# Explicit example of a smooth - but not analytic- closed curve without self-intersections

There exist smooth - but not analytic - closed curves without self-intersections. I just would like to see a simple example of such a curve.

-
Press the "faq" and "how to ask" buttons on top please. –  BS. Apr 12 '11 at 13:11

Let $\psi:\mathbb{R}\to\mathbb{R}$ be a $C^\infty$ function with compact support in $[0,\pi]$ and such that $\psi(t)>0$ if $0 < t < \pi$. Then $$\gamma(t)=\psi(t)(\cos t,\sin t),\quad 0 < t <\pi,\quad \gamma(0)=\gamma(\pi)=(0,0)$$ is such a curve. You can take for instance $$\psi(t)=\exp\Bigl(-\frac{1}{t^2(\pi^2-t^2)}\Bigr).$$

-
$$\gamma:\quad \phi\ \mapsto\ \Bigl(1+\exp{-1\over \pi^2 -\phi^2}\Bigr)\ (\cos\phi,\sin\phi)\qquad(-\pi< \phi< \pi)$$
with filled-in point $(-1,0)$.
Isn't it the case that as $\phi$ approaches $\pi$ and $-\pi$, the expression approaches $(0,0)$, and is not smooth there...? –  Joseph O'Rourke Apr 12 '11 at 12:33