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An Introduction to Computer Animation

An Introduction to Computer Animation. Michiel van de Panne. Overview. Creating Animations Representing Rotations. (1) Creating Animations. algorithm. (parameterized models, physics, optimization). user. data. (motion capture). (keyframes). Representing motion. DOF vs time

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An Introduction to Computer Animation

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  1. An Introduction to Computer Animation Michiel van de Panne

  2. Overview • Creating Animations • Representing Rotations

  3. (1) Creating Animations algorithm (parameterized models,physics, optimization) user data (motion capture) (keyframes)

  4. Representing motion • DOF vs time • cubic polynomial curves x smooth t fast in-out linear • alternative for motion through space: y + v apply arc-length reparameterization t x

  5. world torso RUarm RLarm LUarm Rhand LLarm Lhand RUleg RLleg LUleg LLleg Rfoot Lfoot head trans(0.30,0,0) rot(z, ) (2) Representing Rotations

  6. Transformation Hierarchies glTranslate3f(x,y,0); glRotatef( ,0,0,1); DrawBody(); glPushMatrix(); glTranslate3f(0,7,0); DrawHead(); glPopMatrix(); glPushMatrix(); glTranslate(2.5,5.5,0); glRotatef( ,0,0,1); DrawUArm(); glTranslate(0,-3.5,0); glRotatef( ,0,0,1); DrawLArm(); glPopMatrix(); ... (draw other arm) • Example y x

  7. Rotation DOFs • 2D: 1 DOF • 3D: 3 DOF • 4D: 6 DOF

  8. Transformations • Rotation glRotatef(angle,x,y,z); glRotated(angle,x,y,z);

  9. 3x3 Rotation Matrix

  10. 3x3 Rotation Matrix • 9 elements • 6 constraints • renormalization algorithms • extracting pure rotational component (polar decomp) ... and determinant = 1

  11. Rotations • SO(3) • rotations do not commute • require at least 4 parameters for a smooth parameterization • analogy: surface of the earth • 2D surface, 3 params • combing the hairy ball • camera orientation: view object from any dir

  12. Rotations • orientation vs rotation? • how to specify? • how to interpolate? • 2D vs 3D

  13. pitch yaw roll Fixed Angle Representations • fixed sequence of 3 rotations • RPY orientation: • can use many ordering of axes • Euler angles:

  14. Euler’s Rotation Theorem • can always go from one orientation to anotherwith one rotation about a single axis 180 deg about line in xy-plane where

  15. Quaternions • review of complex numbers • quaternions

  16. Quaternions • rotation of a vector • two successive rotations

  17. Quaternions RH rule • unit quaternions • addition • multiplication

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