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

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keyframe interpolation and speed control

Keyframe Interpolation and Speed Control

Jehee Lee

Seoul National University

controlling the motion along a curve
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
reparameterization
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
reparameterization1
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)

arc length parameterization
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
chord length approximation
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

chord length approximation1
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
computing arc length numerically
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
computing arc length numerically1
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
speed control
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
ease in ease out
Ease-In / Ease-Out
  • Sine interpolation
ease in ease out1
Ease-In / Ease-Out
  • Using sinusoidal pieces for acceleration and deceleration
ease in ease out2
Ease-In / Ease-Out
  • Constant Acceleration
ease in ease out3
Ease-In / Ease-Out
  • Constant Acceleration
    • Parabolic ease-in/ease-out
general distance time functions
General Distance-Time Functions
  • The user may work directly with the distance-time curve
    • Eg). Bezier, B-splines, cubic interpolating splines
summary
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