1. 110/27/2015 01:47 Graphics II 91.547 Animation Introduction and Motion Control Session 6

2. 210/27/2015 01:47 Animation 0 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. 310/27/2015 01:47 Advantages of Computer Animation 0 Eliminates requirements of building models 0 No restriction on camera movement 0 Easy inclusion of shading models 0 Can introduce physical models

4. 410/27/2015 01:47 Animation Taxonomy 0 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 0 Procedural animation 0 Stochastic animation 0 Behavioral animation

5. 510/27/2015 01:47 Motion Control: Keyframing In computer graphics animation, keyframe -> key parameter. Therefore selection of the parameter becomes critical in defining appropriate motion. Interpolating angle Interpolating endpoints

6. 610/27/2015 01:47 Motion control of rigid objects Parameterization of position Rigid Objects Articulated Objects Soft Objects Position Orientation

7. 710/27/2015 01:47 Spline-driven Position Animation Q(u) Equal arc length, s Equal u

8. 810/27/2015 01:47 Arclength Parameterization of Splines Eval. Spline s u s Evaluate numerically (arc length along spline) Spline parameter s x,y,z

9. 910/27/2015 01:47 Arclength Parameterization of Splines Arclength along spline: z x y Integrating gives:

10. 1010/27/2015 01:47 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. 1110/27/2015 01:47 Forward Differencing Approach to Evaluating A Where:

12. 1210/27/2015 01:47 Ease-in, ease-out motion Arc length no longer proportional to time Q(t) Equal t

13. 1310/27/2015 01:47 Velocity Curves t s Velocity Curve t u Bisection Search Eval. Cubic t u s

14. 1410/27/2015 01:47 Velocity Curves s 3 Position Spline t s 1 2 Velocity Curve

15. 1510/27/2015 01:47 Velocity Curves Position Spline t s Velocity Curve Equal time Intervals Gentle Acceleration from Rest Gentle deceleration to Rest

16. 1610/27/2015 01:47 General Kinetic Control (Steketee & Badler 1985) 0 “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: 0 “Kinetic Spline” -Each keyframe number is assigned a time. The kinetic spline interpolates through the resulting pairs:

17. 1710/27/2015 01:47 Parameterization of Orientation: Euler Angles x y z x y z x y z Transformation:

18. 1810/27/2015 01:47 Multiple Ways to Define Rotation with Euler Angles Rx -> Ry -> Rz Rx -> Rz -> Ry Ry -> Rx -> Rz Ry -> Rz -> Rx Rz -> Rx -> Ry Rz -> Ry -> Rx Possible Orderings:

19. 1910/27/2015 01:47 Problems with Euler Angle Parameterization: Gimbal Lock x y z x y z x y z x’ Selecting a  rotation about y removes a degree of freedom.

20. 2010/27/2015 01:47 Problems with Euler Angle Parameterization: Interpolation y x z y x z X roll  

21. 2110/27/2015 01:47 Problems with Euler Angle Parameterization: Interpolation y x z  y x z  y x z

22. 2210/27/2015 01:47 Problems with Euler Angle Parameterization: Interpolation Picture from p. 359 Case 1: Case 2: Generating Interpolated Orientations: Case 1 Case 2

23. 2310/27/2015 01:47 Alternate Approach to Parameterizing Arbitrary Orientation Replace Euler angles with a single angle of rotation about an axial direction defined by the unit vector, n.

24. 2410/27/2015 01:47 Quaternions i j k A quaternion is made up of a scalar plus a vector: We use the notation: Multiplication is defined:

25. 2510/27/2015 01:47 Quaternions Take a pure quaternion (one that has no scalar part): And a unit quaternion: Define:

26. 2610/27/2015 01:47 Moving in and out of Quaternion Space Converts to the transformation matrix: The quaternion:

27. 2710/27/2015 01:47 Interpolating Between Two Quaternions Where:

28. 2810/27/2015 01:47 Interpolating Examples

29. 2910/27/2015 01:47 Interpolating Examples, contd. Pictures from page 367