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Well-matchedness in Euler Diagrams

Mithileysh Sathiyanarayanan and John HowseVisual Modelling Group, University of Brighton, UK

{M.Sathiyanarayanan, John.Howse}@brighton.ac.uk

Euler Diagrams Workshop 2014 Melbourne, Australia

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Motivation

Well-matchedness needs to be considered for developing strategies to transform abstract descriptions into diagrams.

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Euler Diagrams

Euler diagrams represent relationships between sets, including intersection, containment, and disjointness.

These diagrams have become the foundations of various visual languages.

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Venn DiagramsA Venn diagram contains all possible

intersections of curves and shading is used to indicate empty sets.

This Venn diagram has the same semantics as the Euler diagram on the previous slide.

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Euler Diagrams in DetailAn Euler diagram comprises a set of

closed curves drawn in the plane, where each curve has a label. Curve labels can be repeated.

The set of curves with the same label is called a contour. The closed curves partition the plane into minimal regions.

A zone is a set of minimal regions that are all contained by the same curves.

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Example

This diagram has 4 curves3 contours8 minimal regions 5 zones

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Well-formedness Properties

1. All of the curves are simple (they do not self-intersect)

2. No pair of curves runs concurrently.

3. There are no triple points of intersection between the curves.

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Well-formedness Properties

4. Whenever two curves intersect, they cross.

5. Each zone is connected (i.e. consists of exactly one minimal region).

6. Each curve label is used on at most one curve.

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Peirce’s ClassificationPeirce classified syntactic elements

into three categories: icon, index and symbol.

Closed curves are considered to be icons. A label is considered to be an index. Shading is a symbol.

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Well-matchednessPeirce thought that ‘A diagram

ought to be as iconic as possible’. Closely related to iconicity is the notion of well-matched to meaning.

A notation is well-matched to meaning when its syntactic relationships reflect the semantic relationships being represented.

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Well-matchedness

‘C is a subset of A and C is disjoint from B’

Each of these six Euler diagrams represent the statement

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EXAMPLE

Diagram D1 • is well-formed.• well-matched to meaning.

Diagram D2 • contains shading but is well-formed. • it is only partially well-matched to meaning.

In general, an Euler diagram that contains extra zones that are shaded is not (fully) well-matched to meaning.

Gives rise to the concept of well-matchedness at the zone level.

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ZONE LEVEL

Principle 1: An Euler diagram is well-matched at the zone level if it does not contain any extra zones (zones that must be shaded to preserve semantics).

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The curve C is enclosed by the curve A.

The curves C and B are disjoint.

So the diagram is well-matched as far as the curves are concerned.

This gives rise to the concept of well-matchedness at the curve level.

Diagram D3 is well-matched to meaning at the zone level.

EXAMPLE

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CURVE LEVEL

Principle 2: An Euler diagram is well-matched at the curve level if the subset, intersection and disjointness relationships between sets are matched by containment, overlap and disjointness of the curves representing the sets.

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EXAMPLE

Diagram D3 is well-matched to meaning at the zone level and curve level.

D3 is not well-formed -- it contains two disjoint zones.

Having two disjoint regions representing the same set is disconcerting and appears to go against the nature of a well-matched relation.

This gives rise to the concept of well-matchedness at the minimal region level.

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MINIMAL REGION LEVEL

Principle 3: An Euler diagram is well-matched

at the minimal region level if it is well-matched at the zone level and does not contain a disconnected zone.

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EXAMPLE

Diagram D4 is well-matched to meaning at the zone level but not well-matched at the curve and minimal region level.The contour C (consisting of the two curves C) is enclosed by the contour A.

The contour C is disjoint from the contour B.

So at the contour level this diagram is well-matched.

This gives rise to the concept of well-matchedness at the contour level.

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CONTOUR LEVEL

Principle 4:

An Euler diagram is well-matched at the contour level if the subset, intersection and disjointness relationships between sets are matched by containment, overlap and disjointness of the contours representing the sets.

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EXAMPLE

Diagram D5 is well-matched to meaning at the zone and contour level but not well-matched at the curve and minimal region level.

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Diagram D6 is well-matched to meaning at the zone and contour level but not well-matched at the curve and minimal region level.

EXAMPLE

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Each of these four Euler diagrams represent the statement‘C is a subset of the disjoint union of A and B’

MORE EXAMPLES

There is no well-formed Euler diagram without shading that represents this statement.

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Diagram D1 is well-formed but contains shading.

It is not well-matched at any level.

Diagram D2 contains two curves with label C.

It is well-matched at the zone, minimal region and contour level.

It is not well-matched at the curve level.

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Diagram D3 contains a non-simple curve, C. It contains no extra zones and each zone is a minimal region.It is well-matched at the zone and minimal region level. It is also well-matched at the curve level and the contour level.

This diagram is a fairly natural way of representing ‘C is a subset of the disjoint union of A and B’but it contains a very unnatural non-simple curve.

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D4 contains concurrency and triple points.

This diagram is well-matched at all levels.

However, it might be difficult to work out the relationship between curves A and B.

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Finally, we consider two more examples to complete our analysis of the relationship between well-formedness and well-matchedness in Euler diagrams.

MORE EXAMPLES

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Diagram D1 contains a non-simple curve C.

It is well-matched at the zone, curve and contour levels

The zone within the non-simple curve is divided into two minimal regions.

So it is not well-matched at the minimal region level.

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The diagram D2 represents the statement

‘A and B are disjoint and C and D are disjoint’.

Curves A and B touch but do not cross as do curves C and D.

It is well-matched at all levels.

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Conclusion

We have considered the notion of well-matchedness in Euler diagrams, particularly those that break some of the well-formedness properties.

We have identified four levels of well-formedness.

1. Two of these concern curves: the curve and contour levels

2. Two concern regions:

the zone and minimal region level.

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Putting the four levels together we can state a general well-matchedness principle.

Well-matchedness Principle 5:

An Euler diagram is fully well-matched if it well-matched at the zone, minimal region, curve and contour levels.

General Well-matchedness Principle

All well-formed Euler diagrams without shading are well-matched.

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Future Work

Empirical studies will be conducted to inform and validate the well-matchedness of Euler diagrams.

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