Fourier Epicycle Drawing
For $N$ complex sample points, compute $C_k$ via discrete Fourier transform (DFT) and reconstruct $z(t) = \sum_k C_k \exp(2\pi i k t / N)$. With $M$ epicycles the reconstruction error decreases monotonically; full $N/2$ reproduces the path within float precision.
WHAT TO TRY
- Add epicycles one at a time: each is a rotating vector at one Fourier frequency, and chaining them traces the drawing ever more faithfully, a literal picture of a Fourier series.
- The biggest circles carry the coarse shape and the small fast ones add fine detail, exactly how truncating a Fourier series keeps the low frequencies first.
- Watch the reconstruction error fall as the number of epicycles grows: any closed curve is a sum of circles upon circles, the geometric soul of the discrete Fourier transform.