Nathaniel Bowditch studied these curves in 1815. They're also known as Lissajous curves. Such a curve is produced when both the x and y coordinates of a point vary as sine waves, and the ratio of frequencies of the two waves is a ratio of integers.
For example, the curves in the first image have a ratio of 1:2, so in the time it takes a point to one cycle left to right it has gone two cycles up and down. Try to figure out the ratios in the other images. An ellipse is produced when the ratio is 1:1.
These are some images I made back in 1983 with a Tektronix 4105 terminal.