1. Computer Animation Algorithms and Techniques
Technical Background
Orientation
Rick Parent
Computer Animation

2. Orientation Representation
Rotation matrix
Fixed angles: rotate about global coordinate system
Euler angles: rotate about local coordinate system
Axis-angle: arbitrary axis and angle
Quaternions: mathematically handy axis-angle 4-tuple
Exponential map: 3-tuple version of quaternions
Rick Parent
Computer Animation

3. Fixed Angles Rotate around ‘fixed’ (global) axesY Z
Fixed Angles
Rotate around ‘fixed’ (global) axes
Given in pre-specified order: e.g., x, y, z; also could be x, y, x
Rick Parent
Computer Animation

4. Gimbal Lock Fixed angle: e.g., x, y, z Y Y X X Z Z Rick Parent
Computer Animation

5. Gimbal Lock Fixed order of rotations: x, y, z
What do these epsilon rotations do? X Y Z
Rick Parent
Computer Animation

6. Gimbal Lock What is intermediate orientation?X Y Z X Y Z
What is intermediate orientation?
Interpolating FA representations does not produce intuitive rotation because of gimbal lock
Rick Parent
Computer Animation

7. Euler Angles Prescribed order: e.g., x, y, z or x, y, x
Rotate around (rotated) local axes
Note: fixed angles are same as Euler angles in reverse order and vice versa
Rick Parent
Computer Animation

8. Axis-Angle A Rotate about given axis Euler’s Rotation Theorem OpenGLY Z A
Rotate about given axis
Euler’s Rotation Theorem
OpenGL
Fairly easy to interpolate between orientations
Difficult to concatenate rotations
Rick Parent
Computer Animation

9. Axis-angle to rotation matrixY Z A
Concantenate the following:
Rotate A around z to x-z plane
Rotate A’ around y to x-axis
Rotate theta around x
Undo rotation around y-axis
Undo rotation around z-axis
Rick Parent
Computer Animation

10. Axis-angle to rotation matrixY Z A P P’
Rick Parent
Computer Animation

11. Quaternion A Same as axis-angle, but different formY Z A
Same as axis-angle, but different form
Still rotate about given axis
Mathematically convenient form
Note: in this form v is a scaled version of the given axis of rotation, A
Rick Parent
Computer Animation

12. Quaternion Arithmetic
Addition
Multiplication
Inner Product
Length
Rick Parent
Computer Animation

13. Quaternion Arithmetic
Inverse
Unit quaternion
Rick Parent
Computer Animation

14. Vector Represention & Transformation
Rick Parent
Computer Animation

15. Quaternion Geometric Operations
Rick Parent
Computer Animation

16. Unit Quaternion Conversions
Axis-Angle
Rick Parent
Computer Animation

17. Quaternions Avoid gimbal lock Easy to rotate a point
Easy to convert to a rotation matrix
Easy to concatenate – quaternion multiply
Easy to interpolate – interpolate 4-tuples
How about smoothly (in both space and time) interpolate?
Rick Parent
Computer Animation