Lissajous Family

Lissajous curves, named for the French physicist Jules Antoine Lissajous who studied them in the 1850s, are the parametric traces of two perpendicular sinusoidal motions at potentially different frequencies. They were originally produced mechanically by reflecting light off paired tuning forks, and later became the canonical screen image of analog oscilloscopes in lab settings. Their shapes encode frequency ratios and relative phase, making them a standard tool for tuning instruments and visualizing two-channel signals.

Here a single 3:2 curve, (\sin(3t + d), \sin(2t)), is traced over one full period while its phase offset d animates from 0 to 2\pi. The frequency ratio fixes the number of lobes and crossings; the sweeping phase slides the curve through that one-parameter family, shifting the self-intersections and re-braiding the lobes continuously. Rational ratios always produce closed curves; irrational ratios would trace figures that never quite repeat.