Fourier Epicycles

Any closed curve is a sum of rotating vectors — circles riding on circles, each a single frequency. A real complex DFT turns your path into these vectors; chained tip-to-tail and ordered by size, they re-trace the curve with a gold pen. Slide N from one wobbly ellipse up to the exact figure. Pick a preset, or draw your own on the stage.

Motion

Speed 1.0×

Terms N / M

Circles N 1

N=1 is a single circle. As N rises the reconstruction snaps onto your curve. At N=M it is exact (≈ machine precision).

Shape

Show

Skin

Pen position at time t: Σ c[k]·e^(2πi·k·t), with c[k] = (1/M)·Σ z[m]·e^(−2πi·km/M) — a genuine DFT, computed here, no FFT library. Frequencies run 0, +1, −1, +2, −2… so the biggest low circles lead. The badge proves the round-trip is exact, energy is conserved (Parseval), and adding terms only ever lowers the error.