Generating Effective Euler Diagrams

Euler diagrams are used for visualizing categorized data,with applications including crime control, bioinformatics, classification systems and education. Various properties of Euler diagrams have been empirically shown to aid, or hinder, their comprehension by users. Therefore, a key goal is to automatically generate Euler diagrams that possess beneficial layout features whilst avoiding those that are a hindrance.The automated layout techniques that currently exist sometimes produce diagrams with undesirable features. In this paper we present a novel approach, called iCurves, for generating Euler diagrams alongside a prototype implementation. We evaluate iCurves against existing techniques based on the aforementioned layout properties. This evaluation suggests that, particularly when the number of zones is high, iCurves can outperform other automated techniques in terms of effectiveness for users, as indicated by the layout properties of the produced Euler diagrams

Two-Loop Diagrammatics in a Self-Dual Background

Diagrammatic rules are developed for simplifying two-loop QED diagrams with propagators in a constant self-dual background field. This diagrammatic analysis, using dimensional regularization, is used to explain how the fully renormalized two-loop Euler-Heisenberg effective Lagrangian for QED in a self-dual background field is naturally expressed in terms of one-loop diagrams. The connection between the two-loop and one-loop vacuum diagrams in a background field parallels a corresponding connection for free vacuum diagrams, without a background field, which can be derived by simple algebraic manipulations. It also mirrors similar behavior recently found for two-loop amplitudes in N=4 SUSY Yang-Mills theory.Comment: 16 pp, Latex, Axodra

Self-duality, helicity and background field loopology

I show that helicity plays an important role in the development of rules for computing higher loop effective Lagrangians. Specifically, the two-loop Heisenberg-Euler effective Lagrangian in quantum electrodynamics is remarkably simple when the background field has definite helicity (i.e., is self-dual). Furthermore, the two-loop answer can be derived essentially algebraically, and is naturally expressed in terms of one-loop quantities. This represents a generalization of the familiar ``integration-by-parts'' rules for manipulating diagrams involving free propagators to the more complicated case where the propagators are those for scalars or spinors in the presence of a background field.Comment: 12 pages; 1 figure; Plenary talk at QCD2004, Minnesot

How Should We Use Colour in Euler Diagrams?

This paper addresses the problem of how best to use colour in Euler diagrams. The choice of using coloured curves, rather than black curves, possibly with coloured fill is often made in tools that automatically draw Euler diagrams for information visualization as well as when they are drawn manually. We address the problem by empirically evaluating various different colour treatments: coloured or black curves combined with either no fill or coloured fill. By collecting performance data, we conclude that Euler diagrams with coloured curves and no fill significantly outperform all other colour treatments. Most automated layout algorithms adopt colour fill and are, thus, reducing the effectiveness of the Euler diagrams produced. As Euler diagrams can be used in a multitude of areas, ranging from crime control to social network analysis, our results stand to increase the ability of users to accurately and quickly extract information from their visualizations

The State-of-the-Art of Set Visualization

Sets comprise a generic data model that has been used in a variety of data analysis problems. Such problems involve analysing and visualizing set relations between multiple sets defined over the same collection of elements. However, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into six main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address these challenges. Further resources on set visualization are available at http://www.setviz.net

Does the Orientation of an Euler Diagram Affect User Comprehension?

Euler diagrams, which form the basis of numerous visual languages, can be an effective representation of information when they are both well-matched and well-formed. However, being well-matched and well-formed alone does not imply effectiveness. Other diagrammatical properties need to be considered. Information visualization theorists have known for some time that orientation has the potential to affect our interpretation of diagrams. This paper begins by explaining why well-matched and well-formed drawing principles are insufficient and discusses why we should study the orientation of Euler diagrams. To this end an empirical study is presented, designed to observe the effect of orientation upon the comprehension of Euler diagrams. The paper concludes that the orientation of Euler diagrams does not significantly affect comprehension

Evaluating the Comprehension of Euler Diagrams

We describe an empirical investigation into layout criteria that can help with the comprehension of Euler diagrams. Euler diagrams are used to represent set inclusion in applications such as teaching set theory, database querying, software engineering, filing system organisation and bio-informatics. Research in automatically laying out Euler diagrams for use with these applications is at an early stage, and our work attempts to aid this research by informing layout designers about the importance of various Euler diagram aesthetic criteria. The three criteria under investigation were: contour jaggedness, zone area inequality and edge closeness. Subjects were asked to interpret diagrams with different combinations of levels for each of the criteria. Results for this investigation indicate that, within the parameters of the study, all three criteria are important for understanding Euler diagrams and we have a preliminary indication of the ordering of their importance