1. Computer Animation Rick Parent Computer Animation Algorithms and Techniques Physically Based Animation

2. Computer Animation Rick Parent v ave v Physics Review: force, mass, acceleration velocity, position m f a a v’

3. Computer Animation Rick Parent Physics Review: Gravity

4. Computer Animation Rick Parent Physics Review: Other forces Static friction Kinetic friction Viscosity for small objects No turbulence For sphere

5. Computer Animation Rick Parent Physics Review: Spring-damper Hooke’s Law

6. Computer Animation Rick Parent Physics Review: Momentum conservation of momentum (mv) In a closed system, momentum is conserved After collision has same momentum as before collision

7. Computer Animation Rick Parent Elastic Collisions No energy lost (e.g. deformations, heat)

8. Computer Animation Rick Parent Elastic Collisions & Kinetic Energy In elastic collisions, kinetic energy is also conserved As in all collisions: momentum is conserved Solve for velocities

9. Computer Animation Rick Parent Inelastic Collisions Kinetic energy is NOT conserved

10. Computer Animation Rick Parent Inelastic Collisions Kinetic energy lost to deformation and heat Momentum is conserved Coefficient of restitution ratio of velocities before and after collision

11. Computer Animation Rick Parent Center of Mass

12. Computer Animation Rick Parent Physics Review Linear v. angular terms Angular Rotation, orientation rotational (angular) velocity rotational (angular) acceleration Inertia tensor Torque Angular momentum Linear postion velocity acceleration mass Force momentum

13. Computer Animation Rick Parent Change in point due to rotation - Angular velocity On-axis or off axis rotation Angular velocity is the same

14. Computer Animation Rick Parent Delta-orientation due to rotation

15. Computer Animation Rick Parent Change to a point on rotating object

16. Computer Animation Rick Parent Physics Review: Angular Stuff F rr

17. Computer Animation Rick Parent Angular Momentum

18. Computer Animation Rick Parent Inertia Tensor Aka Angular mass

19. Computer Animation Rick Parent Inertia Tensor of particles Discrete version For particle collection

20. Computer Animation Rick Parent Standard Inertia Tensors Symmetric wrt axes Cuboid Sphere

21. Computer Animation Rick Parent Inertia Tensor transformations

22. Computer Animation Rick Parent The Equations If using rotation matrix, will need to orthonormalize updated rotation matrix F r Object attributes M, I object -1

23. Computer Animation Rick Parent Springs Flexible objects Cloth Virtual springs Proportional derivative controllers (PDCs)

24. Computer Animation Rick Parent Spring-mass-damper system f-f

25. Computer Animation Rick Parent Spring-mass system Force V1V1 V2V2 V3V3 E 12 E 23 E 31

26. Computer Animation Rick Parent “Virtual” edge springs system

27. Computer Animation Rick Parent Angular springs Linear spring between vertices Dihedral angular spring

28. Computer Animation Rick Parent Spring mesh Each edge is a spring-damper Angular springs connect every other mass point Each vertex is a point mass Global forces: gravity, wind Diagonal springs for rigidity

29. Computer Animation Rick Parent Virtual springs – soft constraints

30. Computer Animation Rick Parent Proportional (Derivative) Controllers e.g., particle reacts to other forces while trying to maintain position on curve – virtual spring

31. Computer Animation Rick Parent Particle systems Lots of small particles - local rules of behavior Create ‘emergent’ element Particles: Do collide with the environment Do not collide with other partcles Do not cast shadows on other particles Might cast shadows on environment Do not reflect light - usually emit it

32. Computer Animation Rick Parent Particle system source Particle’s birth: constrained and time with initial color and shading (also randomized) Particle’s demise, based on constrained and randomized life span Collides with environment but not other particles Particle’s midlife with modified color and shading

33. Computer Animation Rick Parent Particle system implementation STEPS 1. for each particle 1. if dead, reallocate and assign new attributes 2. animate particle, modify attributes 2. render particles Use constrained randomization to keep control of the simulation while adding interest to the visuals