Pixar mathematics

Pixar's use of harmonic functions (by David Austin) describes mathematical techniques used by Pixar. Incidentally, apparently there exists something called Pixar University, which I learned when I went to the excellent Pixar exhibit at the Oakland Museum of California. As far as I can tell they are not hiring, they're really an internal training program, and anyway I don't know anything about animation. (The exhibit's next stop is in Hong Kong.

The problem that the article addresses is basically this: 3-D animated characters have many "control points" on their surface. Animators would not like to have to specify how all of these move individually, so they don't; they only specify the motion of a subset of these called the "cage". How do we interpolate? When there are just three control points this is obvious -- use barycentric coordinates in both triangles. This is basically exploiting the fact that there's a unique linear transformation taking a generic triangle to another. generic triangle

But what if there are more than three control points? One generalization is "mean value coordinates", which are a linear transformation, but which are problematic when the cage is not convex. Since the cage is often, say, the outline of a human being, this is a real problem! Apparently the newer technique is "harmonic coordinates" -- define the coordinates of the n boundary points of a cage to be (1,0,...,0), (0,1,...,0), ..., (0,0,...,1). Then for any point p in 3-space, let h_i(p) be linear on the edges of the cage, and harmonic (having zero Laplacian) elsewhere. Then the non-cage control point p has harmonic coordinates h₁(p), ..., h_n(p); if you move the cage points then these points keep the same harmonic coordinates but change their coordinates in 3-space. Unfortunately inverting the h-function is a bit harder but it seems worth the trouble.

There's a video explaining this, with lots of images!