1. CSCE 441: Keyframe Animation/Smooth Curves (Cont.)
Jinxiang Chai

2. Key-frame Interpolation
Given parameter values at key frames, how to interpolate parameter values for inbetween frames. θ t

3. Key-frame Interpolation
Given parameter values at key frames, how to interpolate parameter values for inbetween frames. θ t

4. Key-frame Interpolation
Given parameter values at key frames, how to interpolate parameter values for inbetween frames. θ t
Nonlinear interpolation

5. Review: Natural cubic cruves

6. Review: Natural cubic cruves

7. Review: Natural cubic cruves

8. Review: Natural cubic curves

9. Review: Hermite CurvesP1 R1 P4 R4
P1: start position
P4: end position
R1: start derivative
R4: end derivative

10. Review: Hermite CurvesP1 R1 P4 R4

11. Review: Hermite CurvesP1 R1 P4 R4
Herminte basis matrix

12. Review: Hermite CurvesP1 R1 P4 R4
Herminte basis matrix

13. Review: Hermite CurvesP1 R1 P4 R4

14. Review: Hermite CurvesP1 R1 P4 R4

15. Review: Hermite CurvesP1 R1 P4 R4

16. Review: Hermite CurvesP1 R1 P4 R4

17. Review: Hermite CurvesP1 R1 P4 R4
Hermite basis functions

18. Review: Hermite CurvesP1 R1 P4 R4
basis function 1
basis function 1
basis function 1
basis function 1

19. Review: Hermite CurvesP1 R1 P4 R4
*R1
*R4
*P1 +
*P4 + + =

20. Review: Bezier Curves

21. Review: Bezier Curves

22. Review: Bezier Curves

23. Review: Bezier Curves

24. Review: Bezier Curves
*v1
*v2
*v3
*v0 + + + =

25. Review: Different basis functions
Cubic curves:
Hermite curves:
Bezier curves:

26. Complex curves
Suppose we want to draw or interpolate a more complex curve

27. Complex curves
Suppose we want to draw or interpolate a more complex curve
How can we represent this curve?

28. Complex curves Suppose we want to draw a more complex curve
Idea: we’ll splice together a curve from individual segments that are cubic Béziers

29. Complex curves
Suppose we want to draw or interpolate a more complex curve
Idea: we’ll splice together a curve from individual segments that are cubic Béziers

30. Splines
A piecewise polynomial that has a locally very simple form, yet be globally flexible and smooth

31. Splines
There are three nice properties of splines we’d like to have
- Continuity
- Local control
- Interpolation

32. Continuity C0: points coincide, velocities don’t
C1: points and velocities coincide
What’s C2?
- points, velocities and accelerations coincide

33. Continuity Cubic curves are continuous and differentiable
We only need to worry about the derivatives at the endpoints when two curves meet

34. Local control We’d like our spline to have local control
- that is, have each control point affect some well-defined neighborhood around that point

35. Local control We’d like our spline to have local control
- that is, have each control point affect some well-defined neighborhood around that point

36. Local control We’d like our spline to have local control
- that is, have each control point affect some well-defined neighborhood around that point

37. Interpolation Bézier curves are approximating
- The curve does not (necessarily) pass through all the control points
- Each point pulls the curve toward it, but other points are pulling as well
Instead, we may prefer a spline that is interpolating
- That is, that always passes through every control point

38. B-splines We can join multiple Bezier curves to create B-splines
Ensure C2 continuity when two curves meet

39. Derivatives at end points

40. Derivatives at end points

41. Derivatives at end points

42. Derivatives at end points

43. Derivatives at end points

44. Derivatives at end points

45. Continuity in B splines
Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

46. Continuity in B splines
Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

47. Continuity in B splines
Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

48. Continuity in B splines
Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

49. Continuity in B splines
Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

50. Continuity in B splines
Suppose we want to join two Bezier curves (V0, V1, V2,V3) and (W0, W1, W2, W3) so that C2 continuity is met at the joint

51. Continuity in B splines
What does this derived equation mean geometrically?
- What is the relationship between a, b and c, if a = 2b - c?
w2=v1+4v3-4v2

52. de Boor points
Instead of specifying the Bezier control points, let’s specify the corners of the frames that forms a B-spline
These points are called de Boor points and the frames are called A-frames

53. de Boor points
What is the relationship between Bezier control points and de Boor points?
Verify this by yourself!

54. B spline basis matrix

55. Building complex splines
Constraining a Bezier curve made of many segments to be C2 continuous is a lot of work
- for each new segment we have to add 3 new control point
- only one of the control points is really free
B-splines are easier (and C2)
- First specify 4 vertices (de Boor points), then one per segment

56. B splines properties
√ Continuity
√ Local control
x Interpolation

57. Catmull-Rom splines
If we are willing to sacrifice C2 continuity, we can get interpolation and local control.
If we set each derivative to be a constant multiple of the vector between the previous and the next controls, we get a Catmull-Rom spline

58. Catmull-Rom splines

59. Catmull-Rom splines
The segment is controlled by p1,p2,p3,p4
0.5

60. Catmull-Rom splines
The segment is controlled by p1,p2,p3,p4

61. Catmull-Rom splines
The effect of t: how sharply the curve bends at the control points

62. Catmull-Rom splines
The segment is controlled by p1,p2,p3,p4

63. Catmull-Rom splines
From Hermite curves

64. Catmull-Rom splines
From Hermite curves

65. Catmull-Rom splines
From Hermite curves

66. Catmull-Rom splines

67. Catmull-Rom splines

68. Catmull-Rom splines
What do we miss?

69. Catmull-Rom splines ? ?

70. Catmull-Rom splines

71. Catmull-Rom splines
Catmull-Rom splines have C1 continuity (not C2 continuity)
Do not lie within the convex hull of their control points.

72. Catmull-Rom splines
Catmull-Rom splines have C1 continuity (not C2 continuity)
Do not lie within the convex hull of their control points.

73. Catmull-Rom splines properties
X Continuity (C2)
√ Local control
√ Interpolation

74. Keyframe Interpolation
t=50ms
t=0

75. Curves in the animation
Interpolate each parameter separately
Each animation parameter is described by a 2D curve
Q(u) = (x(u), y(u))
Treat this curve as
θ = y(u)
t = x(u)
where θ is a variable you want to animate and t is a time index. We can think of the results as a function: θ(t) θ t u u

76. Keyframing interpolation
is a two-step algorithm:
- Compute the spline parameter u corresponding to the time t:
t=x(u) u

77. Keyframing interpolation
is a two-step algorithm:
- Compute the spline parameter u corresponding to the time t:
how to evaluate the inverse function?

78. Keyframing interpolation
is a two-step algorithm:
Compute the spline parameter u corresponding to the time t:
how to evaluate the inverse function?
- linear function: easy (t=u)
- complex function: finite difference or numerically root finding

79. Keyframing interpolation
is a two-step algorithm:
Compute the spline parameter u corresponding to the time t:
how to evaluate the inverse function?
- linear function: easy
- complex function: finite difference or numerically root finding
Compute the θ corresponding to the spline parameter u:
θ = y(u) u

80. Outline Process of keyframing Key frame interpolation
Hermite and bezier curve
Splines
Speed control

81. Spline parameterization
The spline is parameterized by u (0<=u<=1) and not by time t.

82. Spline parameterization
The spline is parameterized by u (0<=u<=1) and not by time t.
Hard to deal with the questions like
What is the location of point p at time t
What is the velocity of point p at time t
What is the acceleration a of a point p at time t

83. Spline parameterization
Solution: reparameterize the curve in terms of t
- express spline as a function of the arc length s: u=g(s)
- express the arc length s as a function of t: s=f(t)

84. Speed control
Simplest form is to have constant velocity along the path
s=f(t)

85. Arc-length reparameterization
Now we want a way to find u given a particular arclength s: u=g(f(t))
Not possible analytically for most curves (e.g. B-splines)

86. What’s parametric value for the arc length .08?
Finite difference
Sample the curve at small intervals of the parameter and determine the distance between samples
Use theses distances to build a table of arclength for this particular curve: (ui,si)
What’s parametric value for the arc length .08?

87. Finite difference
Sample the curve at small intervals of the parameter and determine the distance between samples
Use theses distances to build a table of arclength for this particular curve: (ui,si)

88. What’s parametric value for the arc length .08?
Finite difference
Sample the curve at small intervals of the parameter and determine the distance between samples
Use theses distances to build a table of arclength for this particular curve: (ui,si)
What’s parametric value for the arc length .08?

89. What’s parametric value for the arc length .08?
Finite difference
Sample the curve at small intervals of the parameter and determine the distance between samples
Use theses distances to build a table of arclength for this particular curve: (ui,si)
What’s parametric value for the arc length .08?

90. Finite difference
Sample the curve at small intervals of the parameter and determine the distance between samples
Use theses distances to build a table of arclength for this particular curve: (ui,si)
What’s parametric value for the arc length .05?
A linear interpolation of 0.00 and 0.05

91. Speed control recipe
Given a time t, lookup the corresponding arclength S in the speed curve
For S, look up the corresponding value of u in the reparameterization table
Evaluate the curve at u to obtain the correct interpolated position for the animated object for the given time t