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.
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 
.
