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Parametric Cartesian equation: x = asin(nt + c), y = bsin(t)
Can also be written in the form: x = acos(wxt - dx), y = acos(wyt - dy)
Lissajous curves or Lissajous figures are sometimes called Bowditch curves after Nathaniel Bowditch who studied them in 1815. They were studied in more detail (independently) by Jules-Antoine Lissajous in 1857.
Lissajous curves tend to look somewhat like the figures projected during
laser light shows, and there is good reason for that.
Lissajous would use sounds of different frequencies to vibrate a
mirror. A beam of light reflected from the mirror would trace patterns which depended on the frequencies of the sounds. This is quite similar to the
laser light shows of today, except Lissajous' setup used a simple beam of light instead of a powerful
laser.
Before the days of
digital frequency meters and
phase-locked loops, Lissajous figures were used to determine the frequencies of sounds or
radio signals. A signal of known frequency was applied to the horizontal axis of an
oscilloscope, and the signal to be measured was applied to the vertical axis. The resulting Lissajous figure was a function of the
ratio of the two frequencies. Compare the resulting figure to a figure from known frequencies, and you could then
extrapolate the
unknown frequency.
Lissajous figures often appeared as props in early
science fiction movies. One of the most obvious examples can be found in the opening sequence of "
The Outer Limits". The pattern of criss-cross lines is actually a Lissajous figure.
The curves can either loop back upon themselves, or repeat forever. They close upon themselves if and only if
wx/
wy is
rational.
A few more examples of Lissajous curves...
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