Lissajous curves

Lissajous curves (pronounced /ˈlɪsəʒuː/) are the family of curves that oscilloscope draws when its two inputs (representing horizontal and vertical shift) are connected to two sinusoidal signals, possibly having different frequency and phase shift. For instance, connecting both inputs to the same source produces a single line that is inclined by 45 degrees. However if the signals are shifted in time by a quarter of they period (90 degrees), oscilloscope will draw a circle instead. More complex shapes can be observed when the signals also differ in frequency. To get a stable picture, the frequencies must not be arbitrary but instead must be related to each other by certain not too complex ration (1:1, 1:2, 4:5 and so on). In the past, these curves were important method of the signal comparison and the trained engineer was able to interpret the picture saying this ratio and the phase shift. They were observable even with the most primitive oscilloscopes without the horizontal sweep generator.

As it is seen from the provided applet, in recent times Lissajous figures can also be easily generated by software. They can also be traced mechanically using harmonograph. The visual form of these curves often remembers a three-dimensional knot, and indeed many knots (including those known as Lissajous knots), project to the plane as Lissajous figures.

In mathematics, a Lissajous curve is the graph of a system of parametric equations

where a and b are frequencies and is the frequency shift. A and B are the amplitudes of the two signals.