7,375 research outputs found
A Survey of Euler Diagrams
No full textEuler diagrams visually represent containment, inpresstersection and exclusion using closed curves. They first appeared several hundred years ago, however, there has been a resurgence in Euler diagram research in the twenty-first century. This was initially driven by their use in visual languages, where they can be used to represent logical expressions diagrammatically. This work lead to the requirement to automatically generate Euler diagrams from an abstract description. The ability to generate diagrams has accelerated their use in information visualization, both in the standard case where multiple grouping of data items inside curves is required and in the area-proportional case where the area of curve intersections is important. As a result, examining the usability of Euler diagrams has become an important aspect of this research. Usability has been investigated by empirical studies, but much research has concentrated on wellformedness, which concerns how curves and other features of the diagram interrelate. This work has revealed the drawability of Euler diagrams under various wellformedness properties and has developed embedding methods that meet these properties.
Euler diagram research surveyed in this paper includes theoretical results, generation techniques, transformation methods and the development of automated reasoning systems for Euler diagrams. It also overviews application areas and the ways in which Euler diagrams have been extended
Properties of Euler Diagrams
No full textEuler diagrams have numerous application areas, with a large variety of languages based on them. In relation to software engineering, such areas encompass modelling and specification including from a formal perspective. In all of these application areas, it is desirable to provide tools to layout Euler diagrams, ideally in a nice way. Various notions of niceness can be correlated with certain properties that an Euler diagram may or may not possess. Indeed, the relevant layout algorithms developed to date produce Euler diagrams that have certain sets of properties, sometimes called well-formedness conditions. However, there is not a commonly agreed definition of an Euler diagram and the properties imposed on them are rarely stated precisely. In this paper, we provide a very general definition of an Euler diagram, which can be constrained in varying ways in order to match the variety of definitions that exist in the literature. Indeed, the constraints imposed correspond to properties that the diagrams may possess. A contribution of this paper is to provide formal definitions of these properties and we discuss when these properties may be desirable. Our definition of an Euler diagram and the formalization of these properties provides a general language for the Euler diagram community to utilize. A consequence of using a common language will be better integration of, and more accessible, research results
Euler Graph Transformations for Euler Diagram Layout
No full textEuler diagrams are frequently used for visualizing information about collections of objects and form an important component of various visual languages. Properties possessed by Euler diagrams correlate with their usability, such as whether the diagram has only simple curves or possesses concurrency. Sometimes, every diagram that represents some given information possesses some undesirable properties, and reducing the number of violations of undesirable properties is beneficial. In this paper we show how to count the number of violations from the reduced Euler graph. We then define various transformations on the Euler graph which can reduce the number of violations of a given property, but sometimes at the expense of increasing the number of violations of another property. These transformations can be used to improve the quality of the drawn diagram, which is important for effective information visualization
Embedding wellformed Euler diagrams
No full textEuler 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
Changing Euler Diagram Properties by Edge Transformation of Euler Dual Graphs
No full textEuler diagrams form the basis of several visual modelling notations, including statecharts and constraint diagrams. Recently, various techniques for automated Euler diagram drawing have been proposed, contributing to the Euler diagram generation problem: given an abstract description, draw an Euler diagram with that description and which possesses certain properties. A common generation method is to find a dual graph from which an Euler diagram is subsequently created. In this paper we define transformations of the dual graph that allow us to alter the properties that the generated diagram possesses. In addition, because the dual graph of a previously generated diagram can be found, our transformations can be used to take such a diagram and produce a new diagram with the same abstract description, but with different properties. As a result, we can produce a variety of different diagrams for any given abstract description, allowing us to choose an Euler diagram that conforms to the properties that a user prefers
Generating Euler Diagrams from Existing Layouts
No full textEuler diagrams have a wide variety of uses, from information visualization to logical reasoning. In all of their application areas, the ability to automatically layout Euler diagrams brings considerable benefits. In this paper, we present a novel approach to Euler diagram generation. We develop certain graphs associated with Euler diagrams in order to allow curves to be added by finding cycles in these graphs. This permits us to build Euler diagrams inductively, adding one curve at a time. Our technique is adaptable, allowing the easy specification, and enforcement, of sets of wellformednesss conditions; we present a series of results that identify properties of cycles that correspond to the wellformedness conditions. This improves upon other contributions towards the automated generation of Euler diagrams which implicitly assume some fixed set of wellformedness conditions must hold. In addition, unlike most of these other generation methods, our technique allows any abstract description to be drawn as an Euler diagram. To establish the utility of the approach, a prototype implementation has been developed
General Euler Diagram Generation
No full textEuler diagrams are a natural method of representing set-theoretic data and have been employed in diverse areas such as Visualizing statistical data, as a basis for diagrammatic logics and for displaying the results of database search queries. For effective use of Euler diagrams in practical computer based applications, the generation of a diagram as a set of curves from an abstract description is necessary. Various practical methods for Euler diagram generation have been proposed, but in all of these methods the diagrams that can be produced are only for it restricted Subset of all possible abstract descriptions. We describe a method for Euler diagram generation, demonstrated by implemented software. and illustrate the advances in methodology via the production of diagrams which were difficult or impossible to draw using previous approaches. To allow the generation of all abstract descriptions we may be reqUired to have some properties of the final diagram that are not considered nice. In particular we permit more than two curves to pass though it single point, permit sonic curve segments to be drawn Concurrently, and permit duplication of curve labels. However, Our method attempts to minimize these bad properties according to it chosen prioritization
Drawing Euler Diagrams with Circles
No full textEuler diagrams are a popular and intuitive visualization tool which are used in a wide variety of application areas, including biological and medical data analysis. As with other data visualization methods, such as graphs, bar charts, or pie charts, the automated generation of an Euler diagram from a suitable data set would be advantageous, removing the burden of manual data analysis and the subsequent task of drawing an appropriate diagram. To this end, various methods have emerged that automatically draw Euler diagrams from abstract descriptions of them. One such method draws some, but not all, abstract descriptions using only circles. We extend that method so that more abstract descriptions can be drawn with circles. Furthermore, we show how to transform any `undrawable' abstract description into a drawable one. Thus, given any abstract description, our method produces a drawing using only circles. The method has been implemented in a software tool available for download
Automatically Drawing Euler Diagrams with Circles
No full textEuler diagrams are used for visualizing categorized data. These categories, together with information about when categories share some datum, can be turned into a succinct diagram description from which an Euler diagram can be generated. Closed curves represent the categories and the relationships between the curves (such as containment) correspond to relationships between the categories (such as subset). A range of automated Euler diagram drawing methods have been proposed but they often produce diagrams that are aesthetically unpleasing, can be computationally complex and most of them cannot draw a diagram for some (often many) given collections of categories. One such method is capable of drawing aesthetically pleasing Euler diagrams, using only circles, and is computationally efficient (being of polynomial time complexity) but it applies to a very restricted subset of collections of categorized data. This paper substantially extends that method so it can always draw an Euler diagram, that is it applies to all collections of categorized data. In particular, we identify a class of diagram descriptions that can be drawn with circles, generalizing previous work. For diagram descriptions outside of this class, we define transformations that can be used to turn them into descriptions inside the drawable with circles class. We demonstrate how such transformations can be done in a general, a process during which many choices must be made. Further, we provide strategies for making particular choices which ensure desirable properties, such as curve containment, are preserved. We have provided a software implementation of the drawing method, which is freely available from www.eulerdiagrams.com/inductivecircles.htm
A Graph Theoretic Approach to General Euler Diagram Drawing
No full textAbstractEuler diagrams are used in a wide variety of areas for representing information about relationships between collections of objects. Recently, several techniques for automated Euler diagram drawing have been proposed, contributing to the Euler diagram generation problem: given an abstract description, draw an Euler diagram with that description and which possesses certain properties, sometimes called well-formedness conditions. We present the first fully formalized, general framework that permits the embedding of Euler diagrams that possess any collection of the six typically considered well-formedness conditions. Our method first converts the abstract description into a vertex-labelled graph. An Euler diagram can then be formed, essentially by finding a dual graph of such a graph. However, we cannot use an arbitrary plane embedding of the vertex-labelled graph for this purpose. We identify specific embeddings that allow the construction of appropriate duals. From these embeddings, we can also identify precisely which properties the drawn Euler diagram will possess and ‘measure’ the number of times that each well-formedness condition is broken. We prove that every abstract description can be embedded using our method. Moreover, we identify exactly which (large) class of Euler diagrams can be generated
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