Lines in space: Combinatorics and algorithms
Chazelle, Bernard
Edelsbrunner, Herbert
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.
Springer
1996
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/4027
Chazelle B, Edelsbrunner H, Guibas L, Sharir M, Stolfi J. Lines in space: Combinatorics and algorithms. <i>Algorithmica</i>. 1996;15(5):428-447. doi:<a href="https://doi.org/10.1007/BF01955043">10.1007/BF01955043</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/BF01955043
info:eu-repo/semantics/altIdentifier/issn/0178-4617
info:eu-repo/semantics/closedAccess