After trying and failing to create the image of a sin wave travelling round a circle mathematically, I decided to instead take an approach that utilized the nature of the way the circle was drawn in the first place.
It is constructed from a series of points, each forming anchor points of a line that interpolates through them. This automatic interpolation creates the form of a sin wave when every other point is displaced. As such, to create the image of a sin wave it was a matter of displacing some these points (with a small fall-off). The circular effect then comes from moving the point, about which the vertices are displaced, in a circular motion.
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