Drawing a Closed Curve with Fourier Epicycles

A closed plane curve can be parametrized by a continuous, 1-periodic complex-valued function z(t) = x(t) + i y(t), with z(t + 1) = z(t). For the piecewise-smooth parametrizations used here, z(t) equals its Fourier series:

z(t) = ∞Σk = −∞ a_k e^2πikt, a_k = ∫₀¹ z(t) e^−2πikt dt.

In this demo, the curve is sampled at N = 511 points z_n = z(n/N), equally spaced along its length. The discrete coefficients are c_k = (1/N) N−1Σn = 0 z_n e^−2πikn/N for integers k with −255 ≤ k ≤ 255. They approximate the continuous coefficients a_k above.

1. The curve and its epicycles

Draw on the left. The right panel shows an M-term Fourier approximation.

Your curve z(t)

presets

Fourier approximation with M = 21 terms

M 21

term order animation period 8 s show circles

target curve circles |c_k| tip of the chain traced approximation

2. Fourier amplitudes |c_k|

Highlighted bars are included in the current approximation. A phase is shown as — when the displayed amplitude is 0.00.

frequencies shown |k| ≤ 40

#	k	\|c_k\| (px)	arg c_k	included

1. The curve and its epicycles

Your curve z(t)

Fourier approximation with M = 21 terms

2. Fourier amplitudes |ck|

2. Fourier amplitudes |c_k|