Fourier Analysis: Drawing Llamas with Circles

If you put a read head at the tip that that could increment the phase of some circle from reading some dot it previously put down, or some similar scheme, could you make circles Turing complete?

This makes me unnecessarily happy and giddy.

I love Fourier series/transforms/DFTs irrationally.

This just makes my heart leap with joy to see creative output with it.

Shameless plug: I made this tool years ago (predates AI) that draws arbitrary images using Fourier circles https://phqb.github.io/circle_drawing/.

Can't you have another set of circles perpendicular to plane, which encode when the pencil draws, allowing for discontinuous regions?

I have a bit of a hard time liking this sentence in an otherwise lovely article: "The velocity is the same as the position but imaginary instead of real."

Well atoms are circles and llamas are made of atoms so real llamas are also made of circles i mean spheres similar thimg anyway