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

(1) | |||

(2) |

sometimes also written in the form

(3) | |||

(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 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.

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1999-05-25