Lissajous figures, Epicycloids, Hypocycloids

Lissajous figures represent perpendicular harmonic motions. They are sometimes used in electronics to determine the relative frequency and phase of two signals.

Named after the French physicist Jules A. Lissajous (1822 - 1880) they are also sometimes called Bowdich curves.

Harmonic motions

In mathematics, harmonic functions are those which can be expressed using sine and cosine functions. The word harmonic also denotes a sequence of numbers whose reciprocals form an arithmetic progression e.g. 1/2, 1/3, 1/4, 1/5, ... .

Lissajous figures

Lissajous Figures are curves traced out by a point which follows two simple harmonic motions in perpendicular directions. The shape of the curve is determined by the relative phase and frequency of the motion.

The point at time t is given by

X = A1.sin(P1+F1.t)
Y = A2.sin(P2+F2.t)

where A1, A2 are the amplitudes, P1, P2 are the relative phases and F1, F2 are the frequencies. If F1 and F2 are aliquot (integrally related) then the curve will be static.

If the frequencies and phases are the same then a diagonal line will be produced. If the frequencies are the same and the phase difference is 180 degrees then the opposite diagonal will be traced. For a phase difference of 90 or 270 degrees (and the same frequencies) a circle will be traced. Other phase differences with the same frequencies produce ellipses.


Cycloids are produced by two circular motions in which the rotation of one vector is relative to the tip of the other vector (or a point on one circle rolling around the inside or outside of another). If both vectors rotate in the same direction then the figure is an epicycloid; if they rotate in opposite directions it is a hypocycloid.

For an epicycloid the point at time t is given by

X = A1.sin(P1+F1.t)+A2.sin(P2+F2.t)
Y = A1.sin(pi/2+P1+F1.t)+A2.sin(pi/2+P2+F2.t)

For an hypocycloid the point at time t is given by

X = A1.sin(P1+F1.t)+A2.sin(P2-F2.t)
Y = A1.sin(pi/2+P1+F1.t)+A2.sin(pi/2+P2-F2.t)

Note that the basic circular motions are given by introducing a phase offset of pi/2 (90 degrees) as described for Lissajous figures above.

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