Aperture of plane curves

Daisuke Kagatsume, Takashi Nishimura

For any given $C^{\infty }$ immersion $\mathbf{r}:S^1\to {\mathbb{R}}^2$ 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)…