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

Authors:
M. McKenna;J. O'Rourke
Affiliations:
Department of Computer Science, The Johns Hopkins University, Baltimore, MD;Department of Computer Science, The Johns Hopkins University, Baltimore, MD
Venue:
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Year:
1988

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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 &Ogr;(n4&agr;(n)) algorithm for building a data structure that permits enumeration of mutually inaccessible classes of such red lines, where &agr;(n) is the inverse Ackermann function. The core of the algorithm is a construction of &Ogr;(n2) 2-D arrangement of hyperbolas, each in &Ogr;(n2&agr;(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 &Ogr;(n4&agr;(n)) algorithms in each case.