Rather than tracing the Lissajous curve itself as illustrated above we can instead choose to draw a motive at each iteration point we touch while drawing the curve. Our motif might just be a straight line at some angle that is itself allowed to rotate. The combination of such lines based on the curve tends to generate caustic curves. See the Lissajous Curve Gallery or a few example below. Some of these simulate the decay effect of harmonograph pendulums.
As always with Processing, it is relatively straightforward to elaborate the basic idea to a point so far beyond earlier conceptions that at first sight it seems to have little relationship with its origins. The "Waves" series of productions in the Generative Art Gallery are based around elaborations of a simple Lissajous curve using variations on a wave motif.