# Introducing 3D Venn and Euler Diagrams

Rodgers, Peter, Flower, Jean, Stapleton, Gem (2012) Introducing 3D Venn and Euler Diagrams. In: Chapman, Peter and Micallef, Luana, eds. Proceedings of the 3rd International Workshop on Euler Diagrams 2012. CEUR-WS , 854. pp. 92-106. CEUR-WS.org

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http://www.cs.kent.ac.uk/pubs/2012/3216

## Abstract

In 2D, Venn and Euler diagrams consist of labelled simple closed curves and have been widely studied. The advent of 3D display and interaction mechanisms means that extending these diagrams to 3D is now feasible. However, 3D versions of these diagrams have not yet been examined. Here, we begin the investigation into 3D Euler diagrams by defining them to comprise of labelled, orientable closed surfaces. As in 2D, these 3D Euler diagrams visually represent the set-theoretic notions of intersection, containment and disjointness. We extend the concept of wellformedness to the 3D case and compare it to wellformedness in the 2D case. In particular, we demonstrate that some data can be visualized with wellformed 3D diagrams that cannot be visualized with wellformed 2D diagrams. We also note that whilst there is only one topologically distinct embedding of wellformed Venn-3 in 2D, there are four such em- beddings in 3D when the surfaces are topologically equivalent to spheres. Furthermore, we hypothesize that all data sets can be visualized with 3D Euler diagrams whereas this is not the case for 2D Euler diagrams, unless non-simple curves and/or duplicated labels are permitted. As this paper is the first to consider 3D Venn and Euler diagrams, we include a set of open problems and conjectures to stimulate further research.

Item Type: Conference or workshop item (Paper) Euler Diagrams, Venn Diagrams Q Science > QA Mathematics (inc Computing science) > QA 76 Software, computer programming, Faculties > Sciences > School of Computing > Computational Intelligence Group Peter Rodgers 21 Sep 2012 09:49 UTC 23 Jan 2020 04:06 UTC https://kar.kent.ac.uk/id/eprint/30797 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-4100-3596