Graphics II 91.547 Animation Introduction and Motion Control

1. Graphics II 91.547AnimationIntroduction and Motion Control Session 6

2. Animation • Traditional 2D animation • Origins in late 1920s • Flat shading • Illusion of 3D produced by fluidity of characters, use of perspective, motion of “virtual camera” • Disney animators pioneered major techniques • “Squash and stretch” • Secondary action • Appeal

3. Advantages of Computer Animation • Eliminates requirements of building models • No restriction on camera movement • Easy inclusion of shading models • Can introduce physical models

4. Animation Taxonomy • Representational animation • Rigid objects • Single, unchanging model for each object • Articulated objects • Rigid subobjects, connected at joints • Motions generally revolute • Soft objects • Model is deformed • Procedural animation • Stochastic animation • Behavioral animation

5. Interpolating endpoints Motion Control: Keyframing In computer graphics animation, keyframe -> key parameter. Therefore selection of the parameter becomes critical in defining appropriate motion. Interpolating angle

6. Motion control of rigid objectsParameterization of position Position Orientation Rigid Objects Articulated Objects Soft Objects

7. Spline-driven Position Animation Equal arc length, s Equal u Q(u)

8. Arclength Parameterization of Splines Eval. Spline Eval. s x,y,z s (arc length along spline) s Evaluate numerically u Spline parameter

9. Arclength Parameterization of Splines Arclength along spline: y Integrating gives: x z

10. Arclength Parameterization of Splines General form of Cubic Spline: Taking the derivatives and integrating: Where: This function will not integrate analytically, so integration must be done numerically.

11. Forward Differencing Approach to Evaluating A Where:

12. Ease-in, ease-out motionArc length no longer proportional to time Equal t Q(t)

13. Velocity Curves Velocity Curve s t u t u Eval. Cubic Bisection Search s t

14. Velocity Curves 3 s Velocity Curve s 2 1 t Position Spline

15. Velocity Curves Position Spline Gentle Acceleration from Rest Velocity Curve s Equal time Intervals Gentle deceleration to Rest t

16. General Kinetic Control(Steketee & Badler 1985) • “Position Spline” • Let the motion parameter to be interpolated be . is specified at n key values, . The position spline is constructed by assigning a keyframe number to each key value and interpolating through the resulting tuples: • “Kinetic Spline” • Each keyframe number is assigned a time. The kinetic spline interpolates through the resulting pairs:

17. y y y x x x z z z Parameterization of Orientation:Euler Angles Transformation: Transformation: Transformation:

18. Multiple Ways to Define Rotation withEuler Angles Rx -> Ry -> Rz Rx -> Rz -> Ry Ry -> Rx -> Rz Ry -> Rz -> Rx Rz -> Rx -> Ry Rz -> Ry -> Rx Possible Orderings:

19. y y y x x x z z z Problems with Euler Angle Parameterization:Gimbal Lock x’ Selecting a p/2 rotation about y removes a degree of freedom.

20. y y x x z z Problems with Euler Angle Parameterization:Interpolation R p X roll p R

21. y y y x x x z z z Problems with Euler Angle Parameterization:Interpolation R R p p R

22. Problems with Euler Angle Parameterization:Interpolation Generating Interpolated Orientations: Case 1: Case 2: Picture from p. 359 Case 1 Case 2

23. Alternate Approach to ParameterizingArbitrary Orientation Replace Euler angles with a single angle of rotation about an axial direction defined by the unit vector, n.

24. j i k Quaternions A quaternion is made up of a scalar plus a vector: We use the notation: Multiplication is defined:

25. Quaternions Take a pure quaternion (one that has no scalar part): And a unit quaternion: Define:

26. Moving in and out of Quaternion Space The quaternion: Converts to the transformation matrix:

27. Interpolating Between Two Quaternions Where:

28. Interpolating Examples

29. Interpolating Examples, contd. Pictures from page 367