Lissajous figures are plane curves formed by the intersection of two sine waves that lie orthogonal to one another:

x(t) = a_{x} sin(b_{x}t + c_{x})

y(t) = a_{y} sin(b_{y}t + c_{y})

When the amplitudes, frequencies, and phases of the two waves differ, the resulting figures are complicated, intermeshing curves. In addition, these special cases occur:

- When the frequencies and phases of the two waves are the same (b
_{x}= b_{y}and c_{x}= c_{y}), the resulting Lissajous figure is a straight line passing through the origin of the x-y axes. If the amplitudes are also the same, then the straight line lies at 45° to the x and y axes. - When the frequencies and amplitudes are the same (b
_{x}= b_{y}and (a_{x}= a_{y}), the Lissajous figures are ellipses. In addition, if the phase difference is 90° or 270°, the ellipses collapse to circles.

Lissajous figures have been used in various parts of science and engineering. A typical application is in signal analysis. Let x(t) represent an unknown sine wave whose amplitude and frequency are to be determined. We feed x(t) into an oscilloscope and apply a know sine wave y(t) othogonal to the unknown x(t), producing a Lissajous figure on the screen. We now adjust the amplitude and frequency of the known wave y(t), changing the shape of the Lissajous figure. When the figure becomes ellipitical, we have matched y(t) to the unknown x(t) and thereby determined the unknown amplitude and frequency.

The figures were first discovered by the American mathematician Nathaniel Bowditch (1773-1838) in 1815, so they are sometimes called Bowditch curves. They were discovered independently by the French mathematician Jules Antoine Lissajous (1822-1880) and studied thoroughly by him in 1857-58.

To create the Lissajous figures appearing on this website, we dampened the amplitudes:

x(t) = a_{x} exp[-k_{x}t] sin(b_{x}t + c_{x})

y(t) = a_{y} exp[-k_{y}t] sin(b_{y}t + c_{y})

The resulting motion models that of a decaying, compound pendulum; that is, a pendulum that can move independently in both the x and y directions.