Animation

1
Animation

• Keyframe
• Skeletal hierarchy
• Inverse Kinematic
• Parametric
• Scripted

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
3
Dynamics

• Linear dynamics
• F ma or a F/m
• at dv/dt d2x/dt2
• vt at dt vtdt dx/dt
• xt vt dt ½ at dt2 xtdt
• xtdt xt vt dt ½ Ft/m dt2

• Angular dynamics
• ? I ? or ? ?/I
• ?tdt ?t ?tdt ½?t/I dt2
• I moment of inertia
• ? torque

4
Conservation of Momentum

• Linear momentum
• p m1v1 m2v2
• v2 m1v1/m2
• (elastic collision)

• Angular momentum
• L I1?1 I2 ?2

v
-v
5
Numerical Integration

• dx/dt ?(x, t)
• Euler xi1 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
  t1
• xpi1 xi h ?(x, t)
• xi1 xi ½h (?(xi, ti) ?(xpi1, ti1))
• 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)
• xi1 xi 1/6 (d1 2d2 2d3 d4)
• Then adapt next step size (h) based on error

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

7
Motion Capture

• Optical markers
• Point cloud
• Tracking
• Marker identification
• Skeletal mapping
• Editing

8
Digital Puppetry

• Real-time performance
• Quick