1. Animation Keyframe Inverse Kinematic Parametric Scripted
Skeletal hierarchy
Inverse Kinematic
Parametric
Scripted
CS4995-1: Animation
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2. Ref: http://www.cs.berkeley.edu/~laura/cs184/quat/quaternion.html
Rotations
Euler angles – rotations about canonical axes (or in planes)
rx, ry, rz or az, el, ro
Order of rotation is important
singularities at 0o and 90o elevation
interpolations are not always “great circle”
Quaterions – rotations about a vector
i  i = -1 ( = j  j = k  k )
i  j = k ; k  i = j ; j  k = i (non-commutative)
conjugate: q = w + xi + yj + zk
magnitude: ||q|| = sqrt(w2 + x2 + y2 + z2 )
interpolations follow “great circle”
Conversion to and from Euler angles v
Ref:
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3. Dynamics Linear dynamics F = ma or a = F/m v Angular dynamics
at = dv/dt = d2x/dt2
vt + at dt = vt+dt = dx/dt
xt + vt dt + ½ at dt2 = xt+dt
xt+dt = xt + vt dt + ½ Ft/m dt2
Angular dynamics
 = I  or  = /I
t+dt = t + tdt + ½t/I dt2
I = moment of inertia
 = torque  v
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4. Conservation of Momentum
Linear momentum
p = m1v1 = m2v2
v2 = m1v1/m2
(elastic collision)
Angular momentum
L = I11 = I2 2 1 v1 2 v2 v -v
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5. Numerical Integration
dx/dt = (x, t)
Euler – xi+1= xi + h(x, t) where h = dt
Fast, but imprecise… error is O(h2)
Multi-step methods: compute intermediate results
Predictor-corrector – average slope of  at t and t+1
xpi+1 = xi + h (x, t)
xi+1 = xi + ½h ((xi, ti) + (xpi+1, ti+1))
Runge-Kutta – 4th-order solution
d1= h (xi, ti)
d2 = h (ti+ ½h, xi + ½ d1)
d3 = h (ti+ ½h, xi + ½ d2)
d4 = h (ti+ h, xi + d3)
xi+1 = xi + 1/6 (d1 + 2d2 + 2d3 + d4)
Then adapt next step size (h) based on error
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6. Collision Detection Bounding volumes
Sphere
Axis aligned bounding box
Oriented bounding box
Bounding polygon
Intersection testing…. Backing out
Space partitioning
Projection of position over time
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7. Motion Capture Optical markers Point cloud Tracking Editing
Marker identification
Skeletal mapping
Editing
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8. Digital Puppetry Real-time performance Quick CS4995-1: Animation
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