Computer Animation Algorithms and Techniques Technical Background Orientation

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

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

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

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

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

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

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

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

Axis-angle to rotation matrix Y A X Z Rick Parent 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 Computer Animation

Axis-angle to rotation matrix Y A P’ X P Z Rick Parent Computer Animation

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

Quaternion Arithmetic Addition Multiplication Inner Product Length Rick Parent Computer Animation

Quaternion Arithmetic Inverse Unit quaternion Rick Parent Computer Animation

Vector Represention & Transformation Vector Transform Rick Parent Computer Animation

Quaternion Geometric Operations Rick Parent Computer Animation

Unit Quaternion Conversions Axis-Angle Rick Parent Computer Animation

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