Drawing Metro Maps using B ezier Curves Martin Fink Lehrstuhl f - - PowerPoint PPT Presentation
Drawing Metro Maps using B ezier Curves Martin Fink Lehrstuhl f ur Informatik I Universit at W urzburg Joint work with Herman Haverkort, Martin N ollenburg, Maxwell Roberts, Julian Schuhmann & Alexander Wolff 1 /21 We
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Martin Fink Lehrstuhl f¨ ur Informatik I Universit¨ at W¨ urzburg
Joint work with Herman Haverkort, Martin N¨
Julian Schuhmann & Alexander Wolff
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[N¨
[Hong et al., 2006], [Stott et al., 2011]
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[Roberts, 2007]
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metro line: polyline with bends (possibly in stations) very schematized
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metro line: polycurve without bends
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metro line: polycurve without bends [Roberts et al., 2012]: improved planning speed
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metro line: polycurve without bends [Roberts et al., 2012]: improved planning speed more artistic demand of Peter Eades! [GD’10]
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parametric curves
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p p′ q′ q parametric curves C: [0, 1] → R2 t → (1 − t)3p + 3(1 − t)2tp′ + 3(1 − t)t2q′ + t3q C
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p p′ q′ q parametric curves C: [0, 1] → R2 t → (1 − t)3p + 3(1 − t)2tp′ + 3(1 − t)t2q′ + t3q leave vertices in direction of tangents C
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p p′ q′ q parametric curves C: [0, 1] → R2 t → (1 − t)3p + 3(1 − t)2tp′ + 3(1 − t)t2q′ + t3q leave vertices in direction of tangents curves share tangents fit smoothly together C
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p p′ q′ q parametric curves C: [0, 1] → R2 t → (1 − t)3p + 3(1 − t)2tp′ + 3(1 − t)t2q′ + t3q leave vertices in direction of tangents curves share tangents fit smoothly together C
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p p′ q′ q parametric curves C: [0, 1] → R2 t → (1 − t)3p + 3(1 − t)2tp′ + 3(1 − t)t2q′ + t3q leave vertices in direction of tangents curves share tangents fit smoothly together C
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p p′ q′ q
− → pC parametric curves C: [0, 1] → R2 t → (1 − t)3p + 3(1 − t)2tp′ + 3(1 − t)t2q′ + t3q leave vertices in direction of tangents
control point = tangent + distance C
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[Brandes, Shubina, Tamassia, Wagner, 2001]: B´ ezier curves used for visualizing train connections.
[Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998]
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[Brandes, Shubina, Tamassia, Wagner, 2001]: B´ ezier curves used for visualizing train connections.
[Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998]
Our approach: – share tangents – move vertices
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[Brandes, Shubina, Tamassia, Wagner, 2001]: B´ ezier curves used for visualizing train connections. [Finkel and Tamassia, 2005]: Force-directed algorithm for drawing graphs with B´ ezier Curves
[Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998]
Our approach: – share tangents – move vertices
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[Brandes, Shubina, Tamassia, Wagner, 2001]: B´ ezier curves used for visualizing train connections. [Finkel and Tamassia, 2005]: Force-directed algorithm for drawing graphs with B´ ezier Curves Method: Control-points as extra vertices.
[Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998]
Our approach: – share tangents – move vertices
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[Brandes, Shubina, Tamassia, Wagner, 2001]: B´ ezier curves used for visualizing train connections. [Finkel and Tamassia, 2005]: Force-directed algorithm for drawing graphs with B´ ezier Curves Method: Control-points as extra vertices.
[Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998]
Our approach: – share tangents – move vertices
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use B´ ezier curves for representing edges
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use B´ ezier curves for representing edges use force-directed approach
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geographic input
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straight-line drawing
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straight-line drawing alternative: use
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approximation by B´ ezier Curves
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approximation by B´ ezier Curves put control points close to vertex use common tangents
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while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification
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while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification stop if displacement is small
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while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification
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while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification sum up to desired changes
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while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification avoid intersections
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while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification merge curves around deg-2 vertex
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while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification merge 4 curves to 2
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– repulsion
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– repulsion – attraction of adjacent vertices
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– repulsion – attraction of adjacent vertices reuse forces from Fruchterman-Rheingold
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– repulsion – attraction of adjacent vertices desired edge length = const · (# intermediate stops + 1)
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– repulsion – attraction of adjacent vertices – attraction to geographic position
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a b ≈ 1 3
use Fruchterman-Rheingold-like spring force a b
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u t vt v move vertex v towards tangent
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u t vt α v move vertex v towards tangent vt rotate tangent towards straight-line
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u t vt α v move vertex v towards tangent vt rotate tangent towards straight-line adding up rotational forces
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u t vt α v move vertex v towards tangent vt rotate tangent towards straight-line law of the lever: force weighted by distance adding up rotational forces
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tangents repelling each other force = const. · 1
α
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intermediate stop
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intermediate stop keep control points
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intermediate stop keep control points different distances for keeping the drawing crossing-free
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intermediate stop keep control points different distances for keeping the drawing crossing-free Tests: not necessary
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merge 4 curves to 2
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merge 4 curves to 2 station = crossing
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merge 4 curves to 2 edge proportions should be kept approximately testing different control point distances necessary
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merge 4 curves to 2 edge proportions should be kept approximately testing different control point distances necessary adds many additional constraints
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merge 4 curves to 2 edge proportions should be kept approximately testing different control point distances necessary adds many additional constraints Tests: performed only once at the end
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without merging edges 90 curves
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with merging edges at intermediate stations 25 curves
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additionally merging edges at interchange stations (degree 4) 9 curves
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works well on smaller networks/sparse regions
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works well on smaller networks/sparse regions dense region like city centers are more problematic minimizing the number of curves is important
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works well on smaller networks/sparse regions dense region like city centers are more problematic minimizing the number of curves is important further minimization with local changes very hard
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works well on smaller networks/sparse regions dense region like city centers are more problematic minimizing the number of curves is important further minimization with local changes very hard Idea: Use global approximation of lines by continuos curves – subject to constraints