This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail in 1857 by Jules Antoine Lissajous (for whom it has been named).
The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. A circle is a simple Lissajous curve.
The general formula of a Lissajous curve is x = A sin(at + δ), y = Bsin(bt). Visually, the ratio a/b determines the number of "lobes" of the figure. For example, a ratio of 3/1 or 1/3 produces a figure with three major lobes. Similarly, a ratio of 5/4 produces a figure with five horizontal lobes and four vertical lobes. Rational ratios produce closed (connected) or "still" figures, while irrational ratios produce figures that appear to rotate. The ratio A/B determines the relative width-to-height ratio of the curve. For example, a ratio of 2/1 produces a figure that is twice as wide as it is high. Finally, the value of δ determines the apparent "rotation" angle of the figure, viewed as if it were actually a three-dimensional curve. For example, δ = 0 produces x and y components that are exactly in phase, so the resulting figure appears as an apparent three-dimensional figure viewed from straight on (0°). In contrast, any non-zero δ produces a figure that appears to be rotated, either as a left–right or an up–down rotation (depending on the ratio a/b).
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