Aperture of plane curves

For any given $C^{\mathrm{\infty }}$ immersion $\mathbf{r}:S^1\to {\mathbb{R}}^2$ such that the set $$\mathcal{N}\mathcal{S}${{\bf r}}=$\mathbb{R}$^2-\cup{s\in S^1}\left({\bf r}(s)+d{\bf r}_{s}(T_s(S^1))\right)$ is not empty, a simple geometric model of crystal growth is constructed. It is shown that our geometric model of crystal growth never formulates a polygon while it is growing. Moreover, it is shown also that our model always dissolves to a point.