Lissajous curves are the family of curves described by the parametric equations
(1)
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(2)
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sometimes also written in the form
(3)
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(4)
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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.
Lissajous curves are a special case of the harmonograph with damping constants .
Special cases are summarized in the following table, and include the line, circle, ellipse, and section of a parabola.
It follows that , gives a parabola from the fact that this gives the parametric equations , which is simply a horizontally offset form of the parametric equation of the parabola .