Lissajous Figures

Theory
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 justice to Lissajous Figures. When the horizontal and vertical sine wave frequencies differ by a fixed amount, this is equivalent to constantly rotating the phase between them. The figure produced by this rotating phase appears to be a rotating 3D figure.
In addition, as in 3D wireframe images, the figure can appear to rotate in either direction, depending on how your brain interprets it. It can also spontaneously reverse the direction of rotation. (In a real 3D wireframe image the image can also appear to rotate around a different axis.)
By constantly increasing or decreasing the phase we produce the equivalent of having a small frequency difference. Technically, this is phase modulation. Large differences in frequency to produce integer multiples of frequencies are produced by multiplying the step size of the angles used to look up the sine value.

Results and Discussion
Cosine Graphs

Sine Graphs:

These graphs were obtained simply by plotting 2000 points with varying phase difference and ratios of ωx: ωy. These patterns can be used to determine frequencies the frequencies of sounds or radio signals by adjusting a known wave to match one of the known figures and ratios. Lissajous figures can be obtained through the plotting of data in Excel, or through observations on an oscilloscope.