They are sometimes known as Bowditch curves after Nathaniel Bowditch, who studied them in 1815. They were studied in more detail (independently) by JulesAntoine Lissajous in 1857 (MacTutor Archive). Lissajous curves have applications in physics, astronomy, engineering, and other sciences. The curves close if is rational.
Lissajous curves are a special case of the harmonograph with damping constants .
Examples of Lissajous Curves, courtesy of Weisstein, Eric W. "Lissajous Curve." From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/LissajousCurve.html
Special cases are summarized in the following table, and include the line, circle, ellipse, and section of a parabola.
parameters

curve

,

line

, ,

circle

, ,

ellipse

,

section of a parabola

The line, circle, ellipse, and section of a parabola; all special cases of the Lissajou Curve courtesy of Weisstein, Eric W. "Lissajous Curve." From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/LissajousCurve.html
REFERENCES:

Cundy, H. and Rollett, A. "Lissajous's Figures." §5.5.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 242244, 1989.

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 7071, 1997.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178179 and 181183, 1972.

MacTutor History of Mathematics Archive. "Lissajou Curves." http://wwwgroups.dcs.stand.ac.uk/~history/Curves/Lissajous.html.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 142, 1991.
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