Chazelle, Bernard; Edelsbrunner, HerbertISTA ; Guibas, Leonidas; Sharir, Micha; Stolfi, Jorge
Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in three-dimensional space. Our main results include: 1. A tight Θ(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to the n given lines. 2. A similar bound of Θ(n3) for the complexity of the set of all lines passing above the n given lines. 3. A preprocessing procedure using O(n2+ε) time and storage, for any ε > 0, that builds a structure supporting O(logn)-time queries for testing if a line lies above all the given lines. 4. An algorithm that tests the "towering property" in O(n4/3+ε) time, for any ε > 0: do n given red lines lie all above n given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space and ε-nets for various geometric range spaces.
Work on this paper by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work on this paper by Herbert Edelsbrunner has been supported by NSF Grant CCR-87-14565. Work on this paper by Leonidas Guibas has been supported by grants from the Mitsubishi and Toshiba Corporations. Work on this paper by Micha Sharir has been supported by ONR Grant N00014-87-K-0129, by NSF Grants DCR-83-20085 and CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the NCRD — the Israeli National Council for Research and Development, and the EMET Fund of the Israeli Academy of Sciences.
428 - 447
Chazelle B, Edelsbrunner H, Guibas L, Sharir M, Stolfi J. Lines in space: Combinatorics and algorithms. Algorithmica. 1996;15(5):428-447. doi:10.1007/BF01955043
Chazelle, B., Edelsbrunner, H., Guibas, L., Sharir, M., & Stolfi, J. (1996). Lines in space: Combinatorics and algorithms. Algorithmica. Springer. https://doi.org/10.1007/BF01955043
Chazelle, Bernard, Herbert Edelsbrunner, Leonidas Guibas, Micha Sharir, and Jorge Stolfi. “Lines in Space: Combinatorics and Algorithms.” Algorithmica. Springer, 1996. https://doi.org/10.1007/BF01955043.
B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, and J. Stolfi, “Lines in space: Combinatorics and algorithms,” Algorithmica, vol. 15, no. 5. Springer, pp. 428–447, 1996.
Chazelle B, Edelsbrunner H, Guibas L, Sharir M, Stolfi J. 1996. Lines in space: Combinatorics and algorithms. Algorithmica. 15(5), 428–447.
Chazelle, Bernard, et al. “Lines in Space: Combinatorics and Algorithms.” Algorithmica, vol. 15, no. 5, Springer, 1996, pp. 428–47, doi:10.1007/BF01955043.