Lissajous curves

That is, this form of parametric equation.

x = A sin(a t + d)
y = B sin(b t)

Why does a/b have to be rational for the curve to be closed?

>>1

I don't know.
Closed curve implies x(t) = x(t + m) and y(t) = y(t + m) for some m.

so Asin(at + d) = Asin(at+d+am) => am = n*Pi some n in the integers


Similar bm = z*Pi some z in the integers.

Therefore

am/bm = a/b = n/z where n and z are integers.



That wasn't that hard.