To no longer be perplexed with spirographs, that are veritably enclosed by a round boundary, whereas Lissajous curves are enclosed by rectangular boundaries.
A Lissajous figure, made by releasing sand from a container at the discontinue of a Blackburn pendulum
A Lissajous curve , also acknowledged as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations
which describe complicated harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in extra detail in 1857 by Jules Antoine Lissajous (for whom it has been named).
The looks of the figure is extremely quiet 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 traces (δ = 0). One other easy Lissajous figure is the parabola (b/a = 2, δ = π/4). Varied ratios have extra subtle curves, that are closed fully if a/b is rational. The visual plot of these curves is on the whole suggestive of a 3-dimensional knot, and indeed many kinds of knots, including those acknowledged as Lissajous knots, challenge to the plane as Lissajous figures.
Visually, the ratio a/b determines the quite numerous of “lobes” of the figure. As an illustration, a ratio of 3/1 or 1/3 produces a figure with three main lobes (take into fable picture). In an identical vogue, a ratio of 5/4 produces a figure with 5 horizontal lobes and 4 vertical lobes. Rational ratios have closed (linked) or “serene” figures, whereas irrational ratios have figures that seem to rotate. The ratio A/B determines the relative width-to-height ratio of the curve. As an illustration, a ratio of 2/1 produces a figure that is twice as wide because it is some distance high. Finally, the price of δ determines the unpleasant “rotation” perspective of the figure, seen as if it were in actuality a 3-dimensional curve. As an illustration, δ = 0 produces x and y substances that are precisely in share, so the ensuing figure appears to be like as an obvious three-dimensional figure seen from straight on (0°). In contrast, any non-zero δ produces a figure that appears to be like to be rotated, both as a left–lawful or an up–down rotation (looking on the ratio a/b).
Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively. This enlighten Lissajous figure was adapted into the emblem for the Australian Broadcasting Company
A circle is a easy Lissajous curve
Lissajous figures the build a = 1, b = N (N is a pure number) and
are Chebyshev polynomials of the main roughly level N. This property is exploited to have a space of facets, called Padua facets, at which a goal may per chance perhaps well also merely be sampled in thunder to compute both a bivariate interpolation or quadrature of the goal over the area [−1,1] × [−1,1].
The relation of some Lissajous curves to Chebyshev polynomials is clearer to recognize if the Lissajous curve which generates every of them is expressed the consume of cosine capabilities in home of sine capabilities.
The animation shows the curve adaptation with repeatedly increasing a/b fragment from 0 to 1 in steps of 0.01 (δ = 0).
Below are examples of Lissajous figures with an queer pure number a, a sharp pure number b, and |a − b| = 1.
Sooner than novel electronic tools, Lissajous curves is prone to be generated mechanically by strategy of a harmonograph.
Lissajous curves may per chance also be generated the consume of an oscilloscope (as illustrated). An octopus circuit may per chance also be mature to cover the waveform photos on an oscilloscope. Two share-shifted sinusoid inputs are utilized to the oscilloscope in X-Y mode and the percentage relationship between the signals is offered as a Lissajous figure.
Within the legit audio world, this arrangement is mature for realtime diagnosis of the percentage relationship between the left and lawful channels of a stereo audio signal. On higher, extra subtle audio mixing consoles an oscilloscope may per chance perhaps well also merely be built-in for this motive.
On an oscilloscope, we insist x is CH1 and y is CH2, A is the amplitude of CH1 and B is the amplitude of CH2, a is the frequency of CH1 and b is the frequency of CH2, so a/b is the ratio of frequencies of the two channels, and δ is the percentage shift of CH1.
A purely mechanical utility of a Lissajous curve with a = 1, b = 2 is within the driving mechanism of the Mars Light form of oscillating beam lamps novel with railroads within the mid-1900s. The beam in some versions traces out a lopsided figure-8 pattern on its facet.
Utility for the case of a = b
In this figure both input frequencies are the same, however the percentage distinction between them creates the form of an ellipse.
Prime: Output signal as a goal of time.
Center: Input signal as a goal of time.
Backside: Ensuing Lissajous curve when output is plotted as a goal of the input.
In this enlighten instance, because the output is 90 levels out of share 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 no doubt may per chance perhaps well also merely have confidence a obvious amplitude and some share shift. Using an oscilloscope that can build one signal in opposition to but every other (versus one signal in opposition to time) to build the output of an LTI system in opposition to the input to the LTI system produces an ellipse that is a Lissajous figure for the special case of a = b. The facet ratio of the ensuing ellipse is a goal of the percentage shift between the input and output, with an facet ratio of 1 (ideal circle) same to a share shift of ±90° and an facet ratio of ∞ (a line) same to a share shift of 0° or 180°.
The figure below summarizes how the Lissajous figure modifications over various share shifts. The proportion shifts are all detrimental so that delay semantics may per chance also be mature with a causal LTI system (assert that −270° is same to +90°). The arrows assert the path of rotation of the Lissajous figure.
A pure share shift impacts the eccentricity of the Lissajous oval. Prognosis of the oval enables share shift from an LTI system to be measured.
A Lissajous curve is mature in experimental checks to safe out if a system may per chance perhaps well also merely be wisely categorised as a memristor. It’s some distance continuously mature to take a look at two various electrical signals: a acknowledged reference signal and a signal to be examined.
In motion photos
Science fiction vogue Lissajous animation
- Lissajous figures were veritably displayed on oscilloscopes supposed to simulate high-tech tools in science-fiction TV shows and movies within the 1960s and 1970s.
- The title sequence by John Whitney for Alfred Hitchcock‘s 1958 feature film Vertigo is primarily based on Lissajous figures.
Lissajous figures are veritably mature in graphic originate as logos. Examples contain:
- The Australian Broadcasting Company (a = 1, b = 3, δ = π/2)
- The Lincoln Laboratory at MIT (a = 3, b = 4, δ = π/2)
- The College of Electro-Communications, Japan (a = 5, b = 6, δ = π/2).
- Disney’s Movies Any place streaming video utility uses a stylized model of the curve
- Facebook‘s rebrand into Meta Platforms shall be a Lissajous Curve (a = 1, b = -2, δ = π/20).
In novel artwork
- The Dadaist artist Max Ernst painted Lissajous figures directly by swinging a punctured bucket of paint over a canvas.
- 3D Java applets depicting the building of Lissajous curves in an oscilloscope:
- Tutorial from the NHMFL
- Physics applet by Chiu-king Ng
- Lissajous Curves: Interactive simulation of graphical representations of musical intervals and vibrating strings
- Appealing Lissajous figures