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. t θ

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

5. Review: Natural cubic cruves

8. Review: Natural cubic curves Properties: - Go through four control points - not good for local control

9. Review: Hermite Curves P1P1 R1R1 P4P4 R4R4 P 1 : start position P 4 : end position R 1 : start derivative R 4 : end derivative

10. Review: Hermite Curves P1P1 R1R1 P4P4 R4R4

11. P1P1 R1R1 P4P4 R4R4 Herminte basis matrix

12. Review: Hermite Curves P1P1 R1R1 P4P4 R4R4 Herminte basis matrix

13. Review: Hermite Curves P1P1 R1R1 P4P4 R4R4

14. P1P1 R1R1 P4P4 R4R4

15. P1P1 R1R1 P4P4 R4R4

16. P1P1 R1R1 P4P4 R4R4

17. P1P1 R1R1 P4P4 R4R4 Hermite basis functions

18. Review: Hermite Curves P1P1 R1R1 P4P4 R4R4 basis function 1basis function 2basis function 3 basis function 4

19. Review: Hermite Curves P1P1 R1R1 P4P4 R4R4 *P 1 *P 4 *R 1 *R 4 +++ =

20. Review: Bezier Curves

24. *v 0 *v 1 *v 2 *v 3 +++ =

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

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 C 0 : points coincide, velocities don’t C 1 : points and velocities coincide What’s C 2 ? - 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 - the curve is always located within the convex hull based on control points. 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 C 2 continuity when two curves meet

39. Derivatives at end points

45. Continuity in B splines Suppose we want to join two Bezier curves (V 0, V 1, V 2,V 3 ) and (W 0, W 1, W 2, W 3 ) so that C 2 continuity is met at the joint

46. Continuity in B splines Suppose we want to join two Bezier curves (V 0, V 1, V 2,V 3 ) and (W 0, W 1, W 2, W 3 ) so that C 2 continuity is met at the joint

47. Continuity in B splines Suppose we want to join two Bezier curves (V 0, V 1, V 2,V 3 ) and (W 0, W 1, W 2, W 3 ) so that C 2 continuity is met at the joint

48. Continuity in B splines Suppose we want to join two Bezier curves (V 0, V 1, V 2,V 3 ) and (W 0, W 1, W 2, W 3 ) so that C 2 continuity is met at the joint

49. Continuity in B splines Suppose we want to join two Bezier curves (V 0, V 1, V 2,V 3 ) and (W 0, W 1, W 2, W 3 ) so that C 2 continuity is met at the joint

50. Continuity in B splines Suppose we want to join two Bezier curves (V 0, V 1, V 2,V 3 ) and (W 0, W 1, W 2, W 3 ) so that C 2 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? b is the middle point of a and c. w 2 =v 1 +4v 3 -4v 2

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 C 2 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 C 2 ) - 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 C 2 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 control points, we get a Catmull-Rom spline

58. Catmull-Rom splines

59. The segment is controlled by p 1,p 2,p 3,p 4 0.5

60. Catmull-Rom splines The segment is controlled by p 1,p 2,p 3,p 4

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 p 1,p 2,p 3,p 4

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

68. What do we miss?

69. Catmull-Rom splines ? ?

71. Catmull-Rom splines have C 1 continuity (not C 2 continuity) Do not lie within the convex hull of their control points.

72. Catmull-Rom splines Catmull-Rom splines have C 1 continuity (not C 2 continuity) Do not lie within the convex hull of their control points.

73. Catmull-Rom splines properties X Continuity (C 2 ) √ Local control √ Interpolation

74. Keyframe Interpolation t=0 t=50ms

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) u u θ t

76. Keyframing interpolation is a two-step algorithm: - Compute the spline parameter u corresponding to the time t: u t=x(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: u θ = y(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 of 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 Time warping function to control speed - positive - monotonic; you cannot reverse the time s=f(t)

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

86. Speed control Assume that the motion slows down at the beginning and end of the motion curve s=f(t)

87. 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)

88. 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: (u i,s i ) What’s parametric value for the arc length.08?

89. 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: (u i,s i )

90. 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: (u i,s i ) Finite difference What’s parametric value for the arc length.08?

91. 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: (u i,s i ) What’s parametric value for the arc length.08?

92. 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: (u i,s i ) What’s parametric value for the arc length.05? A linear interpolation of 0.00 and 0.05

93. 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