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lissajousfigure

lissajous figure

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There are 15 letters in LISSAJOUSFIGURE ( A¹E¹F⁴G²I¹J⁸L¹O¹R¹S¹U¹ )

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LISSAJOUS FIGURE - In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations ...

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Lissajous figure might refer to
In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations* * * * x * = * A * sin * ⁡ * ( * a * t * + * δ * ) * , * * y * = * B * sin * ⁡ * ( * b * t * ) * , * * * {\displaystyle x=A\sin(at+\delta ),\quad y=B\sin(bt),} * 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 (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (b/a = 2, δ = π/4). 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 (see image). Similarly, a ratio of 5/4 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 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). * * Lissajous figures where a = 1, b = N (N is a natural number) and * * * * * δ * = * * * * N * − * 1 * * N * * * * * π * 2 * * * * * {\displaystyle \delta ={\frac {N-1}{N}}{\frac {\pi }{2}}} * 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]. * The relation of some Lissajous curves to Chebyshev polynomials is clearer to understand if the Lissajous curve which generates...

Lissajous figure might refer to

In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations*
*
*
* x
* =
* A
* sin
* ⁡
* (
* a
* t
* +
* δ
* )
* ,
*
* y
* =
* B
* sin
* ⁡
* (
* b
* t
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* ,
*
*
* {\displaystyle x=A\sin(at+\delta ),\quad y=B\sin(bt),}
* 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 (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (b/a = 2, δ = π/4). 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 (see image). Similarly, a ratio of 5/4 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 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).
*
* Lissajous figures where a = 1, b = N (N is a natural number) and
*
*
*
*
* δ
* =
*
*
*
* N
* −
* 1
*
* N
*
*
*
*
* π
* 2
*
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*
* {\displaystyle \delta ={\frac {N-1}{N}}{\frac {\pi }{2}}}
* 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].
* The relation of some Lissajous curves to Chebyshev polynomials is clearer to understand if the Lissajous curve which generates...

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