Lissajous Curve

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

$\displaystyle x(t)$	$\textstyle =$	$\displaystyle A\cos(\omega_x t-\delta_x)$	(1)
$\displaystyle y(t)$	$\textstyle =$	$\displaystyle B\cos(\omega_y t-\delta_y),$	(2)

sometimes also written in the form

$\displaystyle x(t)$	$\textstyle =$	$\displaystyle a\sin(nt+c)$	(3)
$\displaystyle y(t)$	$\textstyle =$	$\displaystyle b\sin t.$	(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.

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 53-54, 1993.

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.