Convex-Arc Drawings of Pseudolines

Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have distance-preserving embeddings into hypercubes [6]. They are also closely related to oriented matroids [11]. We consider here the visualization of arrangements using well-shaped curves. Primarily, we study weak outerplanar pseudoline arrangements. An arrangement is weak if it does not necessarily have a crossing for every pair of pseudolines [12], and outerplanar if every crossing is part of an unbounded face of the arrangement. We show that these arrangements can be drawn with all curves convex, either as polygonal chains with at most two bends per pseudoline or as semicircles above a line. Arbitrary pseudolines can also be drawn as convex curves, but may require linearly many bends.