Lissajous Curve

An interactive Lissajous curve — the looping pattern traced when two perpendicular oscillations of different frequencies and phase combine, as seen on an oscilloscope. Looking for a different curve? Browse the full list of interactive curves.

Lissajous Curve

Drag the "a" and "b" sliders to change the frequency ratio, and "δ" to shift the phase between the two oscillations.

Lissajous Curve — The Details

A Lissajous curve (or Lissajous figure, named after the French physicist Jules Antoine Lissajous, though studied earlier by Nathaniel Bowditch) is the path traced by a point whose $x$ and $y$ coordinates each oscillate sinusoidally, but at different frequencies and phases. The shape of the resulting curve depends entirely on the ratio between the two frequencies and the phase difference between them: a 1:1 ratio traces an ellipse or a straight line depending on the phase, while higher integer ratios produce increasingly intricate looping patterns. Lissajous figures are a classic way to visually compare two frequencies on an oscilloscope — feeding one signal into the horizontal input and another into the vertical input traces the curve live, and audio engineers still use them today to check the phase relationship between stereo channels. The parametric equations are:

$x(t) = A\sin(at + \delta)$

$y(t) = B\sin(bt)$