1. Physics of Hair
Maxim Bovykin

2. Hair Dynamics Used for fur, long hair, grass…
Started with short hair / fur
Evolved into longer, human hair
Proprietary Algorithms

3. Writing hair simulator
Hair does not stretch at all
But, (long) hair bends quite a lot
Upon extreme bending, hair becomes stiff

4. Uniqueness of Hair Dynamics
Hair does not stretch at all
Stiffness Issue in length
But, (long) hair bends quite a lot
Preservation of Angular Momentum
Upon extreme bending, hair becomes stiff
Stiffness issue in bending

5. Hair is a non-linear problem
Rigid Joint (linear in angles)
Non-linear constraints
Mass Spring (linear in position)
Non-linear in angles

6. Mass Spring Friendly for production environment
Robust Collision Response
Efficient
Challenges
Standard mass spring integrator fails for a non-linear problem
Implicit integrator helps, but not fully.

7. Basic Mass Spring Systemxi xj

8. Basic Mass Spring Systemxi xj L

9. Basic Mass Spring Systemxi xj Fj

10. Basic Mass Spring Systemxi xj Fj

11. Basic Mass Spring System

12. How large is k ?

13. k
= 
Hair does not stretch at all
k is close to infinity

14. k =  Explicit integrator diverges on Large dt (smaller dt slow!)
Large k (smaller k looks bad!)

15. k =   implicit integration? Implicit integrator adds stability
Large dt (OK)
Large k (OK)

16. k
= 
 implicit integration?
n+1

17. Implicit integration – a closer look
Filter
Damping

18. Implicit integration – a closer look
Damping
Filter
Implicit integrator adds stability through
Wider filtering of velocity
Artificial Damping

19. = * Implicit integration as a filter Damping Filter Filter
Velocity Field

20. = * = * As k becomes larger, Implicit filter widens Damping Filter
Smaller k * =
larger k

21. = * = * As K, implicit filter makes things move together Smaller k
Stability gain good for cloth, not quite for hair
Too large k results in excessive smoothing
Loss of angular momentum
Too small k results in stretching of hair

22. Implicit integration as a damping force
Filter
Damping
Artificial damping is automatically added
Damping term depends on k and dt
Increase in stability, but loss of accuracy
too much damping as k approaches 

23. k =   implicit integration? Implicit integrator adds stability
Loss of angular momentum
‘Good’ Jacobian (filter) very important

24. k Is infinity! Well, how do we preserve length then?
 use non-linear correction

25. non-linear post correction

26. non-linear post correction

27. non-linear post correction

28. non-linear post correction
Post solve correction
Successive relaxation until convergence
Guaranteed length preservation
Cheap simulation of kinfinity

29. non-linear post correction
How to implement?
Cloth simulation literatures
Provot 1995 (position only)
Bridson 2002 (impulse)
Goldenthel 2007 (CLM)
Hair-specific relaxation possible

30. non-linear post correction
Provot1995
Simulates biphasic spring
Apply correction until convergence (or time limited)
Non-linear, Non-dynamic inverse kinematic procedure

31. non-linear post correction
Bridson2002
Momentum preserving velocity filter
Non-linear Jacobi vs Gauss-Seidel

32. k =  Recap Large k not good Explicit integrator blows up
Loss of angular momentum in implicit solvers

33. k =  Recap Use post-step relaxation Very small k ok
Explicit integrators ok

34. Angle preservation Length preservation isn’t enough
Hair tends to go back to original shape

35. Angle preservation

36. Angle preservation Flexion spring?
Linear spring between distant nodes.

37. Flexion spring (x) Linear Spring  Loss of angular momentum
Ambiguity in direction for angles > 180
Unwanted wrinkles in hair shape

38. Angle preservation Derivation on angles
Two angles suffice (no torsion)
Energy derived from changes in angles

39. Angle preservation 0 F 

40. Angle preservation Derivation on angles
Additional anchor point needed at the root

41. Angle preservation Derivation on angles Full angle required (360 )
=2/3
=4/3
’= 2- 
= 2/3
Derivation on angles
Full angle required (360 )
Axis generation an issue

42. Angle preservation Implicit integration used on angles
Non-linear problem again
Jacobian treatment needed

43. Predictor-corrector scheme
Implicit integration as a predictor for angles
Non-linear corrector for length
Two-pass implicit filtering with non-linear corrector in the middle of integration
Linearized Implicit integrator augmented with a non-linear optimizer

44. Predictor-corrector scheme
Implicit Filter (Predictor)
Sharpener (Corrector)
Implicit Filter (Predictor)

45. 1.First pass-implicit integration
First implicit solve to get new velocity

46. 2.First pass-implicit integration
Advance position with the predicted mid-step velocity

47. 3.Non-linear Correction
Apply non-linear corrector to get position (length) right

48. 4.Impulse Change velocity due to length preservation
Velocity may be out of sync after impulse

49. 5.Second implicit integration
Filters out velocity field
Velocity field in sync again

50. Recap - Simulating Hair with Mass Spring System
Hair dynamics is a non-linear numerical integration problem
Standard mass spring integrator fails
Implicit integrator helps, but not enough.
Predictor-corrector scheme

51. Hair Dynamics – Collision Detection
Simple collision detection algorithm:
Calculate distance between core shapes (line or rectangle)
Subtract appropriate offset (radius of each SSV)

52. Hair Dynamics – hair hair interaction

53. Hair Hierarchy

54. Subdivision of Hair

55. Subdivision of Hair

56. The End - Q & A