Incubator escapee wiki

Lissajous curve

Article Discussion

Template:Short description Template:Distinguish Template:Multiple issues

File:Lissajous figure - sand on paper.jpg
A Lissajous figure, made by releasing sand from a container at the end of a Blackburn pendulum

A Lissajous curve Template:IPAc-en, also known as Lissajous figure or Bowditch curve Template:IPAc-en, is the graph of a system of parametric equations

<math>x=A\sin(at+\delta),\quad y=B\sin(bt),</math>

which describe the superposition of two perpendicular oscillations in x and y directions of different angular frequency (a and b). The resulting family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail in 1857 by Jules Antoine Lissajous (for whom it has been named).<ref>Template:Cite book</ref><ref>Template:Cite book</ref><ref>Template:Cite book</ref> Such motions may be considered as a particular kind of complex harmonic motion.

The appearance of the figure is sensitive to the ratio Template:Math. For a ratio of 1, when the frequencies match a=b, the figure is an ellipse, with special cases including circles (Template:Math, Template:Math radians) and lines (Template:Math). A small change to one of the frequencies will mean the x oscillation after one cycle will be slightly out of synchronization with the y motion and so the ellipse will fail to close and trace a curve slightly adjacent during the next orbit showing as a precession of the ellipse. The pattern closes if the frequencies are whole number ratios i.e. Template:Math is rational.

Another simple Lissajous figure is the parabola (Template:Math, Template:Math). Again a small shift of one frequency from the ratio 2 will result in the trace not closing but performing multiple loops successively shifted only closing if the ratio is rational as before. A complex dense pattern may form see below.

The visual form of such 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 Template:Math determines the number of "lobes" of the figure. For example, a ratio of Template:Sfrac or Template:Sfrac produces a figure with three major lobes (see image). Similarly, a ratio of Template:Sfrac produces a figure with five horizontal lobes and four vertical lobes. Rational ratios produce closed (connected) or "still" figures, while irrational ratios produce figures that appear to rotate. The ratio Template:Math determines the relative width-to-height ratio of the curve. For example, a ratio of Template:Sfrac produces a figure that is twice as wide as it is high. Finally, the value of Template:Math determines the apparent "rotation" angle of the figure, viewed as if it were actually a three-dimensional curve. For example, Template:Math produces Template:Math and Template:Math components that are exactly in phase, so the resulting figure appears as an apparent three-dimensional figure viewed from straight on (0°). In contrast, any non-zero Template:Math produces a figure that appears to be rotated, either as a left–right or an up–down rotation (depending on the ratio Template:Math).

File:Lissajous-Figur 1 zu 3 (Oszilloskop).jpg
Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively. This particular Lissajous figure was adapted into the logo for the Australian Broadcasting Corporation
File:Harmonie-circulaire.gif
A circle is a simple Lissajous curve

Lissajous figures where Template:Math, Template:Math (Template:Math is a natural number) and

<math>\delta=\frac{N-1}{N}\frac{\pi}{2} </math>

are Chebyshev polynomials of the first kind of degree Template:Math. 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 Template:Math.

The relation of some Lissajous curves to Chebyshev polynomials is clearer to understand if the Lissajous curve which generates each of them is expressed using cosine functions rather than sine functions.

<math>x=\cos(t),\quad y=\cos(Nt)</math>

ExamplesEdit

File:Lissajous animation.gif
Animation showing curve adaptation as the ratio Template:Math increases from 0 to 1

The animation shows the curve adaptation with continuously increasing Template:Math fraction from 0 to 1 in steps of 0.01 (Template:Math).

Below are examples of Lissajous figures with an odd natural number Template:Math, an even natural number Template:Math, and Template:Math.

GenerationEdit

Prior to modern electronic equipment, Lissajous curves could be generated mechanically by means of a harmonograph.

Practical applicationEdit

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 Template:Math is CH1 and Template:Math is CH2, Template:Math is the amplitude of CH1 and Template:Math is the amplitude of CH2, Template:Math is the frequency of CH1 and Template:Math is the frequency of CH2, so Template:Math is the ratio of frequencies of the two channels, and Template:Math is the phase shift of CH1.

A purely mechanical application of a Lissajous curve with Template:Math, Template:Math 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 Template:MathEdit

File:LissajousTechnion.png
In this figure both input frequencies are identical, but the phase difference between them creates the shape of an ellipse.
File:Circular Lissajous.gif
Top: Output signal as a function of time.
Middle: Input signal as a function of time.
Bottom: Resulting Lissajous curve when output is plotted as a function of the input.
In this particular example, because the output is 90 degrees out of phase from the input, the Lissajous curve is a circle, and is rotating counterclockwise.

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 Template:Math. 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 ±90° and an aspect ratio of ∞ (a line) corresponding to a phase shift of 0° or 180°.Template:Cn

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° is equivalent to +90°). The arrows show the direction of rotation of the Lissajous figure.Template:Cn

File:Lissajous phase.svg
A pure phase shift affects the eccentricity of the Lissajous oval. Analysis of the oval allows phase shift from an LTI system to be measured.

In engineeringEdit

A Lissajous curve is used in experimental tests to determine if a device may be properly categorized as a memristor.Template:Citation needed It is also used to compare two different electrical signals: a known reference signal and a signal to be tested.<ref>Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In popular cultureEdit

In motion picturesEdit

File:Simple Lissajous Animation.ogv
Lissajous animation
  • Lissajous figures were sometimes displayed on oscilloscopes meant to simulate high-tech equipment in science-fiction TV shows and movies in the 1960s and 1970s.<ref name="Information1987">Template:Cite journal</ref>

|CitationClass=web }}</ref>

Company logosEdit

Lissajous figures are sometimes used in graphic design as logos. Examples include:

|CitationClass=web }}</ref>

|CitationClass=web }}</ref>

In modern artEdit

Template:Further

|CitationClass=web }}</ref>

In music educationEdit

Lissajous curves have been used in the past to graphically represent musical intervals through the use of the Harmonograph,<ref>Template:Cite book</ref> a device that consists of pendulums oscillating at different frequency ratios. Because different tuning systems employ different frequency ratios to define intervals, these can be compared using Lissajous curves to observe their differences.<ref>Template:Cite journal</ref> Therefore, Lissajous curves have applications in music education by graphically representing differences between intervals and among tuning systems.

See alsoEdit

NotesEdit

Template:Sister project Template:Reflist

External linksEdit

Interactive demosEdit

Template:Authority control

Retrieved from "https://zalansite.site/mediawiki/index.php?title=Lissajous_curve&oldid=444101"