On lines missing polyhedral sets in 3-space

Abstract

We show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:(1)An $$O(n^3 2^{c\sqrt {\log n} } )$$ upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set ofn lines, wherec is a suitable constant. This bound is almost tight.(2)AnO(n1.5+?) randomized expected time algorithm that tests whether a directionv exists along which a set ofn red lines can be translated away from a set ofn blue lines without collisions. ?>0 is an arbitrary small but fixed constant.(3)An $$O(n^3 2^{c\sqrt {\log n} } )$$ upper bound on the worst-case complexity of theenvelope of lines above a terrain withn edges, wherec is a suitable constant.(4)An algorithm for computing the intersection of two polyhedral terrains in 3-space withn total edges in timeO(n4/3+?+k1/3n1+?+klog2n), wherek is the size of the output, and ?>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelleet al. [5]. The tools used to obtain these results include Plucker coordinates of lines, random sampling, and polarity transformations in 3-space.