Difference between revisions of "Lissajous curve"
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Not to be confused with spirographs, which are generally enclosed by a circular boundary, whereas Lissajous curves are enclosed by rectangular boundaries. | Not to be confused with spirographs, which are generally enclosed by a circular boundary, whereas Lissajous curves are enclosed by rectangular boundaries. | ||
| − | In mathematics, | + | In mathematics,'''Lissajous curve''' /ˈlɪsəʒuː/, also known as '''Lissajous figure''' or '''Bowditch curve''' /ˈbaʊdɪtʃ/, is the graph of a system of [http://wikipedia.org/wiki/Parametric_equation parametric equations] |
[[File:D7e794d715a8a762f0eab306136638b4.png]] | [[File:D7e794d715a8a762f0eab306136638b4.png]] | ||
Which describe complex harmonic motion. This family of curves was investigated by [http://wikipedia.org/wiki/Nathaniel_Bowditch Nathaniel Bowditch] in 1815, and later in more detail by [http://wikipedia.org/wiki/Jules_Antoine_Lissajous Jules Antoine Lissajous]in 1857. | Which describe complex harmonic motion. This family of curves was investigated by [http://wikipedia.org/wiki/Nathaniel_Bowditch Nathaniel Bowditch] in 1815, and later in more detail by [http://wikipedia.org/wiki/Jules_Antoine_Lissajous 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 [http://wikipedia.org/wiki/Ellipse ellipse], with special cases including [http://wikipedia.org/wiki/Circles circles] (A = B, δ = π/2 radians) and 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. 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. | + | The appearance of the figure is highly sensitive to the ratio a/b. For a ratio of 1, the figure is an [http://wikipedia.org/wiki/Ellipse ellipse], with special cases including [http://wikipedia.org/wiki/Circles circles] (A = B, δ = π/2 radians) and [http://wikipedia.org/wiki/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. 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. |
Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively. | Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively. | ||
Lissajous figures where a = 1, b = N (N is a natural number) and | Lissajous figures where a = 1, b = N (N is a natural number) and | ||
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[[File:lissa2.png]] | [[File:lissa2.png]] | ||
Latest revision as of 15:54, 10 October 2013
Not to be confused with spirographs, which are generally enclosed by a circular boundary, whereas Lissajous curves are enclosed by rectangular boundaries.
In mathematics,Lissajous curve /ˈlɪsəʒuː/, also known as Lissajous figure or Bowditch curve /ˈbaʊdɪtʃ/, is the graph of a system of parametric equations
Which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajousin 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 (a/b = 2, δ = π/2). 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.
Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively.
Lissajous figures where a = 1, b = N (N is a natural number) and
