Parametrizing a Closed Curve in \mathbb{R}^3

[Dox]

Hello.

In a certain problem I'm interested on, I need to write a general form of the parametrization of a closed curve on $$\mathbb{R}^3$$.

I thought in parametrize it using a kind of Fourier series. Could it be possible?

Thing become even worse 'cause I'd like to the curve doesn't cross itself.

Every idea is welcome.

Best wishes.

 

[quasar987]

Would it not be sufficient to say that such a curve is a function $r(t)=(x(t),y(t),z(t))$, $t\in [0,1]$ such that r(t1)=r(t2) iff r1=0, r2=1.

 

[Dox]

Would it not be sufficient to say that such a curve is a function $r(t)=(x(t),y(t),z(t))$, $t\in [0,1]$ such that r(t1)=r(t2) iff r1=0, r2=1.

Thanks for your answer Quasar987, but that exactly the point. In order to satisfy the boundary condition I must expand in Fourier series, Isn't it?

Thank you very much.

 

[quasar987]

I don't know what you mean.

Why do you consider the answer I wrote incomplete/inadequate?

 

[Dox]

I mean:

The condition is $$r(0)=r(1)$$.

In general a function satisfying that condition could be expand as a Fourier series, Isn't it? Because sine and cosine are periodic functions.

 

[quasar987]

If the function is not too wild, yes. But again, what does this have to do with the problem?