1. CSCE 441 Computer Graphics: Keyframe Animation/Smooth Curves Jinxiang Chai

2. Outline Keyframe interpolation Curve representation and interpolation - natural cubic curves - Hermite curves - Bezier curves Required readings: 12-6 & 14-1 14-214-3 14-4, & 14-7

3. Computer Animation Animation - making objects moving Compute animation - the production of consecutive images, which, when displayed, convey a feeling of motion.

4. Animation Topics Rigid body simulation - bouncing ball - millions of chairs falling down

5. Animation Topics Rigid body simulation - bouncing ball - millions of chairs falling down Natural phenomenon - water, fire, smoke, mud, etc.

6. Animation Topics Rigid body simulation - bouncing ball - millions of chairs falling down Natural phenomenon - water, fire, smoke, mud, etc. Character animation - articulated motion, e.g. full-body animation - deformation, e.g. face

7. Animation Topics Rigid body simulation - bouncing ball - millions of chairs falling down Natural phenomenon - water, fire, smoke, mud, etc. Character animation - articulated motion, e.g. full-body animation - deformation, e.g. face Cartoon animation

8. Animation Criterion Physically correct - rigid body-simulation - natural phenomenon Natural looking - character animation Expressive - cartoon animation

9. Keyframe Animation

10. Keyframe Interpolation IK can be used to create Key poses t=0 t=50ms

11. Keyframe Interpolation What’s the inbetween motion? t=0 t=50ms

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

13. 2D Animation Highly skilled animators draw the key frames Less skilled (lower paid) animators draw the in- between frames Time consuming process Difficult to create physically realistic animation

14. 3D Animation Animators specify important key frames in 3D Computers generates the in-between frames Some dynamic motion can be done by computers (hair, clothes, etc) Still time consuming; Pixar spent four years to produce Toy Story

15. The Process of Keyframing Specify the keyframes Specify the type of interpolation - linear, cubic, parametric curves Specify the speed profile of the interpolation - constant velocity, ease-in-ease-out, etc Computer generates the in-between frames

16. A Keyframe In 2D animation, a keyframe is usually a single image In 3D animation, each keyframe is defined by a set of parameters

17. Keyframe Parameters What are the parameters? –position and orientation –body deformation –facial features –hair and clothing –lights and cameras

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

19. Inbetween Frames Linear interpolation Cubic curve interpolation

20. Keyframe Interpolation t=0 t=50ms

21. Linear Interpolation Linearly interpolate the parameters between keyframes t x t0t0 t1t1 x0x0 x1x1

22. Linear Interpolation: Limitations Requires a large number of key frames when the motion is highly nonlinear.

23. Cubic Curve Interpolation We can use a cubic function to represent a 1D curve

24. Smooth Curves Controlling the shape of the curve

25. Smooth Curves Controlling the shape of the curve

26. Smooth Curves Controlling the shape of the curve

27. Smooth Curves Controlling the shape of the curve

28. Smooth Curves Controlling the shape of the curve

29. Smooth Curves Controlling the shape of the curve

30. Constraints on the cubics How many points do we need to determine a cubic curve?

31. Constraints on the Cubic Functions How many points do we need to determine a cubic curve?

32. Constraints on the Cubic Functions How many points do we need to determine a cubic curve? 4

33. Constraints on the Cubic Functions How many points do we need to determine a cubic curve? 4

34. Natural Cubic Curves Q(t 1 )Q(t 2 )Q(t 3 ) Q(t 4 )

35. Interpolation Find a polynomial that passes through specified values

36. Interpolation Find a polynomial that passes through specified values

37. Interpolation Find a polynomial that passes through specified values

38. Interpolation Find a polynomial that passes through specified values

39. 2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

40. 2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

41. 2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

42. Limitation? What’s the main limitation of interpolation using natural cubic curves?

43. Limitation? What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve

44. Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve

45. Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve

46. Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG

47. Constraints on the cubic curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG MG

48. Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG M G

49. Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG M? G?

50. Two Issues How to select a new set of control points? How to determine M? M?

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

52. Hermite Curve A Hermite curve is determined by - endpoints P 1 and P 4 - tangent vectors R 1 and R 4 at the endpoints P1P1 R1R1 P4P4 R4R4

53. Hermite Curve A Hermite curve is determined by - endpoints P 1 and P 4 - tangent vectors R 1 and R 4 at the endpoints Use these elements to control the curve, i.e. construct control vector P1P1 R1R1 P4P4 R4R4 MhMh GhGh

54. Hermite Basis Matrix Given desired constraints: - endpoints meet P 1 and P 4 Q(0) = [0 0 0 1 ] · M h · G h = P 1 Q(1) = [1 1 1 1 ] · M h · G h = P 4 - tangent vectors meet R 1 and R 4

55. Tangent Vectors

57. Hermite Basis Matrix Given desired constraints: - endpoints meet P 1 and P 4 Q(0) = [0 0 0 1 ] · M h · G h = P 1 Q(1) = [1 1 1 1 ] · M h · G h = P 4 - tangent vectors meet R 1 and R 4 Q’(0) =[0 0 1 0] · M h · G h =R 1 Q’(1) =[3 2 1 0] · M h · G h =R 4

58. Hermite Basis Matrix Given desired constraints: - endpoints meet P 1 and P 4 Q(0) = [0 0 0 1 ] · M h · G h = P 1 Q(1) = [1 1 1 1 ] · M h · G h = P 4 - tangent vectors meet R 1 and R 4 Q’(0) =[0 0 1 0] · M h · G h =R 1 Q’(1) =[3 2 1 0] · M h · G h =R 4 So how to compute the basis matrix M h ?

59. Hermite Basis Matrix We can solve for basis matrix M h MhMh

60. Hermite Basis Matrix We can solve for basis matrix M h MhMh

61. Hermite Basis Matrix P1P1 R1R1 P4P4 R4R4

62. Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points P1P1 P3P3 P2P2 P4P4

63. Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points -The curve starts at P1 going towards P2 and Arrives at P4 coming from the direction P3 - usually does not pass through P2 or P3; these points are only there to provide directional information. P1P1 P3P3 P2P2 P4P4

64. Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points P1P1 P3P3 P2P2 P4P4

65. Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points How to compute the basis matrix M b ? P1P1 P3P3 P2P2 P4P4

66. Bezier Basis Matrix Establish the relation between Hermite and Bezier control vectors

67. Bezier Basis Matrix Establish the relation between Hermite and Bezier control vectors

68. Bezier Basis Matrix Establish the relation between Hermite and Bezier control vectors M hb GbGb

69. Bezier Basis Matrix For Hermite curves, we have For Bezier curves, we have

70. Bezier Basis Matrix For Hermite curves, we have For Bezier curves, we have

71. Bezier Basis Matrix P1P1 P3P3 P2P2 P4P4

72. Hermite Basis Function Let’s define B as a product of T and M B h (t) indicates the weight of each element in G h

73. Hermite Basis Function Let’s define B as a product of T and M B h (t) indicates the weight of each element in G h What’s function of this red curve?

74. Hermite Basis Function Let’s define B as a product of T and M B h (t) indicates the weight of each element in G h What’s function of this red curve? 2t 3 -3t 2 +1

75. Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials

76. Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of this red curve?

77. Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of this red curve? B 1 = -t 3 +3t 2 -3t+1

78. Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of this red curve? B 2 = -t 3 +3t 2 -3t+1

79. Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of the Bezier curve? B 1 P 1 +B 2 P 2 +B 3 P 3 +B 4 P 4

80. How to interpolate a 3D curve x y z o

81. x y z o Bezier curve

82. Try this online at - Move the interpolation point, see how the others (and the point on curve) move - Control points (can even make loops) Bezier java applet http://www.cse.unsw.edu.au/~lambert/splines/

83. Cubic curves: Hermite curves: Bezier curves: Different basis functions

84. Complex curves Suppose we want to draw a more complex curve

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

86. Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control

87. Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers

88. Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers

89. Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers

90. Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers Why cubic? - Lowest dimension with control for the second derivative - Lowest dimension for non-planar polynomial curves

91. Next lecture Spline curve and more key frame interpolation