李撒如图究竟是个什么东西——Lissajous curve

Lissajous Curve

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.

parameters	curve
,	line
,  ,	circle
,  ,	ellipse
,	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 

.