Michael Kass, John Anderson
The Spacetime Constraints formulation attempts to marry
the realism of physical simulation with the controllability
of keyframe animation, but the resulting nonlinear optimization
problems are generally extremely complicated and slow to solve. Here we
explore the range of Spacetime Constraints problems that give
rise to quadratic optimization functions solvable with linear
systems of equations. We find that they generalize traditional
splines to encompass oscillatory solutions.
These problems can be solved at full frame rates, giving animators a keyframe
animation tool with built in knowledge of a physical model.
In addition to the splines themselves, we also introduce a new
analysis method to extract oscillatory behavior from physical simulations
in a way that can be connected naturally to the splines.
It turns out that in order to have sufficient control of the
frequency response of splines, we solve the Spacetime
Constraints problems over the domain of complex numbers. As a consequence,
our solutions have an imaginary part in addition to the real part.
The imaginary part defines a phase angle that we show is very useful for
controlling and generalizing oscillatory behavior whether
extracted from simulation data or authored by hand.
Additional materials: [WigglySplines.mov]
Available as Pixar Technical Memo #06-06