Fractal Mountains, Splines, and Subdivision Surfaces

1
Fractal Mountains, Splines, and Subdivision
Surfaces

• Jordan Smith
• UC Berkeley
• CS184

2
Outline

• Fractal Mountains
• Review Bézier Curves
• Bsplines and Blossoming
• Subdivision Curves
• Subdivision Surfaces
• Quad mesh (Catmull-Clark scheme)
• Triangle mesh (Loop scheme)

3
Fractals

• Self-similar recursive modeling operators

Sierpinski Triangle
Koch Snowflake
4
Linear Fractal Mountains
Gen 0
Gen 1
Gen 2
Gen 3

• 2 step recursive process
• Subdivide chain by creating edge midpoints
• Randomly perturb midpoint positions

5
Fractal Mountain Surfaces
Gen 0
Gen 1
Gen 2

• 2 step recursive process
• Subdivide triangles at edge midpoints
• Randomly perturb midpoint positions

6
Fractal Mountains Summary

• 2 step recursive process
• Topological split at edge midpoints
• Random perturbation of midpoint positions
• Triangle topological split maintains a water
  tight connected mesh

7
Outline

• Fractal Mountains
• Review Bézier Curves
• Bsplines and Blossoming
• Subdivision Curves
• Subdivision Surfaces
• Quad mesh (Catmull-Clark scheme)
• Triangle mesh (Loop scheme)

8
Review of Bézier CurvesDeCastlejau Algorithm
V2
V3
V1
V4
Insert at t ¾
9
Review of Bézier CurvesDeCastlejau Algorithm
001
011
111
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10
Review of Bézier CurvesDeCastlejau Algorithm
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011
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11
Review of Bézier CurvesDeCastlejau Algorithm
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011
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12
Review of Bézier CurvesDeCastlejau Algorithm
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011
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13
Review of Bézier CurvesDeCastlejau Algorithm
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011
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14
Review of Bézier CurvesDeCastlejau Algorithm
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011
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15
Review of Bézier CurvesDeCastlejau Algorithm
001
011
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16
Bézier Curves Summary

• DeCastlejau algorithm
• Evaluate Position(t) and Tangent(t)
• Subdivides the curve into 2 subcurves with
  independent control polygons
• Subdivision of Bézier curves and convex hull
  property allows for
• Adaptive rendering based on a flatness criterion
• Adaptive collision detection using line segment
  tests

17
Outline

• Fractal Mountains
• Review Bézier Curves
• Bsplines and Blossoming
• Subdivision Curves
• Subdivision Surfaces
• Quad mesh (Catmull-Clark scheme)
• Triangle mesh (Loop scheme)

18
Blossoming of Bsplines
234
345
456
123
Knots
0
1
7
6
5
4
3
2
19
(No Transcript)
20
Blossoming of Bsplines
234
345
33.54
33.53.5
3.53.54
233.5
3.545
456
123
Knots
0
1
7
6
5
4
3
2
3.5
21
Blossoming of Bsplines
234
345
33.54
33.53.5
3.53.54
3.53.53.5
233.5
3.545
456
123
Knots
0
1
7
6
5
4
3
2
3.5
22
Bspline Blossoming Summary

• Blossoming of Bsplines is a generalization of the
  DeCastlejau algorithm
• Control point index triples on the same control
  line share 2 indices with each other
• Inserting a knot (t value)
• Adds a new control point and curve segment
• Adjusts other control points to form a control
  polygon
• Inserting the same t value reduces the parametric
  continuity of the curve
• A control point triple with all 3 indices equal
  is a point on the Bspline curve

23
Bézier is a specific Bspline
234
345
456
123
Knots
0
1
7
6
5
4
3
2
24
Bézier is a specific Bspline
234
345
334
344
444
333
445
233
456
123
Knots
0
1
7
6
5
4
3
2
25
Bézier / Bspline Summary

• Bézier curves are a specific instance of
  non-uniform Bspline curves
• Knot vector is 0,0,0,0,1,1,1,1
• First and last control point are on the curve
• C0 continuity at first and last control point

26
Outline

• Fractal Mountains
• Review Bézier Curves
• Bsplines and Blossoming
• Subdivision Curves
• Subdivision Surfaces
• Quad mesh (Catmull-Clark scheme)
• Triangle mesh (Loop scheme)

27
Subdivision Curves
V20
V30
V40
V10

• Subdivision is a recursive 2 step process
• Topological split
• Local averaging / smoothing

28
Subdivision Curves
E20
V20
V30
E10
E30
V40
V10
E40

• Subdivision is a repeated 2 step process
• Topological split
• Local averaging / smoothing

29
Subdivision Curves
E21
V20
V30
V21
V31
E11
E31
V41
V11
V40
V10
E41

• Subdivision is a repeated 2 step process
• Topological split
• Local averaging / smoothing

30
Subdivision Curves
E21
V20
V30
V21
V31
E11
E31
V41
V11
V40
V10
E41

• Subdivision is a repeated 2 step process
• Topological split
• Local averaging / smoothing

31
Subdivision Curves
E21
V20
V30
V21
V31
E11
E31
V41
V11
V40
V10
E41

• Subdivision is a repeated 2 step process
• Topological split
• Local averaging / smoothing

32
Subdivision of Bspline Curves
234
341
123
412
Knots
4
1
3
2
1
4
3
2
4
33
Subdivision of Bspline Curves
33.54
2.534
344.5
234
341
233.5
3.541
3.544.5
2.533.5
44.51
22.53
4.511.5
1.522.5
411.5
1.523
123
4.512
412
11.52
122.5
Knots
4
1
3
2
1
4
3
2
4
34
Averaging Rules
Vi1
Vi-11
33.54
2.534
344.5
Vi11
234
341
Vi2
233.5
3.541
3.544.5
2.533.5
Vi2 Vi1
Vi-11
44.51
22.53
411.5
4.511.5
1.522.5
1.523
Vi11
123
4.512
412
11.52
122.5
35
Subdivision Curve Summary

• Subdivsion is a recursive 2 step process
• Topological linear split at midpoints
• One local averaging / smoothing operator applied
  to all points
• Double the number of vertices at each step
• Subdivision curves are nothing new
• Averaging rules chosen so that they are simply
  uniform Bspline curves

36
Outline

• Fractal Mountains
• Review Bézier Curves
• Bsplines and Blossoming
• Subdivision Curves
• Subdivision Surfaces
• Quad mesh (Catmull-Clark scheme)
• Triangle mesh (Loop scheme)

37
Subdivision Overview
Control Mesh
Topological Split
Averaging

• Subdivision is a two part process
• Topological split
• Local averaging / smoothing

38
Subdivision Overview
Control Mesh
Generation 1
Generation 2
Generation 3

• Repeated uniform subdivisions of the control mesh
  converge to the limit surface
• Stationary schemes (averaging mask does not
  change)
• Limit position and normal from Eigen analysis

39
Outline

• Fractal Mountains
• Review Bézier Curves
• Bsplines and Blossoming
• Subdivision Curves
• Subdivision Surfaces
• Quad mesh (Catmull-Clark scheme)
• Triangle mesh (Loop scheme)

40
Bspline Surfaces

• A single Bspline surface patch is controlled by a
  regular 4x4 grid of control points

41
Bspline Surfaces

• 2 adjacent patches share 12 control points and
  meet with C2 continuity

42
Bspline Surfaces

• Requires a regular rectangular control mesh grid
  to guarantee continuity (all valence 4 vertices!)
• Subdivision can be performed by knot insertion
  (i.e. blossoming)

43
Catmull-Clark Subdivision Surfaces

• Smooth surfaces over arbitrary topology control
  meshes
• Closed control mesh ? closed limit surface
• Quad mesh generalization of Bsplines
• C1 at non-valence 4 vertices
• C2 every where else (Bsplines)
• Sharp corners can be tagged
• Allows for smooth and sharp features
• Allows for non-closed meshes

44
Catmull-Clark Subdivision
Gen 0
Gen 1
Gen 2

• Extraordinary vertices are generated by
  non-valence 4 vertices and faces in the input
  mesh
• No more extraordinary vertices are created after
  the first generation of subdivision

45
Catmull-Clark Averaging
46
Outline

• Fractal Mountains
• Review Bézier Curves
• Bsplines and Blossoming
• Subdivision Curves
• Subdivision Surfaces
• Quad mesh (Catmull-Clark scheme)
• Triangle mesh (Loop scheme)

47
Loop Subdivision Surfaces
Gen 0
Gen 1
Gen 2
48
Conclusion

• Subdivision is a 2 step recursive process
• Topological split
• Local averaging / smoothing