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Computer Animation
Algorithms and Techniques
Physically Based Animation
Rick Parent
Computer Animation
Physics Review:
force, mass, acceleration
velocity, position
a
f  ma
m
a  f /m
v'  v  a  t
p'  p 
(v  v ' )
1
t  p  vt  at 2
2
2
f
v’
vave
v
a
Rick Parent
Computer Animation
Physics Review: Gravity
Gm1m2
F
d2
Gmearthm2
F
2
d earth radius
F
a
 9 .8 m / s 2
m2
Rick Parent
Computer Animation
Physics Review: Other forces
f st  k st f N
Static friction
f k  kk f N
Kinetic friction
f vis   K vis nv
K vis  6r
Viscosity
for small objects
No turbulence
For sphere
Rick Parent
Computer Animation
Physics Review: Spring-damper
F  k s ( Lcurrent  Lrest )
F  k s ( Lcurrent  Lrest )  k dVspring
Hooke’s Law
Rick Parent
Computer Animation
Physics Review: Momentum
P  mv
conservation of momentum (mv)
In a closed system, momentum is conserved
P  mv  m'v'
After collision has same momentum as before collision
Rick Parent
Computer Animation
Elastic Collisions
No energy lost (e.g. deformations, heat)
mv  m'v'
d mv  dP
F  ma 

dt
dt
Rick Parent
Computer Animation
Elastic Collisions & Kinetic Energy
As in all collisions: momentum is conserved
P  mv  m'v'
In elastic collisions, kinetic energy is also conserved
1 2
1
P   mv   mv2
2
2
Solve for velocities
Rick Parent
Computer Animation
Inelastic Collisions
Kinetic energy is NOT conserved
Rick Parent
Computer Animation
Inelastic Collisions
Kinetic energy lost to deformation and heat
Momentum is conserved
Coefficient of restitution
ratio of velocities before
and after collision
Rick Parent
Computer Animation
Center of Mass
Cmass
1

M
1
 r dm  M
Cmass
Rick Parent
 (r )r dV

  (r )r dV   (r )dV

mi ri

mi
Computer Animation
Physics Review
Linear v. angular terms
Linear
postion
velocity
acceleration
mass
Force
momentum
Rick Parent
Angular
Rotation, orientation
rotational (angular) velocity
rotational (angular) acceleration
Inertia tensor
Torque
Angular momentum
Computer Animation
Change in point due to rotation
 (t )
- Angular velocity
 (t )
 (t )
On-axis or off axis rotation
Angular velocity is the same
p(t )  x(t )  r (t )
 (t ) r (t )
x(t )
Rick Parent
r(t )   (t )  r (t )
r(t )  (t ) r (t ) sin 
Computer Animation
Delta-orientation due to rotation
R(t )  R1 (t ) R2 (t ) R3 (t )
R (t )   (t )  R1 (t )  (t )  R2 (t )  (t )  R3 (t )
 Ay Bz  Az B y   0

 
A  B   Az Bx  Ax Bz    Az
 Ax B y  Ay Bx   Ay

 
 Az
0
Ax
Ay   Bx 
 
 Ax   B y   A* B
0   Bz 
R (t )   (t )* R(t )
Rick Parent
Computer Animation
Change to a point on rotating object
q(t )  R(t )q  x(t )
R(t )q  q(t )  x(t )
q (t )
 (t )
x(t )
q (t )   (t )* R(t )q  v(t )
q (t )   (t )  (q(t )  x(t ))  v(t )
Rick Parent
Computer Animation
Physics Review: Angular Stuff
f  ma
  rF
P  mv
L  r p
dP
f 
dt
Pi  c
dL

dt
Li  c
r
F
r
p  mv

  I
Rick Parent
Computer Animation
Angular
Momentum
q (t )
 (t )
x(t )
L(t )  ((q(t )  x(t ))  mi (q (t )  v(t )))
 ( R(t )q  mi ( (t )  (q(t )  x(t ))))
 (mi ( R(t )q  ( (t )  R(t )q)))
Rick Parent
Computer Animation
Inertia
Tensor
 I xx

I   I xy
 I xz

Aka Angular mass
I xx   ( y  z )dm
2
2
I yy   ( x 2  z 2 )dm
I zz   ( x  y )dm
2
Rick Parent
2
I xy
I yy
I yz
I xz 

I yz 
I zz 
I xy    xydm
I xz    xzdm
I yz    yzdm
Computer Animation
Inertia
Tensor of
particles
 I xx

I   I xy
 I xz

Discrete version
For particle collection
I xy
I yy
I yz
I xz 

I yz 
I zz 
I xx  mi ( yi  zi )
I xy  mi xi yi
I yy  mi ( xi  zi )
I xz  mi xi zi
I zz  mi ( xi  yi )
I yz  mi yi zi
2
2
2
Rick Parent
2
2
2
Computer Animation
Standard Inertia Tensors
Symmetric wrt axes
Cuboid
 I xx
I   0
 0
0
I yy
0
0
0 
I zz 
 M (b 2  c 2 )

0
0
1 

I 
0
M (a 2  c 2 )
0

12
2
2

0
0
M (a  b )

1 0 0
2 MR 

I
0
1
0

5 
0 0 1
2
Sphere
Rick Parent
Computer Animation
Inertia Tensor transformations
 I xx  M (Y 2  Z 2 )
 I xy  MXY
 I xz  MXZ 


I translated    I xy  MXY
I yy  M ( X 2  Z 2 )
 I yz  MYZ 
2
2 
  I xz  MXZ

I

MYZ
I

M
(
X

Y
)
yz
zz

I rotated  RI object R 1
Rick Parent
Computer Animation
 x(t ) 
 R (t )

S (t )  
 P (t )


 L(t ) 
Object attributes
M, Iobject-1
1
1
I rotated  RI object R 1
P (t )
v (t ) 
M
w(t )  I (t ) 1 L(t )
Rick Parent
The Equations
F
r
  rF
 x(t )   v(t ) 
 R(t )  w(t )* R(t )
d
d 


S (t ) 
dt
dt  P(t )  F (t ) 

 

L
(
t
)

(
t
)

 

If using rotation matrix, will need to
orthonormalize updated rotation matrix
Computer Animation
Springs
Flexible objects
Cloth
Virtual springs
Proportional derivative controllers (PDCs)
Rick Parent
Computer Animation
Spring-mass-damper system
f
Rick Parent
-f
Computer Animation
Spring-mass system
V3
E23
E31
Force
Rick Parent
V1
E12
V2
Computer Animation
“Virtual” edge springs system
Rick Parent
Computer Animation
Angular springs
Linear spring between vertices
Dihedral angular spring
Rick Parent
Computer Animation
Spring mesh
Each vertex is a point mass
Each edge is a spring-damper
Diagonal springs for rigidity
Angular springs connect every
other mass point
Global forces: gravity, wind
Rick Parent
Computer Animation
Virtual springs – soft constraints
Rick Parent
Computer Animation
Proportional (Derivative) Controllers
e.g., particle reacts to other forces while trying to
maintain position on curve – virtual spring
d  x(t )  p(t )
F  ks d
Rick Parent
F  k s d  kd vrelative
Computer Animation
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
Rick Parent
Computer Animation
Particle
system
Collides with environment
but not other particles
Particle’s midlife with
modified color and shading
Particle’s demise, based on
constrained and randomized life span
source
Rick Parent
Particle’s birth: constrained and time with
initial color and shading (also randomized)
Computer Animation
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
Rick Parent
Computer Animation