Epicycles

A Fourier series is a sum of rotating circles. Watch them draw.

Shape

Circles used (harmonics)

24

Speed

1.0×

Display

Any closed curve Z(t) equals Σ cₙ·e^i(nt+φₙ) — a stack of circles, each spinning at an integer frequency n, with radius |cₙ| and starting phase φₙ set by the discrete Fourier transform of the path. The tip of the last circle traces the shape. Few circles → a smooth blur; sharp corners need many (notice the ripple — that's the Gibbs phenomenon).