Fourier Epicycles Transform

Visualizes the mathematical foundation of the Fourier Transform in the complex plane. Chains rotating complex exponential phasor gears (epicycles) summing harmonic modes to reconstruct complex geometric curves, complete with live amplitude spectrum bars.

Playground

60 FPS • Canvas 2D

Click + Drag to interact with field

Live Parameters

Harmonic Terms (N) 16

Number of active Fourier epicycle phasor modes

Rotation Speed 0.6

Fundamental angular frequency ω

Mathematical Formulation

40 chars high

Analytical Equation

f(t) = \sum_{k=-N}^{N} c_k e^{i k \omega t}, \quad c_k = \frac{1}{T} \int_0^T f(t) e^{-i k \omega t} dt

Compact Formula

z(t) = sum(c_k * exp(i * k * omega * t))

Renderer Logic

// Complex Fourier Series Phasors
let x = cx, y = cy;
for (let i = 0; i < terms; i++) {
  const theta = h.k * t + h.phase;
  x += h.radius * Math.cos(theta);
  y += h.radius * Math.sin(theta);
}

Open in Playground Zero Dependencies • Standalone

Mathematical Tags

#fourier-transform #complex-analysis #epicycles #phasors #math-study #signal-processing

Author: Harmonic Analysis Core Target: 60 FPS

Fourier Epicycles Transform

Live Parameters

Mathematical Formulation

Mathematical Tags

Export & Embed: Fourier Epicycles Transform

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