Pseudoline Arrangements

Abstract

Pseudoline arrangements generalize in a natural way arrangements of straight lines, discarding the straightness aspect, but preserving their basic topological and combinatorial properties. Elementary and intuitive in nature, at the same time, by the Folkman-Lawrence topological representation theorem (see Chapter 6), they provide a concrete geometric model for oriented matroids of rank 3. After their explicit description by Levi in the 1920’s, and the subsequent development of the theory by Ringel in the 1950’s, the major impetus was given in the 1970’s by Grünbaum’s monograph Arrangements and Spreads, in which a number of results were collected and a great many problems and conjectures posed about arrangements of both lines and pseudolines. The connection with oriented matroids discovered several years later led to further work. The theory is by now very well developed, with many combinatorial and topological results and connections to other areas as for example algebraic combinatorics, as well as a large number of applications in computational geometry. In comparison to arrangements of lines arrangements of pseudolines have the advantage that they are more general and allow for a purely combinatorial treatment. Section 5.1 is devoted to the basic properties of pseudoline arrangements, and Section 5.2 to related structures, such as arrangements of straight lines, configurations (and generalized configurations) of points, and allowable sequences of permutations. (We do not discuss the connection with oriented matroids, however; that is included in Chapter 6.) In Section 5.3 we discuss the stretchability problem. Section 5.4 summarizes some combinatorial results known about line and pseudoline arrangements, in particular problems related to the cell structure of arrangements. Section 5.5 deals with results of a topological nature and Section 5.6 with issues of combinatorial and computational complexity. Section 5.7 with several applications, including sweeping arrangements and pseudotriangulations. Unless otherwise noted, we work in the real projective plane P.