Area-Proportional Drawings of Intersecting Families of Simple Closed Curves

A FISC, or family of intersecting simple closed curves, is a collection of simple closed curves in the plane with the properties that there is some open region common to the interiors of all the curves, and that every two curves intersect in finitely many points or arcs. Let F be a FISC with a set of open regions R. F is said to be area-proportional with respect to weight function ω : R → R if there is a positive constant α such that for any two finite regions, r1 and r2, area(r1)/area(r2) = αω(r1)/ω(r2). We consider F as a directed plane graph, ~ G(F), where the curve intersections are vertices and the curve arcs between vertices are edges. Edges are directed so that each of F ’s curves is traversed in a clockwise fashion. The directed plane dual of ~ G(F), denoted ~ D(F), has edges oriented to indicate inclusion in fewer interiors of the curves. The graph ~ G(F) has an area-proportional drawing with respect to ω if there is some FISC C that is area-proportional to ω and where F can be transformed into C by a continuous transformation of the plane. We describe an O(n|V |) algorithm for creating an area-proportional drawing of ~ G(F) = (V,E) where F is a FISC with n curves and ~ D(F) has only one source and only one sink. For the case of n-Venn diagrams, since |V | ≤ 2 − 2, this yields an O(|V |lg|V |) drawing algorithm.