(For an introduction to Fourier circle drawings, watch e.g. this Mathologer video: https://www.youtube.com/watch?v=qS4H6PEcCCA)
Converting a closed line drawing to its Fourier circle equivalent is easy (for some value of easy, anyway): write an equation for each segment, concatenate them so that they map $[0;2\pi]$ to a closed curve in the complex plane, compute the Fourier integrals for each coefficient, done. Alternately, if the drawing is a point cloud, use a discrete Fourier transformation.
In practice, this algorithm has a problem: it approximates/reproduces not only the curve itself, but also its derivative, i.e. the speed at which the "pen" at the end of the circles moves. This overfitting causes the result to be less than optimal.
As a simple example, assume that your curve is an equilateral triangle. You can get a reasonable approximation of that with just two coefficients ($c_1 = 1, c_{-2} = \frac{1}{4}$ or so), with each leg being mapped mapped to a third of the source interval. If your source image happens to have a different mapping (e.g. assume that the first leg is the image of the interval $[0;\pi]$) the result will require quite a few more values to be as close to the original triangle.
Thus my question is: is there a way to generate these coefficients that is agnostic of the original curve's derivative? The goal, of course, is to arrive at a result that requires a minimum of (… or at least fewer) non-zero coefficients for a given maximum error (or some other reasonable metric; I'm not picky).