Aperture of plane curves

For any given $C^\infty$ immersion ${\bf r}: S^1\to \mathbb{R}^2$ such that the set $\mathcal{NS}_{{\bf r}}=\mathbb{R}^2-\cup_{s\in S^1}\left({\bf r}(s)+d{\bf r}_{s}(T_s(S^1))\right)$ 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.