Lines Through Segments in Three Dimensional Space

We present an efficient output-sensitive algorithm and its exact implementation to solve the following problem: Given a set S of n line segments in threedimensional space, find all the lines that simultaneously intersect quadruples of line segments in S. We refer to this problem as the lines-through-segments problem, or LTS for short. The algorithm properly handles all degenerate cases. Since we do not assume general position, we compute all lines that intersect at least four segments in S. The algorithm runs in O((n + I) log n) time, and requires O(n + I) working space, where I is the output size; I is bounded by O(n). We use Cgal arrangements and in particular its support for two-dimensional arrangements in the plane and on the sphere to efficiently compute the intersecting lines in an exact manner. We also report on the performance of our algorithm and its implementation compared to others. The source code of the LTS program as well as the input examples for the experiments can be obtained from http://acg. cs.tau.ac.il/projects/lts.