Keyframe Interpolation and Speed Control

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.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(u1) to the point p(u2)

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) u1, u2, …, un • 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(ui) and p(ui+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