Lines in space: Combinatorics and algorithms

AbstractQuestions 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 ofn 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 then given lines.2.A similar bound of Θ(n3) for the complexity of the set of all lines passing above then given lines.3.A preprocessing procedure usingO(n2+ɛ) time and storage, for anyε>0, that builds a structure supportingO(logn)-time queries for testing if a line lies above all the given lines.4.An algorithm that tests the “towering property” inO(n2+ɛ) time, for anyε>0; don given red lines lie all aboven given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space andε-nets for various geometric range spaces.

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