Fourier series · Mathematics

FOURIER SERIES SIMULATION

▸ Initializing Fourier coefficients a₀, a_k, b_k…

▸ Building the chain of rotating epicycles

▸ Attaching frequency kω & amplitude A_k = √(a_k²+b_k²)

▸ Placing the pen at the tip of the last circle

▸ Enabling the approximation trace & stacking harmonics…

▸ Ready — Online. ✅

Epicycles & approximation

🌀 Fourier series

amplitude spectrum A_k by order k

Epicycle count (N)—

Fundamental frequency ω—

Rotation angle θ—

Approximation error (RMS)—

Target waveform—

Tip

Each rotating circle = one Fourier term (frequency kω, radius = amplitude). Stacked head-to-tail → the pen at the tip draws the shape. The more epicycles, the closer the trace matches the real function.

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Drag Epicycle count to add/remove harmonics · Speed adjusts the spin rate · switch the Scenario (square · sawtooth · triangle · circle · pulse · heart)

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Pen value over time (approximation vs. true function) Fourier approx.true function