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Animation

Animation. Keyframe Skeletal hierarchy Inverse Kinematic Parametric Scripted. Rotations. Euler angles – rotations about canonical axes (or in planes) rx, ry, rz or az, el, ro Order of rotation is important singularities at 0 o and 90 o elevation

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Animation

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  1. Animation • Keyframe • Skeletal hierarchy • Inverse Kinematic • Parametric • Scripted Page 1

  2. 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: http://www.cs.berkeley.edu/~laura/cs184/quat/quaternion.html Page 2

  3. v Dynamics Angular dynamics •  = I  or  = /I t+dt = t + tdt + ½t/I dt2 I = moment of inertia  = torque Linear dynamics • F = ma or a = F/m 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 Page 3

  4. 1 2 v1 v2 Conservation of Momentum Angular momentum • L = I11 = I2 2 Linear momentum • p = m1v1 = m2v2 • v2 = m1v1/m2 (elastic collision) v -v Page 4

  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 Page 5

  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 Page 6

  7. Motion Capture • Optical markers • Point cloud • Tracking • Marker identification • Skeletal mapping • Editing Page 7

  8. Digital Puppetry • Real-time performance • Quick Page 8

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