In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve , is
the graph of a system of parametric equations
which describe complex harmonic motion. This family of curves was investigated by Nathaniel
Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857.
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 and lines. Another simple Lissajous
figure is the parabola. 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. Visually, the ratio a/b determines the number
of "lobes" of the figure. For example, a ratio of 3/1 or 1/3 produces a figure with three
major lobes. Similarly, a ratio of 5/4 produces a figure with 5 horizontal lobes and 4 vertical
lobes. Rational ratios produce closed or "still" figures, while irrational ratios produce figures
that appear to rotate. The ratio A/B determines the relative width-to-height ratio of the
curve. For example, a ratio of 2/1 produces a figure that is twice as wide as it is high.
Finally, the value of δ determines the apparent "rotation" angle of the figure, viewed as
if it were actually a three-dimensional curve. For example, δ=0 produces x and y components
that are exactly in phase, so the resulting figure appears as an apparent three-dimensional
figure viewed from straight on. In contrast, any non-zero δ produces a figure that appears
to be rotated, either as a left/right or an up/down rotation.
Lissajous figures where a = 1, b = N 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].
Examples The animation below shows the curve adaptation
with continuously increasing fraction from 0 to 1 in steps of 0.01.
Below are examples of Lissajous figures with δ = π/2, an odd natural number a, an even
natural number b, and |a − b| = 1.
Generation Prior to modern electronic equipment, Lissajous
curves could be generated mechanically by means of a harmonograph.
Practical application Lissajous curves can also be generated using
an oscilloscope. 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 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 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 corresponding to a phase shift of and
an aspect ratio of 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. The arrows
show the direction of rotation of the Lissajous figure.
Logos and other uses
Lissajous figures were commonly displayed on oscilloscopes meant to simulate high-tech
equipment in science-fiction TV shows and movies in the 1960s and 1970s.
Lissajous figures are sometimes used in graphic design as logos. Examples include:
The title sequence by John Whitney for Alfred Hitchcock film Vertigo is based on Lissajous
figures. The Australian Broadcasting Corporation
The Lincoln Laboratory at MIT The University of Electro-Communications,
Japan.
In computing, Lissajous figures are in some screen savers.
A Lissajous curve is used in experimental tests to determine if a device may be properly
categorized as a memristor.
See also Rose curve
Lissajous orbit Blackburn pendulum
Lemniscate of Gerono Notes
External links Lissajous Curve at Mathworld
Interactive demos 3D Java applets depicting the construction
of Lissajous curves in an oscilloscope: Tutorial from the NHMFL
Physics applet by Chiu-king Ng
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
Interactive Lissajous curve generator – Javascript applet using JSXGraph
Animated Lissajous figures