Lissajous curve
In mathematics, a
Lissajous curve /ˈlɪsəʒuː/, also known as
Lissajous figure or
Bowditch curve /ˈbaʊdɪtʃ/, is the graph of a system of parametric equations which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857. The appearance of the figure is highly sensitive to the ratio
a/
b. For a ratio of 1, the figure is an ellipse, with special cases including circles and lines. Another simple Lissajous figure is the parabola. Other ratios produce more complicated curves, which are closed only if
a/
b is rational. 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. 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 5 horizontal lobes and 4 vertical lobes.