In the previous post, I said that Lissajous curves are the result of plotting a curve whose *x* and *y* coordinates come from (undamped) harmonic oscillators. If we add a little bit of dampening, multiplying our cosine terms by negative exponentials, the resulting curve is called a **harmonograph**.

Here’s a bit of Mathematica code to draw harmonographs.

harmonograph[f1_, f2_, ph1_, ph2_, d1_, d2_, T_] :=
ParametricPlot[{
Cos[f1 t + ph1] Exp[-d1 t],
Cos[f2 t + ph2] Exp[-d2 t]
}, {t, 0, T}]

For example,

harmonograph[3, 4, 0, Sqrt[2], 0.01, 0.02, 100]

is a slight variation on one of the curves from the previous post, adding dampening terms exp(-0.01*t*) and exp(-0.01*t*). The result is the following.

You can get a wide variety of graphs by playing with the parameters. A couple years ago I wrote about an attempt to reproduce the background of the image in the poster for Alfred Hitchcock’s movie Vertigo. The curve is another example of a harmonograph.

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