Interactive Spacetime Constraints: Wiggly Splines

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.

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Available as Pixar Technical Memo #06-06

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