**Testing**- To get a feeling for elliptic fourier analysis with the SHAPE software I made a set of test shapes:

Then I determined their outline with ChainCoder.exe and calculated the elliptic spectra with CHC2NEF.exe. The I made a plot with OpenOffice Calc. These are the results with some analysis:

Note: Below I've replaced the primary component of 1.0 with 0 otherwise the "harmonics" would be totally invisible. That's why you see nothing in place 1.

**Circle**- Theoretically a perfect circle should not have a spectrum. It should only have the first component. But my hand-drawn digtized circle made in MS-paint is not perfectly symmetrical and it has rough edges. So there are still some "harmonics" but these are much fainter than the harmonics of the other shapes (a factor of 100: 10^-3 instead of 10^-1). I assume that I'm just seeing "random noise" and "quantization noise" in this spectrum.

**Triangle**- One would expect the order-3 harmonics to be dominant for a triangular shape but that is not the case at all. There is no immediately visible correlation between a figure and its spectrum. This is even more obvious if one looks at the shape as more harmonics are added. The second harmonic is already sufficient for a nice triangular shape. I edited the .nef output files by hand and plotted them with NefViewer.exe.

Note: Below I've replaced the primary component of 1.0 with 0 otherwise the "harmonics" would be totally invisible. That's why you see nothing in place 1.

But if we look at the spectral components then we see that 10 harmonics is not really sufficient for a nice sharp triangle. We need at least 20 harmonics.

**Star**- Here you would expect that the order 2, 3 and 6 harmonics would be dominant. And you would expect that the spectrum of the triangle looks limilar to the spectrum of the star. But things are not that intuitive.

**Square**- Here there's a surprising similarity between normal fourier spectra and 2-D fourier spectra. A square wave spectrum has only odd harmonics. And this 2-D spectrum has also only odd harmonics!

**Rectangle**- There is a lot of similarity between the square and the rectangle. Also only odd harmonics, but the signs and ratios of the harmonic components are different from the square. It is interesting to see how the rectangle is constructed from the different harmonics.

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