Fourier Series Wave - 3D Epicycle Arm Machine

Machining the arms...

Presets

Live readouts

Graphs

One period: target vs partial sum

Amplitude spectrum r_n (volts)

RMS error vs number of harmonics N (log)

Power captured vs N (Parseval)

Scope: series vs target over time

Numerical checks

running...

How to use

Choose a waveform and the number of harmonics N. Left: the chain of rotating arms, one per harmonic; the tip height is the series value. The yellow dot hands its height to the scrolling plot, orange is the series and dashed grey is the exact waveform. Try the Gibbs presets, the sigma window and the narrow pulse. Drag to orbit, wheel to zoom, camera buttons for presets. The bars on the bench show the first 20 harmonic amplitudes. Space pauses, R resets.

Models and baked PBR textures made in Blender; Three.js r160 animates every part from the shared maths module.

The science

A periodic signal f(x) of period 2π can be written as a sum of sinusoids whose coefficients are projections of f onto each harmonic:

f(x) ~ a0/2 + Σ [a_n cos(nx) + b_n sin(nx)], a_n = (1/π)∫ f cos(nx) dx, b_n = (1/π)∫ f sin(nx) dx

Each term equals r_n sin(nx + ψ_n) with r_n = sqrt(a_n²+b_n²). That is a vector of length r_n turning at n times the fundamental rate, so the sum is a chain of rotating arms ("epicycles") and the series value is the height of the last tip. In the 3D machine the first 20 harmonics are real arms; any higher ones are folded into one grey residual arm so the tip is still exact. Here the coefficients are computed numerically (8192-point midpoint rule) for the waveform you choose, with x = 2π f t.

Known spectra. Square wave: b_n = 4/(nπ) for odd n only. Sawtooth: 2/(nπ) for all n. Triangle: 8/(n²π²) for odd n (it converges much faster because it is continuous). Full-wave rectified sine contains only even harmonics plus DC.

Gibbs phenomenon. Near a jump the partial sum overshoots by a fixed fraction of the jump: (Si(π)/π - 1/2) = 8.95%, however large N is; only the width of the overshoot shrinks like 1/N. The Lanczos sigma factor sinc(nπ/(N+1)) multiplies each term and reduces the overshoot to about 1.2% at the price of a slightly blurred edge.

Parseval. The mean power of the signal (into 1 ohm) equals the sum of the power in its harmonics: <f²> = (a0/2)² + ½ Σ r_n². The "power captured" readout is the partial sum of that series divided by the exact mean square.

Simplifications. The 80-harmonic cap means the sharpest edges are always slightly rounded; fundamental frequencies are slowed to 0.05 to 1.5 Hz so the motion can be followed (real audio signals are hundreds of times faster). The coefficient integrals are numerical, not symbolic; the checks below compare them with the closed forms.