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Lissajous curves

Name: Anonymous 2007-09-05 20:40 ID:0neokMJ2

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?

Name: Anonymous 2007-09-05 21:42 ID:1gnttBTZ

/b/ is irrational

Name: Anonymous 2007-09-05 23:31 ID:kKEY828e

So that there is some multiple of a and some multiple of b that makes them at equal angles.

Like, if one was irrational, and one wasn't, then you couldn't find a t that would make them line up

That's my guess anyways.  lol idk I'm in biochemistry and shit

Name: Anonymous 2007-09-06 8:01 ID:iCni4AsK

>>3
Thanks but I am supposed to prove it somehow and what if a and b are rationals?

Name: Anonymous 2007-09-06 8:20 ID:6N2JXoT2

>>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.

Name: Anonymous 2007-09-06 17:35 ID:iCni4AsK

>>5
Thank you. Though I am still not sure that x(t) = x(t + m) and y(t) = y(t + m) is enough since the curve can end up at the same point but point in a different direction.

Name: Anonymous 2007-09-06 18:38 ID:6N2JXoT2

>>6


x(t) = x(t+m) for all t, that's how it's closed. This isn't a specific point.

At least I'm pretty certain it being closed <=> it is periodic

Name: Anonymous 2007-09-06 19:22 ID:IU7S1JFw

Who is Lissa Jous? And why does she have so many curves?

Name: Anonymous 2007-09-06 19:42 ID:iCni4AsK

>>7
Ohh, I get it now. I was imagining the t there to be a constant.

Name: Anonymous 2007-09-08 23:37 ID:Pefl57RC

more like lissaJEWS amirite LOL
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