Drawing Area-Proportional Venn-3 Diagrams with Convex Polygons

Area-proportional Venn diagrams are a popular way of visualizing the relationships between data sets, where the set intersections have a specified numerical value. In these diagrams, the areas of the regions are in proportion to the given values. Venn-3, the Venn diagram consisting of three intersecting curves, has been used in many applications, including marketing, ecology and medicine. Whilst circles are widely used to draw such diagrams, most area specifications cannot be drawn in this way and, so, should only be used where an approximate solution is acceptable. However, placing different restrictions on the shape of curves may result in usable diagrams that have an exact solution, that is, where the areas of the regions are exactly in proportion to the represented data. In this paper, we explore the use of convex shapes for drawing exact area proportional Venn-3 diagrams. Convex curves reduce the visual complexity of the diagram and, as most desirable shapes (such as circles, ovals and rectangles) are convex, the work described here may lead to further drawing methods with these shapes. We describe methods for constructing convex diagrams with polygons that have four or five sides and derive results concerning which area specifications can be drawn with them. This work improves the state-of-the-art by extending the set of area specifications that can be drawn in a convex manner. We also show how, when a specification cannot be drawn in a convex manner, a non-convex drawing can be generated.