Generating Euler diagrams

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Generating Euler diagrams
Jean Flower and John Howse University of
Brighton, UK j.a.flower_at_brighton.ac.uk john.howse
_at_brighton.ac.uk Visual Modelling
Group www.it.brighton.ac.uk/research/vmg
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Background constraint diagrams
diagram
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Constraint diagrams
abstract diagram
concrete diagram
There are three contours graph, tree and
other, and four zones one outside all contours,
one just in graph, one in graph and tree and one
in all three contours, and one existential
spider...
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translation
abstract diagram
concrete diagram
read a diagram
There are three contours graph, tree and
other, and four zones one outside all contours,
one just in graph, one in graph and tree and one
in all three contours, and one existential
spider...
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Euler diagrams
abstract diagram
concrete diagram
read a diagram
graph
There are three contours graph, tree and
other, and four zones one outside all contours,
one just in graph, one in graph and tree and one
in all three contours.
tree
other
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Euler diagrams
abstract diagram
concrete diagram
read a diagram
A
B
There are three contours A, B, C and four
zones ,A,A,B,A,B,C
C
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Well-formed diagrams
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An algorithm
This paper presents work on an algorithm for
creating a well-formed Euler diagram from an
abstract Euler diagram. A central idea is that
of the dual graph of a concrete Euler diagram.
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The dual graph
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The dual graph
Notice dual edges AB,ABC or A,AB or
null, C Bipartite.
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Factoring abstraction
contours A, B, C zones , A, A, B, A, B,
C
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Factoring twice
contours A, B, C zones , A, A, B, A,
B, C
objective
strategy
nodes , A, A, B, A, B, C edges A,A,
B,.
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Factoring twice
objective
abstract Euler diagram
concrete Euler diagram
strategy
abstract dual graph
concrete dual graph
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The first inverse
objective
abstract Euler diagram
concrete Euler diagram
strategy
abstract dual graph
concrete dual graph
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Building an abstract dual graph

• Given a abstract Euler diagram,

contours A, B, C zones , A, A, B, A,
B, C
Nodes of the abstract dual graph match
zones Edges pair off nodes with labels differing
by a single letter
nodes , A, A, B, A, B, C edges
A,A,B, null, A,.
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Structure of dual graphs
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Connectivity conditions

• The dual of a concrete Euler diagram is always a
  connected graph.
• Also inside/outside a given contour.

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Connectivity conditions
contours A, B, C zonesnull, A, B, A,B,C
disconnected dual
undrawable abstract diagrams
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A more subtle example
contours A, B, C, D zones null, A, A,B,
A,B,C, B,C
disconnected outside A
undrawable abstract diagrams
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The second inverse
objective
abstract Euler diagram
concrete Euler diagram
strategy
abstract dual graph
concrete dual graph
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Planar dual graphs

• Dual graphs are planar graphs
  (no edge-crossings)
• Generated duals need plane representations
• Known algorithms - how to make use of rich dual
  graph structure? (bipartite )

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Absent dual edges
shaded zones give dual nodes A,C, D and C,
D but the zones are not topologically
adjacent including all dual edges from
abstract dual yields a non-planar graph
A
B
D
C
We may have to remove some dual edges before
using the dual graph to draw an Euler diagram.
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The third inverse
objective
abstract Euler diagram
concrete Euler diagram
strategy
abstract dual graph
concrete dual graph
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From a concrete plane dual graph to an Euler
Diagram
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From a concrete plane dual graph to an Euler
Diagram
B
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From a concrete plane dual graph to an Euler
Diagram
B
C
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From a concrete plane dual graph to an Euler
Diagram
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From a concrete plane dual graph to an Euler
Diagram
A
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From a concrete plane dual graph to an Euler
Diagram
A
needs visual skill to manipulate contours
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Constructing contours from straight line segments
convex face
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Creating a dual with convex faces
AB
C
B
AC
B
A
C
A
ABC
C
A
C
null
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Creating a dual with convex faces
C
AC
AB
C
B
AC
A
B
A
C
A
ABC
C
A
null
C
null
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Creating a dual with convex faces
C
AC
AB
C
B
AC
A
B
A
C
A
ABC
C
A
null
C
null
A
AB
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Constructing inside arcs
C
AC
AC
AB
C
A
B
AC
A
B
A
C
A
ABC
C
A
null
AB
C
null
A
ABC
AB
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Constructing inside arcs
C
AC
AC
AB
C
A
B
AC
A
B
A
C
A
ABC
C
A
null
AB
C
null
A
ABC
AB
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Constructing inside arcs
C
AC
AC
AB
C
A
B
AC
A
B
A
C
A
ABC
C
A
null
AB
C
null
A
ABC
AB
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Inside arcs
A
A
C
C
A
B
null
A
C
B
C
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Constructing outside arcs
C
C
A
B
null
A
B
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Constructing outside arcs
A
null
A
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Constructing outside arcs
null
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Word conditions in faces

• Unwanted zones

avoid construction of new zones
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Word conditions in faces

• triple point problems
• will occur
• if
• reading around
• face boundary gives
• ABCABC
• also avoid AFFBCADEDEBC etc.

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Word conditions in faces

• If a word-obstruction appears in a planar dual
  graph, is it worth trying to come up wth a
  different planar representation without the
  word-problem?
• Can these word-obstructions be resolved in any
  way?

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Overview
objective
abstract Euler diagram
concrete Euler diagram
convex faces boundary words
connectivity issues
strategy
abstract dual graph
concrete dual graph
planarity issues
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Questions and current work
Whats the best planarising algorithm for an
abstract dual graph? How can we overcome the
fact that the word-conditions are a feature of
the plane dual, and not a feature of the abstract
dual? How can we make the outcome look
nicer (less convoluted - easier to read by eye)