Fourier Epicycle Drawing Estimator & "The Sound of Shapes" Studio

Complex DFT Epicycle Estimator · z(t) = Σ X_k e^{i ω_k t} (Drag Canvas to Draw Custom Contour)

Active Circles: N = 50 L₂ Energy: 99.62% RMS Error: 0.0184

DFT: X_k = (1/M) Σ_n=0..M-1 (x_n + i y_n) e^{-i 2π k n / M}

Instantaneous Speed |dz/dt|: 248.4 px/s → FM Pitch: 312.6 Hz

Synchronized Lissajous XY Oscilloscope & Orthogonal Projections x(t), y(t) Phase Lock: Δφ Orthogonal

Complex Harmonic Amplitude & Phase Spectrum |X_k| (Sorted by Energy) k = ±1, ±2, -3, +5

"The Sound of Shapes" — WebAudio Perimeter-Velocity & Additive Harmonic Wave Synth Standby (Visualizing Wave)