30 Arrangements

Given a finite collection S of geometric objects such as hyperplanes or spheres in R , the arrangement A(S) is the decomposition of R into connected open cells of dimensions 0, 1, . . . , d induced by S. Besides being interesting in their own right, arrangements of hyperplanes have served as a unifying structure for many problems in discrete and computational geometry. With the recent advances in the study of arrangements of curved (algebraic) surfaces, arrangements have emerged as the underlying structure of geometric problems in a variety of “physical world” application domains such as robot motion planning and computer vision. This chapter is devoted to arrangements of hyperplanes and of curved surfaces in low-dimensional Euclidean space, with an emphasis on combinatorics and algorithms. In the first section we introduce basic terminology and combinatorics of arrangements. In Section 30.2 we describe substructures in arrangements and their combinatorial complexity. Section 30.3 deals with data structures for representing arrangements and with special refinements of arrangements. The following two sections focus on algorithms: algorithms for constructing full arrangements are described in Section 30.4, and algorithms for constructing substructures in Section 30.5. In Section 30.6 we discuss the relation between arrangements and other structures. Several applications of arrangements are reviewed in Section 30.7. Section 30.8 deals with robustness issues when implementing algorithms and data structures for arrangements and Section 30.9 surveys software implementations. We conclude in Section 30.10 with a brief review of Davenport-Schinzel sequences, a combinatorial structure that plays an important role in the analysis of arrangements.