Well-matchedness in Euler Diagram
-
Upload
mithileysh-sathiyanarayanan -
Category
Engineering
-
view
144 -
download
0
description
Transcript of Well-matchedness in Euler Diagram
![Page 1: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/1.jpg)
1
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
![Page 2: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/2.jpg)
2
Motivation
Well-matchedness needs to be considered for developing strategies to transform abstract descriptions into diagrams.
![Page 3: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/3.jpg)
3
Euler Diagrams
Euler diagrams represent relationships between sets, including intersection, containment, and disjointness.
These diagrams have become the foundations of various visual languages.
![Page 4: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/4.jpg)
4
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.
![Page 5: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/5.jpg)
5
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.
![Page 6: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/6.jpg)
6
Example
This diagram has 4 curves3 contours8 minimal regions 5 zones
![Page 7: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/7.jpg)
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.
![Page 8: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/8.jpg)
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.
![Page 9: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/9.jpg)
9
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.
![Page 10: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/10.jpg)
10
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.
![Page 11: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/11.jpg)
11
Well-matchedness
‘C is a subset of A and C is disjoint from B’
Each of these six Euler diagrams represent the statement
![Page 12: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/12.jpg)
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.
![Page 13: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/13.jpg)
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).
![Page 14: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/14.jpg)
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
![Page 15: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/15.jpg)
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.
![Page 16: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/16.jpg)
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.
![Page 17: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/17.jpg)
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.
![Page 18: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/18.jpg)
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.
![Page 19: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/19.jpg)
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.
![Page 20: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/20.jpg)
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.
![Page 21: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/21.jpg)
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
![Page 22: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/22.jpg)
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.
![Page 23: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/23.jpg)
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.
![Page 24: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/24.jpg)
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.
![Page 25: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/25.jpg)
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.
![Page 26: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/26.jpg)
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
![Page 27: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/27.jpg)
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.
![Page 28: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/28.jpg)
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.
![Page 29: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/29.jpg)
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.
![Page 30: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/30.jpg)
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
All well-formed Euler diagrams without shading are well-matched.
![Page 31: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/31.jpg)
31
Future Work
Empirical studies will be conducted to inform and validate the well-matchedness of Euler diagrams.
![Page 32: Well-matchedness in Euler Diagram](https://reader033.fdocuments.in/reader033/viewer/2022052410/54860b48b4af9fa16e8b4618/html5/thumbnails/32.jpg)
32
THANK YOU FOR LISTENING!