Risch algorithm

This is another one of those postings that I make almost entirely for my own benefit. Today at lunch I was having a discussion with Tom about a particular problem that I was having trouble understanding: namely, understanding why the formulas for optimal sizing of pinhole camera aperatures were, well, what they are. Where do the numbers come from and why? It turns out that you can express the problem in terms of integrals of complex exponentials defined over the aperature, which seems obvious in most respects, but left me with the desire to actually compute some analytic integrals, and let’s face it, it’s been two decades since I did anything more complicated than ordinary polynomials. Tom mentioned the Risch algorithm which I had never heard of before, and while my brain seems too tired to absorb it at the moment, it seems like the kind of thing I’ll look into later, so this is just a reminder.

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Addendum: the image generated by a point light source by a finite aperature at a given wavelength assumes the form of an Airy disk. I wrote the equations out in gnuplot to make the following graph.

If you want some information about how Pixar made Cars, check out the comments on this thread on Digg. True, most of them are completely incorrect, but probably at no greater rate than the commercial news media.

Addendum: when you see totals like “17 hours per frame”, they almost certainly (and in this case do) refer to rendering a single frame on a single machine. Cars runs 1:56:00 long or so, at 24 frames per second. If we accept the 17 hours per frame, that’s 167,000 frames or so, which works out to 324 years of rendering for a single machine. Assuming 3000 machines, that’s about 40 days of rendering continuously.

This presumes that the frames are rendered only once (unrealistic) and that you can achieve 100% utilization (also likely unrealistic) and that all the machines are dedicated to the one production (also unrealistic).

Shameless plug: Cars opens tomorrow. Go out and see it. Sadly, this is the first Pixar feature which doesn’t have my name in the credits. I’m over it. Really.

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