Circle-like parametrization of closed curves.

Question

Can every closed, smooth, non-intersecting curve in the plane be parametrized as

\begin{align} x(t)=a\cos\alpha(t),\:\:\:y(t)=b\sin\alpha(t), \end{align}

being $a$ and $b$ constants and $\alpha(t)$ a continuous function such that $\cos\alpha(0)=\cos\alpha(L)$ and $\sin\alpha(0)=\sin\alpha(L)$, with $L$ some parameter ?

Answer 1

Depending on the range of $\alpha$, this can only parametrise an ellipse, or part of an ellipse. To realize this, you just need to note that, with such a representation, you have that $$ \frac{x(t)^2}{a^2} + \frac{y(t)^2}{b^2} = 1 $$ Were you maybe thinking of $$x(t) = a(t) \cos t, \quad y(t) = b(t) \sin t \quad ?$$