Convex-Arc Drawings of Pseudolines

Abstract

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar-each crossing is incident to an unbounded face-and simple-each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.

Polyline drawings with topological constraints

“…Lemma 1 and the result of Eppstein et al [11] proving the existence of an arrangement of n pseudolines with curve complexity Ω(n) imply the following.…”

“…Denote by L N the arrangement of N pseudolines defined by Eppstein et al [11]. By the argument above, any polyline drawing that partially preserves the topology of the graph G L N contains a sub-drawing of G L N that fully preserves its topology and that hence has curve complexity Ω(N ) by Lemma 1.…”

“…Drawing Γ fully preserves the topology of G if the planarization of G (i.e., the planar simple topological graph obtained from G by replacing crossings with dummy vertices) and the planarization of Γ have the same planar embedding. Eppstein et al [11] prove the existence of a simple arrangement of n pseudolines that, when drawn with polylines, it requires at least one pseudoline to have Ω(n) bends. It is not hard to see that the result by Eppstein et al implies the existence of an n-vertex simple topological graph such that any polyline drawing that fully preserves its topology has curve complexity Ω(n) (see Corollary 2 in Section 2).…”