Hi GK,

That is some really nice work with the Fourier character display! Producing that display using analogue circuitry looks impossible at first glance, but we can all see that it is quite practical. The smooth curves give the characters a certain aesthetic appeal, too.

Now, I was thinking about how this works, and also about the Rossler experiment last year, and I wondered: would it be possible to apply this technique to the synthesis of three dimensional objects? After a certain degree of head scratching, it appears that the maths works out, though I think building a physical implementation will be on the edge of practicality, due to the overall complexity required.

For a 5th order approximation, the character generator requires ten frequencies plus a DC term. With these, it is possible to parameterise a curve as a function of one variable (s), in two dimensions, with a total of 22 coefficients.

A 3D surface can be parameterised as a function of two variables (u and v). Therefore, instead of five frequencies, 25 will be required for the same order synthesis (5 x u x 5 x v). Allowing for the sin, cos, and DC components, a total of 121 terms would need to be synthesised. To cater for the three axes, up to 363 coefficients might be required (although many shapes would only need a relatively sparse coefficient matrix).

The coefficients can be implemented fairly simply using resistors, so the main obstacle to using this technique would appear to be the generation of the signals at all of the different frequencies and phases. To generate them directly would require 100 analogue multipliers. It might be feasible to mix all the u harmonics and all of the v harmonics, multiply the sums, and then filter off the individual (m*u * n*v) components, but I suspect that this would involve a similar level of complexity.

Ideally, the low frequency parameter would change in discrete steps, while the high frequency parameter swept continuously, to give a rectangular U-V grid. However, I think it would be simpler if each parameter changed continuously. This would result in a slightly skewed U-V grid, similar to a television raster scan. It would also be desirable to be able to switch the U and V signals, in order to give grid lines in both directions.

I don't have any plans to actually try to build this - it would be huge! However, if anyone is crazy enough to give it a go, I might be able to offer some suggestions as to how to go about it.

I have attached a couple of Matlab simulations of this method. Note that it is not limited to solids of revolution, though these are the only ones that I have managed to figure out the coefficients for so far