The presentation was given in EULER DIAGRAMS Workshop 2014, Melbourne, Australia.

1. Well-matchedness in Euler
Diagrams
Mithileysh Sathiyanarayanan and John Howse
Visual Modelling Group, University of Brighton, UK
{M.Sathiyanarayanan, John.Howse}@brighton.ac.uk
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Euler Diagrams Workshop 2014
Melbourne, Australia

2. Motivation
Well-matchedness needs to be considered
for developing strategies to transform
abstract descriptions into diagrams.
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3. 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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4. Venn Diagrams
A 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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5. Euler Diagrams in Detail
An 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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6. Example
This diagram has
4 curves
3 contours
8 minimal regions
5 zones
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7. 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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8. 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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9. Peirce’s Classification
Peirce 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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10. Well-matchedness
Peirce 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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11. 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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12. 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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13. 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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14. 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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15. 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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16. 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. 16

17. 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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18. 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. 18

19. 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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20. 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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21. 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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22. 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. 22

23. 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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24. 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. 24

25. 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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26. 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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27. 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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28. 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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29. 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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30. 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
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All well-formed Euler diagrams without shading
are well-matched.

31. Future Work
Empirical studies will be conducted to
inform and validate the well-matchedness
of Euler diagrams.
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32. THANK YOU FOR
LISTENING!
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