for a set of fat triangles, and for a set of triangles stabbed by a common line. In both cases our technique requires near-linear preprocessing and storage, and answers a query in about n 2/3 time. This improves the best known result of close to n 3/4 query time for general triangles. The second algorithm handles stone-throwing amid arbitrary triangles in 3-space, where the curves along which we shoot are vertical parabolas, which are trajectories of stones thrown under gravity. We present an algorithm that answers stone-throwing queries in about n 3/4 time, using near linear storage and preprocessing. As far as we know, this is the first nontrivial solution of this problem. Several extensions of both algorithms are also presented.
This work was supported by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing), and is part of the second author’s Ph.D. dissertation, prepared under the supervision of the first author in Tel Aviv University. Work by Micha Sharir was also supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the U.S.-Israel Binational Science Foundation, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K.: Applications of a new space partition technique. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1991. LNCS, vol. 519, pp. 379–392. Springer, Heidelberg (1991)
Agarwal, P.K.: Intersection and Decomposition Algorithms for Planar Arrangements. Cambridge University Press, New York (1991)
Agarwal, P.K.: Ray shooting and other applications of spanning trees with low stabbing number. SIAM J. Comput. 21, 540–570 (1992)
Agarwal, P.K., Matoušek, J.: Ray shooting and parametric search. SIAM J. Comput. 22(4), 794–806 (1993)
Agarwal, P.K., Matoušek, J.: On range searching with semi-algebraic sets. Discrete Comput. Geom. 11, 393–418 (1994)
Agarwal, P.K., Sharir, M.: Applications of a new space-partitioning technique. Discrete Comput. Geom. 9, 11–38 (1993)
Agarwal, P.K., Sharir, M.: Ray shooting amidst convex polyhedra and polyhedral terrains in three dimensions. SIAM J. Comput. 25, 100–116 (1996)
Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L.J., Hershberger, J., Sharir, M., Snoeyink, J.: Ray shooting in polygons using geodesic triangulations. Algorithmica 12, 54–68 (1994)
Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M., Stolfi, J.: Lines in space: Combinatorics and algorithms. Algorithmica 15, 428–447 (1996)
Chazelle, B., Guibas, L.J.: Visibility and intersection problems in plane geometry. Discrete Comput. Geom. 4, 551–581 (1989)
de Berg, M., Halperin, D., Overmars, M., Snoeyink, J., van Kreveld, M.: Efficient ray shooting and hidden surface removal. Algorithmica 12, 30–53 (1994)
Dobkin, D.P., Kirkpatrick, D.G.: Determining the separation of preprocessed polyhedra – a unified approach. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 400–413. Springer, Heidelberg (1990)
Koltun, V.: Almost tight upper bounds for vertical decompositions in four dimensions. In: Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci., pp. 56–65 (2001)
Matoušek, J.: Reporting points in halfspaces. Comput. Geom. Theory Appl. 2(3), 169–186 (1992)
Mohaban, S., Sharir, M.: Ray shooting amidst spheres in three dimensions and related problems. SIAM J. Comput. 26, 654–674 (1997)
Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995)
Stolfi, J.: Oriented Projective Geometry: A Framework for Geometric Computations. Academic Press, New York (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sharir, M., Shaul, H. (2003). Ray Shooting and Stone Throwing. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_43
Download citation
DOI: https://doi.org/10.1007/978-3-540-39658-1_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20064-2
Online ISBN: 978-3-540-39658-1
eBook Packages: Springer Book Archive