Lissajous figures are patterns generated by the junction of a pair of sinusoidal waves with axes that are perpendicular to one another. Jules-Antione Lissajous studied these figures by producing sounds of different frequencies which were used to vibrate a mirror. A beam of light was then reflected from the mirror to produce trace patterns which were dependent on the frequencies. This setup used by Lissajous is similar to what is used today to project laser light shows. Today, Lissajous figures are generated with an oscilloscope, a type of cathode ray tube that provides a picture of electric signals in the form of a graph. Before digital frequency meters and phase-locked loops, Lissajous figures were initially used to determine the frequencies of sounds or radio signal. A signal of known frequency would be applied to the horizontal axis of an oscilloscope and the signal to be measured was applied to the vertical axis. The amplitude and frequency of the known wave are adjusted, changing the shape of the figure until a known figure of a specific ratio will be obtained, and these ratios will be used to determine the unknown variables. Lissajous figures are plane curves represented by:
x = Axcos(ωxt + Øx)
y = Aycos(ωyt + Øy)
Where: Ax & Ay – amplitudes; t – time; ω - angular velocity; Ø – phase difference When the amplitudes, frequencies, and phases of the two waves differ, complicated intermeshing curves are produced. There are however a few special cases which occur: 1. When the frequencies and phases of the two waves are the same, the resulting 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. 2. When the frequencies and amplitudes are the same (ωx = ωy and Ax = Ay), the Lissajous figures are ellipses. In addition, if the phase difference is 90° or 270°, the ellipses collapse to circles. Mere static pictures however do not do...