waves

Lissajous Figures

Watch beautiful 3D parametric curves traced by two perpendicular harmonic oscillators. Adjust frequency ratio and phase to see patterns shift from lines to ellipses to intricate knots.

Lissajous

A-Level

Lissajous figures are parametric curves x = A·sin(a·t + δ), y = B·sin(b·t). The shape depends on the frequency ratio a:b and phase difference δ. When the ratio is rational, the curve closes; irrational ratios produce an endlessly dense pattern.

Freq Ratio A3

Freq Ratio B2

Phase90°

Amplitude A2.0

📐 Formulas

x = A\sin(at+\delta),\; y = B\sin(bt)Parametric equations

📋 Syllabus

AQA3.6.1.2