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Drawing Euler diagrams with circles: The theory of piercings
 IEEE TRANS. ON VISUALISATION AND COMPUTER GRAPHICS
, 2011
"... Euler diagrams are effective tools for visualizing set intersections. They have a large number of application areas ranging from statistical data analysis to software engineering. However, the automated generation of Euler diagrams has never been easy: given an abstract description of a required Eu ..."
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Cited by 11 (7 self)
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Euler diagrams are effective tools for visualizing set intersections. They have a large number of application areas ranging from statistical data analysis to software engineering. However, the automated generation of Euler diagrams has never been easy: given an abstract description of a required Euler diagram, it is computationally expensive to generate the diagram. Moreover, the generated diagrams represent sets by polygons, sometimes with quite irregular shapes that make the diagrams less comprehensible. In this paper, we address these two issues by developing the theory of piercings, where we define single piercing curves and double piercing curves. We prove that if a diagram can be built inductively by successively adding piercing curves under certain constraints, then it can be drawn with circles, which are more esthetically pleasing than arbitrary polygons. The theory of piercings is developed at the abstract level. In addition, we present a Java implementation that, given an inductively pierced abstract description, generates an Euler diagram consisting only of circles within polynomial time.
Generating Euler Diagrams from Existing Layouts
, 2008
"... Euler diagrams have a wide variety of uses, from information visualization to logical reasoning. In the case of software engineering, they form the basis of a number of notations, such as state charts and constraint diagrams. In all of their application areas, the ability to automatically layout Eu ..."
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Cited by 9 (5 self)
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Euler diagrams have a wide variety of uses, from information visualization to logical reasoning. In the case of software engineering, they form the basis of a number of notations, such as state charts and constraint diagrams. In all of their application areas, the ability to automatically layout Euler diagrams brings considerable benefits. There have been several recent contributions towards the automatic generation and layout of Euler diagrams, all of which start from an abstract description of the diagram and produce a collection of closed curves embedded in the plane. In this paper, we are concerned with producing layouts by modifying existing ones. This type of layout approach is particularly useful in domains where we require an updated, or modified, diagram such as in a logical reasoning context. We provide two methods to add a curve to an Euler diagram in order to create a new diagram. The first method is guaranteed to produce layouts that meet specified wellformedness conditions that are typically chosen by others who produced generation algorithms; these conditions are thought to correlate well accurate user interpretation. We also overview a second method that can be used to produce a layout of any abstract description.
Drawing Euler Diagrams with Circles
"... Abstract. Euler 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 dia ..."
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Cited by 9 (8 self)
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Abstract. Euler 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. 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, allowing sets to be represented by multiple curves. Furthermore, we show how to transform any ‘undrawable ’ abstract description into a drawable one by adding in extra zones. Thus, given any abstract description, our method produces a drawing using only circles. A software implementation of the method is available for download. 1
Diagrammatic Reasoning System with Euler Circles: Theory and Experiment Design
 in Proceedings of the 5th international conference on Diagrammatic Representation and Inference (Diagrams 08), Lecture Notes In Artificial Intelligence
, 2008
"... This paper is concerned with Euler diagrammatic reasoning. Prooftheory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by many logicians. ..."
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Cited by 9 (7 self)
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This paper is concerned with Euler diagrammatic reasoning. Prooftheory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by many logicians. Euler diagrams were introduced in the 18th century by Leonhard Euler [1768]. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint, and there are only few prooftheoretical investigations. Accordingly, in order to ll this gap, we formalize an Euler diagrammatic inference system and prove the soundness and completeness theorems with respect to a formal settheoretical semantics. We further consider, from a prooftheoretical viewpoint, the structure of diagrammatic proofs and manners of their construction. Contents 1
A Survey of Euler Diagrams
, 2013
"... Euler diagrams visually represent containment, intersection and exclusion using closed curves. They first appeared several hundred years ago, however, there has been a resurgence in Euler diagram research in the twentyfirst century. This was initially driven by their use in visual languages, where ..."
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Cited by 6 (1 self)
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Euler diagrams visually represent containment, intersection and exclusion using closed curves. They first appeared several hundred years ago, however, there has been a resurgence in Euler diagram research in the twentyfirst 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 areaproportional 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.
Changing Euler Diagram Properties by Edge Transformation of Euler Dual Graphs
 VL/HCC 2009
"... Euler 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 ..."
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Cited by 5 (4 self)
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Euler 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 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. 1
Drawing AreaProportional Euler Diagrams Representing Up To Three Sets
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
"... Areaproportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visual ..."
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Cited by 5 (3 self)
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Areaproportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact areaproportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn2 and Venn3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler3 diagrams in an areaproportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where nonconvex curves are necessary, our method draws an appropriate diagram using nonconvex polygons. Thus, we are now always able to automatically visualize data for up to three sets.
A General Method for Drawing AreaProportional Euler Diagrams
, 2011
"... Areaproportional Euler diagrams have many applications, for example they are often used for visualizing data in medical and biological domains. There have been a number of recent research efforts to automatically draw Euler diagrams when the areas of the regions are not considered, leading to a ran ..."
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Cited by 4 (4 self)
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Areaproportional Euler diagrams have many applications, for example they are often used for visualizing data in medical and biological domains. There have been a number of recent research efforts to automatically draw Euler diagrams when the areas of the regions are not considered, leading to a range of different drawing techniques. By contrast, substantially less progress has been made on the problem of automatically drawing areaproportional Euler diagrams, although some partial results have been derived. In this paper, we considerably advance the stateoftheart in areaproportional Euler diagram drawing by presenting the first method that is capable of generating such a diagram given any areaproportional specification. Moreover, our drawing method is sufficiently flexible that it allows one to specify which of the typically enforced wellformedness conditions should be possessed by the tobedrawn Euler diagram.
Euler Graph Transformations for Euler Diagram Layout
 IEEE Symposium on Visual Languages and Human Centric Computing (VL/HCC 2010
, 2010
"... Abstract — Euler 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 c ..."
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Cited by 4 (3 self)
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Abstract — Euler 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. KeywordsEuler diagrams; Venn diagrams; graph transformations. I.
An Heuristic for the Construction of Intersection Graphs
 13TH INTERNATIONAL CONFERENCE ON INFORMATION VISUALISATION (IV09), BARCELONA: SPAIN
, 2009
"... Most methods for generating Euler diagrams describe the detection of the general structure of the final drawing as the first step. This information is generally encoded using a graph, where nodes are the regions to be represented and edges represent adjacency. A planar drawing of this graph will the ..."
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Cited by 3 (0 self)
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Most methods for generating Euler diagrams describe the detection of the general structure of the final drawing as the first step. This information is generally encoded using a graph, where nodes are the regions to be represented and edges represent adjacency. A planar drawing of this graph will then indicate how to draw the sets in order to depict all the set intersections. In this paper we present an heuristic to construct this structure, the intersection graph. The final Euler diagram can be constructed by drawing the sets boundaries around the nodes of the intersection graph, either manually or automatically.