Computer Animation Algorithms and Techniques

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Computer AnimationAlgorithms and Techniques
Physically Based Animation
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Physics Review force, mass, acceleration velocity
, position
m
v
vave
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Physics Review Gravity
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Physics Review Other forces
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Physics Review Spring-damper
Hookes Law
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Physics Review Momentum
conservation of momentum (mv)
In a closed system, momentum is conserved
After collision has same momentum as before
collision
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Elastic Collisions
No energy lost (e.g. deformations, heat)
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Elastic Collisions Kinetic Energy
As in all collisions momentum is conserved
In elastic collisions, kinetic energy is also
conserved
Solve for velocities
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Inelastic Collisions
Kinetic energy is NOT conserved
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Inelastic Collisions
Kinetic energy lost to deformation and heat
Momentum is conserved
Coefficient of restitution ratio of velocities
before and after collision
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Center of Mass
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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 mo
mentum
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Change in point due to rotation
- Angular velocity
On-axis or off axis rotation Angular velocity is
the same
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Delta-orientation due to rotation
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Change to a point on rotating object
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Physics Review Angular Stuff
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Angular Momentum
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Inertia Tensor
Aka Angular mass
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Inertia Tensor of particles
Discrete version For particle collection
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Standard Inertia Tensors
Symmetric wrt axes
Cuboid
Sphere
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Inertia Tensor transformations
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The Equations
F
r
Object attributes
M, Iobject-1
If using rotation matrix, will need to
orthonormalize updated rotation matrix
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Springs
Flexible objects Cloth Virtual springs Proportiona
l derivative controllers (PDCs)
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Spring-mass-damper system
f
-f
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Spring-mass system
V3
E23
E31
V2
V1
E12
Force
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Virtual edge springs system
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Angular springs
Linear spring between vertices
Dihedral angular spring
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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
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Virtual springs soft constraints
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Proportional (Derivative) Controllers
e.g., particle reacts to other forces while
trying to maintain position on curve virtual
spring
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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
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Particle system
Collides with environment but not other particles
Particles midlife with modified color and shading
Particles demise, based on constrained and
randomized life span
Particles birth constrained and time with
initial color and shading (also randomized)
source
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Particle system implementation

• STEPS
• for each particle
• if dead, reallocate and assign new attributes
• animate particle, modify attributes
• render particles

Use constrained randomization to keep control of
the simulation while adding interest to the
visuals