Let $\alpha:\mathbb{S}^1\to \mathbb{R}^2$ be a $C^2$ curve satisfying $|\dot \alpha|=1$, and suppose that there exists a constant $c \in \mathbb{S}^1$ such that $ \alpha(\theta+c)=R(\theta)\alpha(\theta) $ holds for every $\theta$, where $R:\mathbb{S}^1 \to \text{SO}(2)$ is $C^1$.
Must $R$ be a constant map?
Differentiating, we get
$ \alpha'(\theta+c)=R'(\theta)\alpha(\theta)+R(\theta)\alpha'(\theta), $ thus $$ 1=| \alpha'(\theta+c)|^2=|R'\alpha|^2+1+2\langle R'\alpha,R\alpha'\rangle. $$ So $$ |R'\alpha|^2+2\langle R'\alpha,R\alpha'\rangle=0. $$
I am not sure how to continue.