Lissajous Curve

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Lissajous curve

Lissajous figure on an oscilloscope,

displaying a 1:3 relationship between the

frequencies of the vertical and horizontal

sinusoidal inputs, respectively.

From Wikipedia, the free encyclopedia

In mathematics, a Lissajous curve /lsu/, also known as Lissajous figureor Bowditch curve/badt/, is the graph of a system of parametric equations

which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditchin 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 (A= B, = /2 radians) and lines (= 0). Another simple Lissajousfigure is the parabola (a/b= 2, = /2). Other ratios produce more complicated curves, which areclosed only if a/bis rational. The visual form of these curves is often suggestive of a three-dimensionalknot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane asLissajous figures.

Lissajous figures where a= 1, b= N(Nis a natural number)and

are Chebyshev polynomials of the first kind of degree N. Thisproperty is exploited to produce a set of points, called Paduapoints, at which a function may be sampled in order tocompute either a bivariate interpolation or quadrature of thefunction over the domain [-1,1][-1,1].

Contents

1 Examples2 Generation

2.1 Practical application

3 Application for the case of a= b

4 Popular culture

5 See also

6 Notes

7 External links

7.1 Interactive demos

Examples

The animation below shows the curve adaptation with continuously increasing fraction from 0 to 1 in

steps of 0.01. (=0)

Lissajous curve - Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Lissajous

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Below are examples of Lissajous figures with = /2, an odd natural number a, an even natural numberb, and |a b| = 1.

a= 1, b= 2 (1:2) a= 3, b= 2 (3:2) a= 3, b= 4 (3:4) a= 5, b= 4 (5:4)

Generation

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

Practical application

Lissajous curves can also be generated using an oscilloscope(as illustrated). An octopus circuit can beused to demonstrate the waveform images on an oscilloscope. Two phase-shifted sinusoid inputs areapplied 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 xis CH1 and yis CH2, Ais amplitude of CH1 and Bis amplitude of CH2,ais frequency of CH1 and bis frequency of CH2, so a/bis a ratio of frequency of two channels, finally, 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 theMars Light type of oscillating beam lamps popular with railroads in the mid-1900s. The beam in someversions traces out a lopsided figure-8 pattern with the "8" lying on its side.


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In this figure both input frequencies are

identical, but the phase variance between

them creates the shape of an ellipse.

Top: Input signal as a function of time,

Middle: Output signal as a function of time.

Bottom: resulting Lissajous curve whenoutput 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.

Application for the case of a= b

When the input to an LTI system is sinusoidal, the output issinusoidal with the same frequency, but it may have adifferent amplitude and some phase shift. Using anoscilloscope that can plot one signal against another (asopposed to one signal against time) to plot the output of anLTI 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 thephase shift between the input and output, with an aspectratio of 1 (perfect circle) corresponding to a phase shift of

and an aspect ratio of (a line) corresponding to aphase shift of 0 or 180 degrees. The figure belowsummarizes how the Lissajous figure changes over differentphase shifts. The phase shifts are all negative so that delaysemantics can be used with a causal LTI system (note that270 degrees is equivalent to +90 degrees). The arrowsshow the direction of rotation of the Lissajous figure.


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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.

Popular culture

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

The Alfred Hitchcock film Vertigo's title sequence is based on Lissajous figures[1]

The Australian Broadcasting Corporation (a= 1, b= 3, = /2)[2]

The Lincoln Laboratory at MIT (a= 4, b= 3, = 0)[3]

The University of Electro-Communications, Japan (a= 3, b= 4, = /2).

In computing, Lissajous figures are in some screen savers.

See also

Rose curve

Lissajous orbit

Blackburn pendulum

Lemniscate of Gerono


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This page was last modified on 18 October 2013 at 20:17.

Text is available under the Creative Commons Attribution-ShareAlike License;

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organization.

Atoms

Notes

^ "Did 'Vertigo' Introduce Computer Graphics to Cinema?" (http://rhizome.org/editorial/2013/may/9/did-

vertigo-introduce-computer-graphics-cinema).

1.

^ "The ABC's of Lissajous figures" (http://www.abc.net.au/science/holo/liss.htm).2.

^ "Lincoln Laboratory Logo" (http://www.ll.mit.edu/about/History/logo.html). MIT Lincoln Laboratory. 2008.

Retrieved 2008-04-12.

3.

External links

Lissajous Curve at Mathworld (http://mathworld.wolfram.com/LissajousCurve.html)

Interactive demos

3D Java applets depicting the construction of Lissajous curves in an oscilloscope:

Tutorial (http://www.magnet.fsu.edu/education/tutorials/java/lissajous/index.html) from the

NHMFLPhysics applet (http://phy.hk/wiki/englishhtm/Lissajous.htm) by Chiu-king Ng

Interactive Lissajous Curves in Java (http://gerdbreitenbach.de/lissajous/lissajous.html) graphical

representations of musical intervals, beats, interference, and vibrating strings

Simple HTML5 Lissajous curve generator (http://devadutta.net/lissajous/) allows controls for A

and B as integers from 1 to 12 each

Interactive Lissajous curve generator (http://jsxgraph.uni-bayreuth.de/wiki/index.php

/Lissajous_curves) Javascript applet using JSXGraph

Animated Lissajous figures (http://ibiblio.org/e-notes/html5/lis/lissa5.htm)

Retrieved from "http://en.wikipedia.org/w/index.php?title=Lissajous_curve&oldid=577758742"

Categories: Curves Trigonometry


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Lissajous Curve

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