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Registered Member #72
Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659
ShawnLG wrote ...
This is so cool. How did they do it?
Using a suitable graphical tool like Matlab, or Python+Numpy+Matplotlib, fling down a few random discrete points in the complex plane. Take the FFT to the frequency domain. Pad the frequency data up to (say) 8 or 16 times its length with zeroes, and transform back. This will give you a smooth interpolated band-limited curve between the points you originally specified. Now that the maths has been taken care of, put your artistic hat on, move existing points and add more until the interpolated curve draws the figure you want. It's rather like pinning a springy steel wire at a few defined points so that the curve in between them goes where you want (in fact the steel wire model produces a parametric cubic spline curve interpolation, which for reasonable length arcs between points and to the casual eye is indistinguishable from the FFT's band-limited interpolation).
Rather like the difference between artistically crafting a good 16x16 icon with a 16 colour palette, and reproducing a 256x256 jpeg, it's possible to get the machine to grunt through from any line drawing to the epicyclic coefficients, if you use a lot of them. But there's a particular artistic economy in molding your points to get an effective picture using the smallest number of coefficients you can. Remember the curve will be continuous, so there's a manual intervention right at the start to get a suitable closed line drawing.
Now when you have the desired line, read off the epicycle amplitudes and phases from the frequency domain Fourrier coefficients.
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Fourier Transform draws Homer Simpson
