A Lissajous figure is a curve formed by the superposition of two perpendicular simple harmonic motions. Such a curve can, by definition, be represented by a pair of parametric equations of the form \[x=A\sin (\alpha t + \phi)\] \[y=B\sin \beta t\]The following Lissajous figure, for example, is formed by setting \(A=B=10\), \(\alpha=3\), \(\beta =2\), and \(\phi=0\):

The constants \(A\) and \(B\) determine the scale of the figure, with the entire graph being contained in a box of dimension \(2A\) by \(2B\).

The entertainment value of these curves lies in playing around with the values of \(\frac {\alpha}{\beta}\) and \(\phi\), which results in a tremendous diversity of aesthetically pleasing patterns. Remembering that simple harmonic motion can be thought of as a one-dimensional projection of an object moving uniformly in a circle, the following table shows the patterns formed using a variety of values for the above constants:

You can approximate a Lissajous curve with a *harmonograph*, which uses two or more pendulums to control the movement of a pen relative to a drawing surface. In this video, for example, a harmonograph is set up with three degrees of freedom (two for the pen and one for the drawing surface)…

For real pendulums, of course, the equations above would have to be modified to include a damping factor, and the amplitude of the motions would have to be reasonably small to approximate the simple harmonic motion condition.

And if somebody asks, you can even pretend you’re studying all this for practical reasons, since in the professional audio world, Lissajous curves are used all the time for realtime analysis of the phase relationship between the left and right channels of a stereo audio signal 😉

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