1. Keyframe Interpolation and Speed Control Jehee Lee Seoul National University

2. Controlling the Motion Along a Curve A parametric function P(t) = (x(t), y(t), z(t)) defines a motion –The parameter t is time –The position at time t is given by x(t), y(t), and z(t) This function defines both –Spatial trajectory, and –Speed of movement along the trajectory It is often very difficult for animators to design a curve that gives desired trajectory and speed simultaneously

3. Reparameterization A parametric function P(u) = (x(u), y(u), z(u)) defines a motion –The parameter u is not actually time –A parameter u(t) is a function of time t Typical process of keyframing –First, P(u) is designed to specify the trajectory –Reparameterization function u(t) is designed later to reflect proper speed and timing –demo2_jingle_bells.avidemo2_jingle_bells.avi

4. Reparameterization How do we determine u(t) ? –Arc length parameterization u(s) –Speed control s(t) Distance-time function Ease-In/Ease-Out The length along the space curve p(u) from the point p(u 1 ) to the point p(u 2 )

5. Arc Length Parameterization The length of the curve –A cubic polynomial p(u) cannot, in general, be parameterized by arc length in a closed form

6. Chord Length Approximation Approximation by chord length –Sample the curve at a multitude of parametric values Ex) u 1, u 2, …, u n –Estimate the arc length by computing the linear distance through the sequence of samples u(s) is monotonically increasing with respect to s

7. Chord Length Approximation Adaptive sampling –Add a new sample at the midpoint between two adjacent points p(u i ) and p(u i+1 ) if the total length changes above given tolerance –Repeat until there is no more point to add

8. Computing Arc Length Numerically Numerical integration –Evenly spaced sample intervals Trapezoidal rule (piecewise linear) Simson’s rule (piecewise quadratic) –Unevenly spaced sample intervals Gaussian quadrature is commonly used Adaptive sampling is also possible

9. Computing Arc Length Numerically Finding u given s –Can be formulated as a root finding problem –Newton-Raphson iteration is commonly used Solution is unique if dp(u)/du is not identically zero over some interval

10. Speed Control Speed control function relates an equally spaced parametric value (e.q., time) to arc length –Input: time t –Output: arc length s(t) –it is a distance-time function Normalized arc length –Arc length divided by the total length –Varies from 0 to 1 –Sometimes, the normalized arc length parameter will still be referred to simply as the arc length

11. Ease-In / Ease-Out Sine interpolation

12. Ease-In / Ease-Out Using sinusoidal pieces for acceleration and deceleration

13. Ease-In / Ease-Out Constant Acceleration

14. Ease-In / Ease-Out Constant Acceleration –Parabolic ease-in/ease-out

15. General Distance-Time Functions The user may work directly with the distance- time curve –Eg). Bezier, B-splines, cubic interpolating splines

16. Summary Decouple trajectory and parameterization –Arc length parameterization –Speed control Commercial animation systems provides UI for designing space curves and speed control curves separately Timing actually affects trajectory Timing is often specified by performance demo.wmv