The Community for Technology Leaders
RSS Icon
Issue No.07 - July (2011 vol.17)
pp: 1020-1032
Gem Stapleton , University of Brighton, Brighton
John Howse , University of Brighton, Brighton
Peter Rodgers , University of Kent, Kent
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.
Automated diagram drawing, Euler diagrams, diagrammatic reasoning, information visualization.
Gem Stapleton, John Howse, Peter Rodgers, "Drawing Euler Diagrams with Circles: The Theory of Piercings", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 7, pp. 1020-1032, July 2011, doi:10.1109/TVCG.2010.119
[1] S. Chow and F. Ruskey, "Drawing Area-Proportional Venn and Euler Diagrams," Proc. Conf. Graph Drawing, pp. 466-477, Sept. 2003.
[2] J. Flower, J. Howse, and J. Taylor, "Nesting in Euler Diagrams: Syntax, Semantics and Construction," Software and Systems Modelling, vol. 3, pp. 55-67, Mar. 2004.
[3] P. Rodgers, L. Zhang, G. Stapleton, and A. Fish, "Embedding Wellformed Euler Diagrams," Proc. 12th Int'l Conf. Information Visualization, pp. 585-593, 2008.
[4] A. Verroust and M.-L. Viaud, "Ensuring the Drawability of Euler Diagrams for up to Eight Sets," Proc. Third Int'l Conf. Theory and Application of Diagrams, pp. 128-141, 2004.
[5] J. Flower and J. Howse, "Generating Euler Diagrams," Proc. Second Int'l Conf. Theory and Application of Diagrams, pp. 61-75, Apr. 2002.
[6] P. Simonetto, D. Auber, and D. Archambault, "Fully Automatic Visualisation of Overlapping Sets," Computer Graphics Forum, vol. 28, no. 3, pp. 967-974, 2009.
[7] J. Flower, A. Fish, and J. Howse, "Euler Diagram Generation," J. Visual Languages and Computing, vol. 19, pp. 675-694, 2008.
[8] G. Stapleton, J. Howse, P. Rodgers, and L. Zhang, "Generating Euler Diagrams from Existing Layouts," Proc. Workshop Layout of Software Eng. Diagrams, 2008.
[9] S. Kent, "Constraint Diagrams: Visualizing Invariants in Object Oriented Modelling," Proc. Conf. Object-Oriented Programming, Systems, Languages and Applications (OOPSLA '97), pp. 327-341, Oct. 1997.
[10] S.-K. Kim and D. Carrington, "Visualization of Formal Specifications," Proc. Sixth Asia Pacific Software Eng. Conf., pp. 102-109, 1999.
[11] I. Oliver, J. Howse, G. Stapleton, E. Nuttila, and S. Törmä, "Expressing Ontologies Using Diagrammatic Logics," Proc. Int'l Semantic Web Conf. (Posters and Demos), http://kcap09.stanford. edu/shareposterDemos , 2010.
[12] Y. Zhao and J. Lövdahl, "A Reuse Based Method of Developing the Ontology for E-Procurement," Proc. Nordic Conf. Web Services, pp. 101-112, 2003.
[13] P. Artes and B. Chauhan, "Longitudinal Changes in the Visual Field and Optic Disc in Glaucoma," Progress in Retinal and Eye Research, vol. 24, no. 3, pp. 333-354, 2005.
[14] R. DeChiara, U. Erra, and V. Scarano, "VennFS: A Venn Diagram File Manager," Proc. Seventh Int'l Conf. Information Visualization (IV '03), pp. 120-126, 2003.
[15] E. Hammer, Logic and Visual Information. CSLI Publications, 1995.
[16] H. Kestler, A. Muller, J. Kraus, M. Buchholz, T. Gress, H. Liu, D. Kane, B. Zeeberg, and J. Weinstein, "Vennmaster: Area-Proportional Euler Diagrams for Functional GO Analysis of Microarrays," BMC Bioinformatics, vol. 9, p. 67, 2008.
[17] T. Quick, C. Nehaniv, K. Dautenhahn, and G. Roberts, "Sensorimotor Information Flow in Genetic Regulatory Network Driven Control Systems," Technical Report Research Note RN/05/29, Univ. College London.
[18] S.-J. Shin, The Logical Status of Diagrams. Cambridge Univ. Press, 1994.
[19] N. Swoboda and G. Allwein, "Using DAG Transformations to Verify Euler/Venn Homogeneous and Euler/Venn FOL Heterogeneous Rules of Inference," J. Software and System Modeling, vol. 3, no. 2, pp. 136-149, 2004.
[20] J. Flower, P. Rodgers, and P. Mutton, "Layout Metrics for Euler Diagrams," Proc. Seventh Int'l Conf. Information Visualisation, pp. 272-280, 2003.
[21] P. Rodgers, L. Zhang, and A. Fish, "General Euler Diagram Generation," Proc. Int'l Conf. Theory and Application of Diagrams, Sept. 2008.
[22] P. Simonetto and D. Auber, "An Heuristic for the Construction of Intersection Graphs," Proc. 13th Int'l Conf. Information Visualisation, 2009.
[23] G. Stapleton, P. Rodgers, J. Howse, and L. Zhang, "Inductively Generating Euler Diagrams," to be published in IEEE Trans. Visualization and Computer Graphics, 2009.
[24] S. Chow, "Generating and Drawing Area-Proportional Euler and Venn Diagrams," PhD dissertation, Univ. of Victoria, 2007.
[25] G. Stapleton, P. Rodgers, J. Howse, and J. Taylor, "Properties of Euler Diagrams," Proc. Workshop Layout of Software Eng. Diagrams, pp. 2-16, 2007.
[26] A. Fish and J. Flower, "Abstractions of Euler Diagrams," Proc. First Int'l Workshop Euler Diagrams, vol. 134, pp. 77-101, 2005.
[27] A. Fish and G. Stapleton, "Formal Issues in Languages Based on Closed Curves," Proc. Distributed Multimedia Systems, Int'l Workshop Visual Languages and Computings, pp. 161-167, 2006.
70 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool