lissajous

Lissajous figures are a family of planar curves traced by a point moving in two perpendicular harmonic motions with a constant phase difference. Named after the French physicist Jules Antoine Lissajous, they were studied in the 19th century to visualize relationships between oscillations.

Mathematically, they are described by x(t) = A sin(ωx t + δx) and y(t) = B sin(ωy t + δy), with amplitudes A, B, angular frequencies ωx, ωy, and phase offsets δx, δy. Often the form x = A sin(ωx t) and y = B sin(ωy t + φ) is used, where φ is the phase difference.

The resulting figure depends on the ratio ωx/ωy. If this ratio is rational, p/q, the curve is closed and contains a finite number of lobes; if irrational, the trace never closes and densely fills a region between the bounding curves.

Adjusting φ and the amplitudes rotates, distorts, or converts the shape into a straight line, ellipse, circle, or other patterns. In the special case ωx = ωy and A = B, φ = π/2 yields a circle; other φ values yield ellipses or lines.

Applications include signal analysis with oscilloscopes, where the pattern indicates the relative phase and frequency ratio of two signals. They also serve as a visual demonstration of harmonic motion in physics and teaching environments.

History: Jules Antoine Lissajous introduced the curves in the 1850s through optical demonstrations using vibrating forks and mirrors.