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Lissajous Curve


LissajousCurves

Lissajous curves are the family of curves described by the parametric equations

x(t)=Acos(omega_xt-delta_x)
(1)
y(t)=Bcos(omega_yt-delta_y),
(2)

sometimes also written in the form

x(t)=asin(omegat+delta)
(3)
y(t)=bsint.
(4)

They are sometimes known as Bowditch curves after Nathaniel Bowditch, who studied them in 1815. They were studied in more detail (independently) by Jules-Antoine Lissajous in 1857 (MacTutor Archive). Lissajous curves have applications in physics, astronomy, and other sciences. The curves close iff omega_x/omega_y is rational.

Lissajous curves are a special case of the harmonograph with damping constants beta_1=beta_2=0.

LissajousSpecial

Special cases are summarized in the following table, and include the line, circle, ellipse, and section of a parabola.

parameterscurve
omega=1, delta=0line
a=b, omega=1, delta=pi/2circle
a!=b, omega=1, delta=pi/2ellipse
omega=2, delta=pi/2section of a parabola

It follows that omega=2, delta=pi/2 gives a parabola from the fact that this gives the parametric equations (acos(2t),sint)=(a(1-2sin^2t),sint)=(a-2asin^2t,sint), which is simply a horizontally offset form of the parametric equation of the parabola (u^2/(4a),u).


See also

Harmonograph, Simple Harmonic Motion

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References

Cundy, H. and Rollett, A. "Lissajous's Figures." §5.5.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 242-244, 1989.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 70-71, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178-179 and 181-183, 1972.MacTutor History of Mathematics Archive. "Lissajous Curves." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lissajous.html.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 142, 1991.

Referenced on Wolfram|Alpha

Lissajous Curve

Cite this as:

Weisstein, Eric W. "Lissajous Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LissajousCurve.html

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