Drawing Metro Maps using Bézier Curves

Transcription

1 Drawing Metro Maps using Bézier Curves Martin Fink Lehrstuhl für Informatik I Universität Würzburg Joint work with Herman Haverkort, Martin Nöllenburg, Maxwell Roberts, Julian Schuhmann & Alexander Wolff 1 /21

2 We have... [Nöllenburg and Wolff, 2011] [Hong et al., 2006], [Stott et al., 2011] 2 /21

3 We want to create... [Roberts, 2007] 3 /21

4 Octilinear vs. Curvy Drawings metro line: polyline with bends (possibly in stations) very schematized 4 /21

5 Octilinear vs. Curvy Drawings metro line: polycurve without bends 4 /21

6 Octilinear vs. Curvy Drawings metro line: polycurve without bends [Roberts et al., 2012]: improved planning speed 4 /21

7 Octilinear vs. Curvy Drawings metro line: polycurve without bends [Roberts et al., 2012]: improved planning speed more artistic demand of Peter Eades! [GD 10] 4 /21

8 (Cubic) Bézier Curves parametric curves 5 /21

9 (Cubic) Bézier Curves parametric curves p q C p q C: [0, 1] R 2 t (1 t) 3 p + 3(1 t) 2 tp + 3(1 t)t 2 q + t 3 q 5 /21

10 (Cubic) Bézier Curves parametric curves leave vertices in direction of tangents p q C p q C: [0, 1] R 2 t (1 t) 3 p + 3(1 t) 2 tp + 3(1 t)t 2 q + t 3 q 5 /21

11 (Cubic) Bézier Curves parametric curves leave vertices in direction of tangents p q C p q C: [0, 1] R 2 curves share tangents t (1 t) 3 p + 3(1 t) 2 tp fit + smoothly 3(1 t)ttogether 2 q + t 3 q 5 /21

12 (Cubic) Bézier Curves parametric curves leave vertices in direction of tangents p q C p q C: [0, 1] R 2 curves share tangents t (1 t) 3 p + 3(1 t) 2 tp fit + smoothly 3(1 t)ttogether 2 q + t 3 q 5 /21

13 (Cubic) Bézier Curves parametric curves leave vertices in direction of tangents p q C p q C: [0, 1] R 2 curves share tangents t (1 t) 3 p + 3(1 t) 2 tp fit + smoothly 3(1 t)ttogether 2 q + t 3 q 5 /21

14 (Cubic) Bézier Curves r C (p) { p C q p C parametric curves leave vertices in direction of tangents our representation: control point = tangent + distance p q C: [0, 1] R 2 t (1 t) 3 p + 3(1 t) 2 tp + 3(1 t)t 2 q + t 3 q 5 /21

15 Previous Work [Brandes, Shubina, Tamassia, Wagner, 2001]: Bézier curves used for visualizing train connections. [Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998] 6 /21

16 Previous Work [Brandes, Shubina, Tamassia, Wagner, 2001]: Bézier curves used for visualizing train connections. [Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998] Our approach: share tangents move vertices 6 /21

17 Previous Work [Brandes, Shubina, Tamassia, Wagner, 2001]: Bézier curves used for visualizing train connections. [Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998] Our approach: share tangents move vertices [Finkel and Tamassia, 2005]: Force-directed algorithm for drawing graphs with Bézier Curves 6 /21

18 Previous Work [Brandes, Shubina, Tamassia, Wagner, 2001]: Bézier curves used for visualizing train connections. [Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998] Our approach: share tangents move vertices [Finkel and Tamassia, 2005]: Force-directed algorithm for drawing graphs with Bézier Curves Method: Control-points as extra vertices. 6 /21

19 Previous Work [Brandes, Shubina, Tamassia, Wagner, 2001]: Bézier curves used for visualizing train connections. [Brandes, Shubina, Tamassia, 2000], [Brandes, Wagner, 1998] Our approach: share tangents move vertices [Finkel and Tamassia, 2005]: Force-directed algorithm for drawing graphs with Bézier Curves Method: Control-points as extra vertices. our approach: control points are no vertices 6 /21

20 Our Approach use Bézier curves for representing edges 7 /21

21 Our Approach use Bézier curves for representing edges use force-directed approach 7 /21

22 Excursion: Force-Directed Algorithms 8 /21

23 Excursion: Force-Directed Algorithms 8 /21

24 Excursion: Force-Directed Algorithms 8 /21

25 Excursion: Force-Directed Algorithms 8 /21

26 Excursion: Force-Directed Algorithms 8 /21

27 Excursion: Force-Directed Algorithms modify drawing iteratively forces: desired changes 8 /21

28 Excursion: Force-Directed Algorithms modify drawing iteratively forces: desired changes Spring Force desired edge lenght l 8 /21

29 Excursion: Force-Directed Algorithms modify drawing iteratively forces: desired changes Spring Force desired edge lenght l 8 /21

30 Excursion: Force-Directed Algorithms modify drawing iteratively forces: desired changes Spring Force desired edge lenght l Task: Find bad instance for spring force 8 /21

31 Our Approach use Bézier curves for representing edges use force-directed approach 9 /21

32 Obtaining an Initial Drawing with Curves geographic input 10/21

33 Obtaining an Initial Drawing with Curves straight-line drawing 10/21

34 Obtaining an Initial Drawing with Curves straight-line drawing alternative: use octilinear drawing 10/21

35 Obtaining an Initial Drawing with Curves approximation by Bézier Curves 10/21

36 Obtaining an Initial Drawing with Curves approximation by Bézier Curves put control points close to vertex use common tangents 10/21

37 Structure of the Algorithm while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification 11/21

38 Structure of the Algorithm while the drawing changes compute forces on vertices stop if displacement is small compute forces on curves apply forces simplify the drawing perform post-processing simplification 11/21

39 Structure of the Algorithm while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing perform post-processing simplification 11/21

40 Structure of the Algorithm while the drawing changes compute forces on vertices compute forces on curves apply forces sum up to desired changes simplify the drawing perform post-processing simplification 11/21

41 Structure of the Algorithm while the drawing changes compute forces on vertices compute forces on curves apply forces simplify the drawing avoid intersections perform post-processing simplification 11/21

42 Structure of the Algorithm 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 11/21

43 Structure of the Algorithm 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 11/21

44 Forces Vertices repulsion 12/21

45 Forces Vertices repulsion attraction of adjacent vertices 12/21

46 Forces Vertices repulsion attraction of adjacent vertices reuse forces from Fruchterman-Rheingold 12/21

47 Forces Vertices repulsion attraction of adjacent vertices desired edge length = const (# intermediate stops + 1) 12/21

48 Forces Vertices repulsion attraction of adjacent vertices attraction to geographic position 12/21

49 Forces Shape of Curves 13/21

50 Forces Shape of Curves 13/21

51 Forces Shape of Curves 13/21

52 Forces Shape of Curves 13/21

53 Forces Shape of Curves a b distances: a b 1 3 use Fruchterman-Rheingold-like spring force 13/21

54 Forces Straightening Curves 14/21

55 Forces Straightening Curves 14/21

56 Forces Straightening Curves straight-line segment = simplest curve 14/21

57 Forces Straightening Curves straight-line segment = simplest curve Task: How would you do it? 14/21

58 Forces Straightening Curves straight-line segment = simplest curve v t t move vertex v towards tangent u v 14/21

59 Forces Straightening Curves straight-line segment = simplest curve v t t move vertex v towards tangent rotate tangent towards straight-line α u v 14/21

60 Forces Straightening Curves adding up rotational forces straight-line segment = simplest curve v t t move vertex v towards tangent rotate tangent towards straight-line α u v 14/21

61 Forces Straightening Curves adding up rotational forces straight-line segment = simplest curve v t t move vertex v towards tangent rotate tangent towards straight-line α u v law of the lever: force weighted by distance 14/21

62 Forces Angular Resolution 15/21

63 Forces Angular Resolution 15/21

64 Forces Angular Resolution α tangents repelling each other force = const. 1 α 15/21

65 Avoiding Intersections 16/21

66 Avoiding Intersections 16/21

67 Merging Curves intermediate stop 17/21

68 Merging Curves intermediate stop keep control points 17/21

69 Merging Curves intermediate stop keep control points different distances for keeping the drawing crossing-free 17/21

70 Merging Curves intermediate stop keep control points different distances for keeping the drawing crossing-free Tests: not necessary 17/21

71 Merging Curves in Interchange Stations merge 4 curves to 2 18/21

72 Merging Curves in Interchange Stations merge 4 curves to 2 station = crossing 18/21

73 Merging Curves in Interchange Stations merge 4 curves to 2 edge proportions should be kept approximately testing different control point distances necessary 18/21

74 Merging Curves in Interchange Stations merge 4 curves to 2 edge proportions should be kept approximately testing different control point distances necessary Question: What could be the problem? 18/21

75 Merging Curves in Interchange Stations merge 4 curves to 2 edge proportions should be kept approximately testing different control point distances necessary adds many additional constraints 18/21

76 Merging Curves in Interchange Stations 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 18/21

77 Test Case Vienna octilinear map 19/21

78 Test Case Vienna without merging edges 90 curves 19/21

79 Test Case Vienna with merging edges at intermediate stations 25 curves 19/21

80 Test Case Vienna additionally merging edges at interchange stations (degree 4) 9 curves 19/21

81 Montréal 20/21

82 Sydney 21/21

83 London 22/21

84 Observations and Conclusion works well on smaller networks/sparse regions 23/21

85 Observations and Conclusion works well on smaller networks/sparse regions dense region like city centers are more problematic minimizing the number of curves is important 23/21

86 Observations and Conclusion 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 23/21

87 Observations and Conclusion 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 23/21