SuperHelices for Predicting the Dynamics of Natural Hair

1
Super-Helices for Predicting the Dynamics of
Natural Hair

• F. Bertails, B.Audoly, M. Cani,
• B. Querleux, F. Leroy, J. Leveque
• Siggraph 2006

2
Outline

• Cosserat Curve
• Super-Helices
• Kinematics
• Dynamics equation
• Application Validation
• Result
• Conclusion

3
Cosserat Curve

• First introduced to the CG field by D. K.Pai
  Eurographics 2002
• A continuous coordinate frame E(s)

4
Cosserat Curve (Cont.)

• Relative Coordinate
• means cross product
• K is spatial angular velocity
• n0 is spatial linear velocity
• We can control the curve with two vector k n0

5
Outline

• Cosserat Curve
• Super-Helices
• Kinematics
• Dynamics equation
• Application Validation
• Result
• Conclusion

6
Super-Helices

• Use several segments of Cosserat curve to model
  hair
• Deform as a helix over every segment
• Lagrangian formulation of hair dynamics
• Validation of physical model against real hair.

7
Outline

• Cosserat Curve
• Super-Helices
• Kinematics
• Dynamics equation
• Application Validation
• Result
• Conclusion

8
Notation

• Center line of the strand
• Local frame of the strand

9
Kinematics

• A time function of angular velocity
• is the torsion and curvature,
• We can control local frame with

10
Structure

• Divide the strand into N segment
• These segment may have different
• lengths .
• is constant along each segment.

11
Constraint

• Fixed length of the strand
• Clamp one end of the strand into the head
• Integrate the strand from s0

12
Reconstruction

• Plug into
• We can obtain the centerline and the
  frame
• How to get ?

13
Lagrangian Mechanics

• Introduced by Joseph Louis Lagrange in 1788
• The difference between the kinetic energy (T) and
  the potential energy (U)

14
Lagrange Equation

• Lagrange equation can reduce to Newton's second
  law
• Lagrange equation can work with a set of
  generalized coordinates, instead of Cartesian
  coordinate
• Make the problem easy

15
Lagrange Equation (Cont.)

• When no external force
• With external force

16
Generalized Coordinate
Ki

• The strand divided into N segment,
• indexed by Q.
• Write as the curvature over the
  segment at time t
• We collect the numbers into a vector
  of size 3N, as generalized coordinate.

17
Dynamic equations

• The kinetic energy T
• The internal energy U
• The dissipation energy D

18
Dynamic equations (Cont.)

• The kinetic energy T
• The internal energy U
• The dissipation energy

19
Dynamic equations (Cont.)

• The right side is the generalized force

Gravity
Air Friction
Interaction force
20
Dynamic equations

• Simplify the lagrange equation, we got
• Ma stiffness friction dissipation force
  external force

21
Dynamic equations (Cont.)

• The inertia matrix (mass inertia of each
  segment)
• The stiffness matrix filled with bending
  torsional stiffnesses. (recoverer the strand)
• is the rest position
• A collects all remaining terms including air drag
  dissipation force. (D)

22
Solve dynamic equations

• Ma stiffness dissipation force external
  force
• We can get from this equation, and then ,
  ,

23
Outline

• Cosserat Curve
• Super-Helices
• Kinematics
• Dynamics equation
• Application Validation
• Result
• Conclusion

24
Hair Geometry

• Modified guild strands
• Full interpolation near the scalp
• No interpolation near the tips.

25
(No Transcript)
26
Hair mass stiffness

• Density
• Stiffness
• Chemical and Physical Behavior of Human Hair
  Robbins,2002

27
Natural Curliness

• is the radius
• the step measured near the tips

28
(No Transcript)
29
Collision detection

• Real-time simulation of self-collisions for
  virtual intestinal surgery. Raghupathi et al.
  2003

30
Contact and Friction

• Internal friction cofficient
• Air-hair friction cofficient v between

31
Contact and friction (Cont.)

• With another object
• Contact performed through penalty forces
• A tangential viscous friction force
• An isotropic friction force
• With hair strands
• Penalty force

32
Friction (Cont.)
33
Visual Comparisons Result

• Performance
• 3GHz Pentinum 4 processor
• 10 strands can be simulated in real-time
• Full hair model need 0.3s to 3 s per frame.
• Video

34
Conclusion

• Adv.
• External forces do not have to be applied at
  specific points such as nodes.
• Allow bigger time step about 1/30 sec.
• Disadv.
• Compute Time is O(N2)
• Degree of freedom restricted by segment
• More DOF, more work