Subdivision Curves

1
Subdivision Curves Surfaces and Fractal
Mountains.

• CS184 Spring 2011

2
Outline

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

3
Review of Bézier CurvesDeCastlejau Algorithm
V2
V3
V1
V4
Insert at t ¾
4
Review of Bézier CurvesDeCastlejau Algorithm
001
011
Ignore funny notation at vertices! ( CS 284
stuff )
111
Insert at t ¾
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5
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
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¾11
111
Insert at t ¾
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6
Review of Bézier CurvesDeCastlejau Algorithm
001
011
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Insert at t ¾
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7
Review of Bézier CurvesDeCastlejau Algorithm
001
011
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00¾
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111
Insert at t ¾
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8
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
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Curve position and tangent for t ¾
111
Insert at t ¾
000
9
Subdivision of Bézier Curves
001
011
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This also yields all control points for
subdivision into 2 Bezier curves
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Insert at t ¾
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10
Subdivision of Bézier Curves
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011
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Convex Hull Property!
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Insert at t ¾
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11
Bézier Curves Summary

• DeCastlejau algorithm is good for
• Evaluating position(t) and tangent(t),
• Subdividing the curve into 2 subcurves with
  their own control polygons.
• Subdivision of Bézier curves and their convex
  hull property allows for
• Adaptive rendering based on a flatness criterion,
• Adaptive collision detection using line segment
  tests.

12
Outline

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

13
Subdivision Curves
V20
V30
An approximating scheme
Limit curve
V40
V10

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

14
Subdivision Curves
E20
V20
V30
E10
E30
V40
V10
E40

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

15
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

16
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

17
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

18
Results in a B-spline Curve
Knots
4
1
3
2
1
4
3
2
4
19
Subdivision Curve Summary

• Subdivsion is a recursive 2 step process
• Topological split at midpoints,
• Local averaging/smoothing operator applied.
• Doubles the number of vertices at each step
• Subdivision curves are nothing new
• Suitable averaging rules can yield uniform
  B-spline curves.

20
Outline

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

21
Subdivision Overview
Control Mesh
Topological Split
Averaging
Limit Surface

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

22
Subdivision Overview
Control Mesh
Generation 1
Generation 2
Generation 3

• Repeated uniform subdivisions of the control mesh
  converge to the limit surface.
• Limit surface can be calculated in closed form
  for stationary schemes (averaging mask does not
  change).

23
Outline

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

24
B-spline Surfaces

• A cubic B-spline surface patch is controlled by a
  regular 4x4 grid of control points

25
B-spline Surfaces

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

26
B-spline Surfaces

• Requires a regular rectangular control mesh grid
  and all valence-4 vertices to guarantee
  continuity.

27
Catmull-Clark Subdivision Surface

• Yields smooth surfaces over arbitrary topology
  control meshes.
• Closed control mesh ? closed limit surface.
• Quad mesh generalization of B-splines
• C1 at non-valence-4 vertices,
• C2 everywhere else (B-splines).
• Also Sharp corners can be tagged
• Allows for smooth and sharp features,
• Allows for non-closed meshes.

28
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 additional extraordinary vertices are created
  after the first generation of subdivision.

29
Catmull-Clark Averaging
(simple averaging)
n valence
30
Outline

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

31
Loop Subdivision Surface
Gen 0
Gen 1
Gen 2
n valence
32
Summary

• Subdivision is a 2 step recursive process
• Topological split,
• Local averaging / smoothing.
• It is an easy way to make smooth objects
• of irregular shape
• of topologies other than rectangular (torus).

33
Outline

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

34
Fractals

• Self-similar recursive modeling operators

Sierpinski Triangle
Koch Snowflake
35
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(proportional
  to subdivided edge length).

36
Fractal Mountain Surfaces
Gen 0
Gen 1
Gen 2

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

37
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.
• Useful to make uneven, natural terrain.
• Often a low-order subdivision is good enough to
  control terrain-following vehicles.