Aperture of plane curves

Abstract

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

11 pages