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 {Mathematical expression} upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set of n lines, where c is a suitable constant. This bound is almost tight. (2) An O(n1.5+ε) randomized expected time algorithm that tests whether a direction v exists along which a set of n red lines can be translated away from a set of n blue lines without collisions. ε>0 is an arbitrary small but fixed constant. (3) An {Mathematical expression} upper bound on the worst-case complexity of the envelope of lines above a terrain with n edges, where c is a suitable constant. (4) An algorithm for computing the intersection of two polyhedral terrains in 3-space with n total edges in time O(n4/3+ε+k1/3 n1+ε+klog2 n), where k is the size of the output, and ε>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelle et al. [5]. The tools used to obtain these results include Plücker coordinates of lines, random sampling, and polarity transformations in 3-space. © 1994 Springer-Verlag New York Inc.