Spacetime Constraints

by Andrew Witkin and Michael Kass

This is a paper from graphics talking about how to implement an animation of objects, such as luxo lamp from Pixar (see details of Lamp Jr. on wikipedia).

First thing is about a spacetime particle. We get the ODE for the particle using Newton's law and typically we should solve the ODE using some numerical methods. We have some initial value constraints and also an objective function (minimum feul that exhausted in the procedure). This is actually a variational optimization problem. The numerical solution is obtained by discretization. The derivatives are approximated via finite differences. And then the objective function actually turns out to be a quadratic form. So we get a linear constrained quadratic optimization problem.

For a complex object, the ODE is obtained via Lagrange dynamics (hmm... almost forget it). So actually we may solve a similar problem (but much more complicated, you have to analyse case by case).