The appearance of the figure is highly sensitive to the ratio a/b. For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (a/b = 2, δ = π/2). Other ratios produce more complicated curves, which are closed only if a/b is rational. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures.
Lissajous figures where a = 1, b = N (N is a natural number) and
are Chebyshev polynomials of the first kind of degree N. This property is exploited to produce a set of points, called Padua points, at which a function may be sampled in order to compute either a bivariate interpolation or quadrature of the function over the domain [-1,1]×[-1,1].
The animation below shows the curve adaptation with continuously increasing fraction from 0 to 1 in steps of 0.01. (δ=0)
Prior to modern electronic equipment, Lissajous curves could be generated mechanically by means of a harmonograph.
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.
On an oscilloscope, we suppose x is CH1 and y is CH2, A is amplitude of CH1 and B is amplitude of CH2, a is frequency of CH1 and b is frequency of CH2, so a/b is a ratio of frequency of two channels, finally, δ is the phase shift of CH1.
A purely mechanical application of a Lissajous curve with a=1, b=2 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 with the "8" lying on its side.
Application for the case of a = b
When the input to an LTI system is sinusoidal, the output is sinusoidal with the same frequency, but it may have a different amplitude and some phase shift. Using an oscilloscope that can plot one signal against another (as opposed to one signal against time) to plot the output of an LTI system against the input to the LTI system produces an ellipse that is a Lissajous figure for the special case of a = b. The aspect ratio of the resulting ellipse is a function of the phase shift between the input and output, with an aspect ratio of 1 (perfect circle) corresponding to a phase shift of and an aspect ratio of (a line) corresponding to a phase shift of 0 or 180 degrees. The figure below summarizes how the Lissajous figure changes over different phase shifts. The phase shifts are all negative so that delay semantics can be used with a causal LTI system (note that −270 degrees is equivalent to +90 degrees). The arrows show the direction of rotation of the Lissajous figure.
- Lissajous figures are sometimes used in graphic design as logos. Examples include:
- In computing, Lissajous figures are in some screen savers.
|This article needs additional citations for verification. (November 2010)|
- "Did 'Vertigo' Introduce Computer Graphics to Cinema?".
- "The ABC's of Lissajous figures".
- "Lincoln Laboratory Logo". MIT Lincoln Laboratory. 2008. Retrieved 2008-04-12.
||This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (November 2010)|
|Wikimedia Commons has media related to Lissajous curves.|
- 3D Java applets depicting the construction of Lissajous curves in an oscilloscope:
- Interactive Lissajous Curves in Java – graphical representations of musical intervals, beats, interference, and vibrating strings
- Simple HTML5 Lissajous curve generator – allows controls for A and B as integers from 1 to 12 each
- Animated Lissajous figures