On lines missing polyhedral sets in 3-space

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(n ^1.5+ε) 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. ε>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(n ^4/3+ε+k ^1/3 n ^1+ε+klog² n), wherek 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 Chazelleet al. 5].

The tools used to obtain these results include Plücker coordinates of lines, random sampling, and polarity transformations in 3-space.