Math and Movies (Animation at Pixar) | Hacker News Reader

Math and Movies (Animation at Pixar)

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58 pointsnazri14 comments

I find it interesting that the interviewer almost calls out on the disappointing fact that the surface smoothness is essentially just "blobized" interpolation: "It doesn't seem very subtle." It's like when a director has run the film through some colorizing filter to give it a desired tone and you can tell to the point of it becoming distracting.

The interviewer was trying to figure out why the subdivision process doesn't turn everything into a blob -- which is understandable, given that the examples he was shown were squares turning into circles and cubes turning into spheres. I think he's trying to understand why, for example, Geri's head doesn't turn into a sphere.

DeRose doesn't say so explicitly, but the reason is that the position of any given point on the infinitely subdivided surface is only determined by the positions of a handful of points in the control mesh. That's why, for example, the derivation for the final position of B given at the end of the video only depends on its neighbors A and C, and not any other points.

Thus, if you want more control over a portion of a subdivision surface, you can just use a denser control mesh in that portion of the model. You can create arbitrarily hard edges by moving control vertices closer together. At a microscopic scale, the edge is still smooth and blobby, but it looks sharp at a distance.

You can also make creases in subdivision surfaces by using different subdivision weights over the surface of the model, which may be what DeRose is getting at when he answers with "magic numbers". But it's my understanding that most modern CG is done using subdivision with the standard Catmull-Clark weights over the entire model, instead relying on the density of the input mesh to specify detail.

This is correct. Explicit/per vertex subdivision weights are basically never used in high end CG. Geometric detail is used, since it is simple and universal. Geometry density is no longer a significant factor in interactivity and not the most significant factor in rendering.

In case someone is interested in the "eigenanalysis," the recursion formula is:

    [1/2, 1/2, 0  ] [A^n]     [A^n+1]
    [1/8, 3/4, 1/8] [B^n]  =  [B^n+1]
    [0,   1/2, 1/2] [C^n]     [C^n+1]
    \_____   _____/
           M

And the question is, given [A^0,B^0,C^0], find [A^∞,B^∞,C^∞], which is equivalent to computing the infinite power of M. Waaaat? Enter the eigendecomposition.

The eigenvalues of M are 1/4, 1/2, and 1. If you compute M^∞, the 1/4 and 1/2 "eigenspaces" will disappear, so you're left with the subspace of the eigenvalue 1. http://bit.ly/eigenex001 M^∞ = QL^∞Q.inv(), hence the [1,4,1] appears... very cool.

Sometimes procrastinating by reading HN actually helps with your work---today I'm working on problem sets for book 2 http://gum.co/noBSLA

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