Let $\gamma : [0, a] \rightarrow \mathbb{R}^2 \setminus \{(0,0)\}$ be a smooth, closed curve with unit-speed ( i.e $||\dot{\gamma }(t)|| = 1$) such that $\gamma ^{(n)}(0) = \gamma ^{(n)}(a)$ for all $n \in \mathbb{N}$. Furthermore assume that $\gamma '(t)$ and $\gamma (t)$ are linearly independent for all $t \in [0,a]$.
Let $r: \mathbb{R}^2 \setminus \{(0,0)\} \rightarrow S^1$ with $r(x) = x / ||x||$. Show that $r \circ \gamma$ is a regular curve (i.e. $\frac{d}{dt} r \circ \gamma (t) \neq 0$ for all $t \in [0,a]$).
I see how the assumptions are necessary and how that statement makes sense. But I haven't been able to proof that $r \circ \gamma $ is regular. Any help is welcome!