Date: 31 May 2005

Arrangements of curves in the plane — topology, combinatorics, and algorithms

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Abstract

Arrangements of curves in the plane are of fundamental significance in many problems of computational and combinatorial geometry (e.g. motion planning, algebraic cell decomposition, etc.). In this paper we study various topological and combinatorial properties of such arrangements under some mild assumptions on the shape of the curves, and develop basic tools for the construction, manipulation, and analysis of these arrangements. Our main results include a generalization of the zone theorem of [EOS], [CGL] to arrangements of curves (in which we show that the combinatorial complexity of the zone of a curve is nearly linear in the number of curves), and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.

Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under grant CCR-8714566. Work on this paper by the third and sixth authors has been supported by Office of Naval Research Grant N00014-82-K-0381, by National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the sixth author has also been supported by a research grant from the NCRD — the Israeli National Council for Research and Development. Work by the fourth author has been supported by National Science Foundation Grant DMS-8501947.