Drawing Graphs in Euler Diagrams

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Drawing GraphsinEuler Diagrams

• Automatic layout of graph-enhanced Euler diagrams

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Terminology

• Contours
• Zones

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Euler and Venn Diagrams
Euler diagram
Euler diagram Venn diagram
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Graph-Enhanced Euler Diagrams
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Initial Layout of Euler Diagram

• Fully automatic
• Polygonal contours
• Correct
• Poor layout

Ref Generating Euler Diagrams (Flower Howse)
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Refined Layout of Euler Diagram

• Multicriteria hillclimbing
• Bezier smoothing
• Quite slow

Ref Layout Metrics for Euler Diagrams (Flower,
Rodgers, Mutton)
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Placement of Nested Diagrams
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Well-formed Euler Diagrams

• Zones
• We do not allow disconnected zones
• Modelled by a sequence of line segments
• Simply connected if isotopic to a disc
• Can also be non-simply connected

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Initial Node Placement
Get bounding box of containing zone Pick random
y-coordinate to draw horizontal line Build an
ordered set of crossing points Pick a random
crossing pair Place the node randomly between the
chosen pair
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Why Random Initial Placement?
When each zone contains more than one node, the
random scattering is useful when applying the
force-directed refinement method.
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Refinement of Node Placement
Force-directed refinement moves nodes away from
zone boundaries
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Ensuring Node Separation
Repulsive forces acting between nodes ensures
node-node separation
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Force Model

• Repulsive force between two nodes
• M c/d
• Repulsive force between a line segment and a node
• M2 lc/d2
• M set of nodes in the zone
• c constant
• d distance
• l length of segment

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Adding Edges
The graphs are completed by adding edges to the
diagram after the nodes have been placed
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Edge Layout

• Desirable characteristics of graphs
• Reduced number of edge crossings
• Reduced overall edge length
• For zones containing more than one node, we can
  think of their locations as candidate locations
• Swapping the location of pairs of nodes does not
  change the meaning of the diagram, but can help
  achieve the desirable characteristics

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Quality Metrics
Crossings 0.0000 Length 0.0000 Total 0.0000
Crossings 1.0000 Length 0.1143 Total 1.1143
Crossings 0.0000 Length 0.0940 Total 0.0940
Crossings 0.0000 Length 0.1515 Total 0.1515
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Example of Node Swapping
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Other Supported Edge Types
Our implementation can also detect edge crossings
between more complicated types of edges
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Embedding without Smoothing
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Embedding with Smoothing
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Good and Bad Layouts
Good Layouts
Bad Layout
Nodes trapped in local a minima while the force
model is being simulated Swapping node locations
cannot remove the edge crossing
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Automated Proofs

• Our tool can generate visualizations of automatic
  proof sequences using spider diagrams

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Summary and Further Work

• We have created a tool for automatic layout of
  graphs in Euler diagrams
• This can be applied to automatically generated
  proof sequences for ease of understanding
• We have since created a system to animate changes
  to preserve the users mental map and to maintain
  structural similarity