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| {{Distinguish2|[[spirograph]]s, which are generally enclosed by a circular boundary, whereas Lissajous curves are enclosed by rectangular boundaries}}
| | Art Teacher (Private Tuition ) Nigel Gartman from Alma, has lots of hobbies and interests including ghost hunting, new launch property singapore and butterfly watching. Recalls what a fantastic spot it was having visited Fertö / Neusiedlersee Cultural Landscape.<br><br>Also visit my blog :: [http://drupal.12thirty4.com/gmaps/node/4865 http://drupal.12thirty4.com] |
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| In [[mathematics]], a '''Lissajous curve''' {{IPAc-en|ˈ|l|ɪ|s|ə|ʒ|uː}}, also known as '''Lissajous figure''' or '''Bowditch curve''' {{IPAc-en|ˈ|b|aʊ|d|ɪ|tʃ}}, is the graph of a system of [[parametric equation]]s
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| : <math>x=A\sin(at+\delta),\quad y=B\sin(bt),</math>
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| which describe [[complex harmonic motion]]. This family of [[curve]]s was investigated by [[Nathaniel Bowditch]] in 1815, and later in more detail by [[Jules Antoine Lissajous]] in 1857.
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| 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'' = [[pi|π]]/2 [[radian]]s) and [[line (mathematics)|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 number|rational]]. The visual form of these curves is often suggestive of a three-dimensional [[knot (mathematical)|knot]], and indeed many kinds of knots, including those known as [[Lissajous knot]]s, project to the plane as Lissajous figures.
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| 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 (see image). Similarly, a ratio of 5/4 produces a figure with 5 horizontal lobes and 4 vertical lobes. Rational ratios produce closed (connected) 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 (0°). In contrast, any non-zero ''δ'' produces a figure that appears to be rotated, either as a left/right or an up/down rotation (depending on the ratio ''a''/''b'').
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| [[Image:Lissajous-Figur 1 zu 3 (Oszilloskop).jpg|thumb|250px|right|Lissajous figure on an [[oscilloscope]], displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively.]]
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| Lissajous figures where ''a'' = 1, ''b'' = ''N'' (''N'' is a [[natural number]]) and
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| : <math>\delta=\frac{N-1}{N}\frac{\pi}{2}\ </math>
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| 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].
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| ==Examples==
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| The animation below shows the curve adaptation with continuously increasing <math>\frac{a}{b}</math> fraction from 0 to 1 in steps of 0.01. (δ=0)
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| [[File:Lissajous animation.gif|center]]
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| Below are examples of Lissajous figures with ''δ'' = ''π''/2, an odd [[natural number]] ''a'', an even [[natural number]] ''b'', and |''a'' − ''b''| = 1.
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| {{center|<gallery>
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| Image:Lissajous_curve_1by2.svg|{{center|''a'' = 2, ''b'' = 1 (2:1)}}
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| Image:Lissajous_curve_3by2.svg|{{center|''a'' = 3, ''b'' = 2 (3:2)}}
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| Image:Lissajous_curve_3by4.svg|{{center|''a'' = 3, ''b'' = 4 (3:4)}}
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| Image:Lissajous_curve_5by4.svg|{{center|''a'' = 5, ''b'' = 4 (5:4)}}
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| </gallery>}}
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| ==Generation==
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| Prior to modern electronic equipment, Lissajous curves could be generated mechanically by means of a [[harmonograph]].
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| === Practical application ===
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| Lissajous curves can also be generated using an [[oscilloscope]] (as illustrated). An [[Analog signature analysis|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.
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| 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.
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| 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.
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| 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 rail-roads in the mid-1900s. The beam in some versions traces out a lopsided figure-8 pattern with the "8" lying on its side.
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| ==Application for the case of ''a'' = ''b''==
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| [[Image:LissajousTechnion.png|thumb|250px|right|In this figure both input frequencies are identical, but the phase variance between them creates the shape of an [[ellipse]].]]
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| [[File:Circular Lissajous.gif|thumb|250px|right|Top: Input signal as a function of time, Middle: Output 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.]]
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| 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 <math>\pm90^\circ</math> and an aspect ratio of <math>\infty</math> (a line) corresponding to a phase shift of 0 or 180 degrees.
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| The figure below summarizes how the Lissajous figure changes over different phase shifts. The phase shifts are all negative so that [[propagation delay|delay]] [[semantics]] can be used with a [[causal system|causal]] LTI system (note that −270 degrees is equivalent to +90 degrees). The arrows show the direction of rotation of the Lissajous figure.
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| [[Image:Lissajous phase.png|thumb|center|600px|A pure phase shift affects the [[eccentricity (mathematics)|eccentricity]] of the Lissajous oval. Analysis of the oval allows phase shift from an [[LTI system]] to be measured..]]
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| ==Popular culture==
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| * Lissajous figures are sometimes used in [[graphic design]] as [[logotype|logo]]s. Examples include:
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| ** The Alfred Hitchcock film [[Vertigo (film)|Vertigo]]'s title sequence is based on Lissajous figures<ref>{{cite web|url=http://rhizome.org/editorial/2013/may/9/did-vertigo-introduce-computer-graphics-cinema|title=Did 'Vertigo' Introduce Computer Graphics to Cinema?}}</ref>
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| ** The [[Australian Broadcasting Corporation]] (''a'' = 1, ''b'' = 3, ''δ'' = π/2)<ref>{{cite web|url=http://www.abc.net.au/science/holo/liss.htm|title=The ABC's of Lissajous figures}}</ref>
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| ** The [[Lincoln Laboratory]] at [[MIT]] (''a'' = 4, ''b'' = 3, ''δ'' = 0)<ref>{{cite web|url=http://www.ll.mit.edu/about/History/logo.html|title=Lincoln Laboratory Logo|publisher=MIT Lincoln Laboratory|year=2008|accessdate=2008-04-12}}</ref>
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| ** The [[University of Electro-Communications]], Japan (''a'' = 3, ''b'' = 4, ''δ'' = π/2).
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| * In computing, Lissajous figures are in some [[screen saver]]s.
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| ==See also==
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| * [[Rose curve]]
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| * [[Lissajous orbit]]
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| * [[Blackburn pendulum]]
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| * [[Lemniscate of Gerono]]
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| ==Notes==
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| {{Refimprove|date=November 2010}}{{reflist}}
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| ==External links==
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| {{More footnotes|date=November 2010}}{{Commons category|Lissajous curves}}
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| * [http://mathworld.wolfram.com/LissajousCurve.html Lissajous Curve at Mathworld]
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| === Interactive demos ===
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| * 3D Java applets depicting the construction of Lissajous curves in an oscilloscope:
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| ** [http://www.magnet.fsu.edu/education/tutorials/java/lissajous/index.html Tutorial] from the [[NHMFL]]
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| ** [http://phy.hk/wiki/englishhtm/Lissajous.htm Physics applet] by Chiu-king Ng
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| * [http://gerdbreitenbach.de/lissajous/lissajous.html Interactive Lissajous Curves in Java] – graphical representations of musical intervals, beats, interference, and vibrating strings
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| * [http://devadutta.net/lissajous/ Simple HTML5 Lissajous curve generator] – allows controls for A and B as integers from 1 to 12 each
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Lissajous_curves Interactive Lissajous curve generator] – Javascript applet using JSXGraph
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| * [http://ibiblio.org/e-notes/html5/lis/lissa5.htm Animated Lissajous figures]
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| [[Category:Curves]]
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| [[Category:Trigonometry]]
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Art Teacher (Private Tuition ) Nigel Gartman from Alma, has lots of hobbies and interests including ghost hunting, new launch property singapore and butterfly watching. Recalls what a fantastic spot it was having visited Fertö / Neusiedlersee Cultural Landscape.
Also visit my blog :: http://drupal.12thirty4.com