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Subdivision Curves

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Title: 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
000
5
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
00¾
¾11
111
Insert at t ¾
000
6
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾1
¾11
111
Insert at t ¾
000
7
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
111
Insert at t ¾
000
8
Review of Bézier CurvesDeCastlejau Algorithm
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
Curve position and tangent for t ¾
111
Insert at t ¾
000
9
Subdivision of Bézier Curves
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
This also yields all control points for
subdivision into 2 Bezier curves
111
Insert at t ¾
000
10
Subdivision of Bézier Curves
001
011
0¾1
0¾¾
00¾
¾¾¾
¾¾1
¾11
Convex Hull Property!
111
Insert at t ¾
000
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.
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