Embedding wellformed Euler diagrams

Euler diagrams are collections of labelled closed curves. They are often used to represent information about the relationship between sets and, as such, they have numerous applications including: visualizing biological data, diagrammatic logics, and visual database querying. Various methods to automatically generate Euler diagrams have been proposed recently. Typically, the generation process starts with an abstract description of an Euler diagram, which is then converted to a planar dual graph. Finally, the process attempts to embed the Euler diagram from the dual graph. This paper describes a method for embedding wellformed Euler diagrams from dual graphs. There are several mechanisms to generate dual graphs but, prior to the novel work described here, no general method for embedding a wellformed Euler diagram from a dual graph had been demonstrated. The method in this paper achieves an embedding of any wellformed Euler diagram. The method first triangulates the dual graph. Then, using the faces of the triangulated graph, an edge labelling technique identifies the vertices of polygons which form the closed curves of the Euler diagram. The method is demonstrated by a Java implementation. In addition, this paper discusses a number of layout improvements that can be explored for this embedding method.