Arrangements of lines in 3-space: A data structure with applications

Abstract

Let an arrangement of blue lines in 3-space be fixed, and imagine a movable red line entangled in the arrangement. We show an O(n4α(n)) algorithm for building a data structure that permits enumeration of mutually inaccessible classes of such red lines, where α(n) is the inverse Ackermann function. The core of the algorithm is a construction of O(n2) 2-D arrangements of hyperbolas, each in O(n2α(n)) time. The algorithm is applied to stabbing 3-polytopes, enumerating pairwise-visible face pairs, enumerating 2-D projections of convex 4-polytopes, and other problems, resulting in O(n4α(n)) algorithms in each case.