Prior to modern electronic equipment, Lissajous curves could be generated mechanically by means of a
harmonograph.
Acoustics John Tyndall produced Lissajous curves by attaching a small mirror to a
tuning fork, and shining a bright light on the mirror. This produced a vertically oscillating bright dot. He then applied a rotating mirror to reflect the dot, producing a spread out curve. He used this technique as an analog oscilloscope to observe and quantify the oscillation patterns of a tuning fork. Later,
Helmholtz produced a Lissajous curve as follows. He made an "oscillation microscope" by attaching one lens of a microscope to a tuning fork, so that it oscillated in one direction. He attached a bright dot of paint on a violin string. Then he viewed the dot through the microscope while the string vibrated in the other direction, and saw a Lissajous curve. This is called the "Helmholtz motion".
Practical application Lissajous curves can also be generated using an
oscilloscope (as illustrated). An
octopus circuit can be used to demonstrate the
waveform images on an oscilloscope. Two phase-shifted sinusoid inputs are applied to the oscilloscope in X-Y mode and the phase relationship between the signals is presented as a Lissajous figure. In the professional audio world, this method is used for realtime analysis of the phase relationship between the left and right channels of a stereo audio signal. On larger, more sophisticated audio mixing consoles an oscilloscope may be built-in for this purpose. On an oscilloscope, we suppose is CH1 and is CH2, is the amplitude of CH1 and is the amplitude of CH2, is the frequency of CH1 and is the frequency of CH2, so is the ratio of frequencies of the two channels, and is the phase shift of CH1. A purely mechanical application of a Lissajous curve with , is in the driving mechanism of the
Mars Light type of oscillating beam lamps popular with railroads in the mid-1900s. The beam in some versions traces out a lopsided figure-8 pattern on its side. ==Application for the case of
a =
b==