Arrangement of Lines - Other Types of Arrangement

Other Types of Arrangement

A hyperbolic line arrangement combinatorially equivalent to a chord diagram used by Ageev (1996) to show that triangle-free circle graphs may sometimes need 5 colors.

A pseudoline arrangement is a family of curves that share similar topological properties with a line arrangement. These can be defined most simply in the projective plane as simple closed curves any two of which meet in a single crossing point. A pseudoline arrangement is said to be stretchable if it is combinatorially equivalent to a line arrangement; it is complete for the existential theory of the reals to distinguish stretchable arrangements from non-stretchable ones.

Arrangements of lines in the hyperbolic plane have also been studied. Any finite set of lines in the Euclidean plane has a combinatorially equivalent arrangement in the hyperbolic plane (e.g. by enclosing the vertices of the arrangement by a large circle and interpreting the interior of the circle as a Klein model of the hyperbolic plane). However, in hyperbolic line arrangements lines may avoid crossing each other without being parallel; the intersection graph of the lines in a hyperbolic arrangement is a circle graph. The corresponding concept to hyperbolic line arrangements for pseudolines is a weak pseudoline arrangement, a family of curves having the same topological properties as lines such that any two curves in the family either meet in a single crossing point or have no intersection.

More generally, geometers have studied arrangements of other types of curves in the plane, arrangements of hyperplanes in higher dimensional spaces, and of other more complicated types of surface. Arrangements in complex vector spaces have also been studied; since complex lines do not partition the complex plane into multiple connected components, the combinatorics of vertices, edges, and cells does not apply to these types of space, but it is still of interest to study their symmetries and topological properties.