For $\alpha \in (0,2\pi)$, consider the region $W_{\alpha}={(rcos(\theta),rsin(\theta))}$ $\in$ $\mathbb{R}^{2}$: $ r \in [0,2\pi, \theta \in [0,\alpha)$ in $\mathbb{R}^{2}$. For which $\alpha$ $\in (0.2\pi)$ is $W_{\alpha}$ diffeomorphic to $W_{\frac{\pi}{2}}?$
So, I understand that any $W_{\alpha}$ for which $\alpha > \pi$ cannot be diffeomorphic to $W_{\frac{\pi}{2}}$ as the two unit orthogonal vectors in the first quadrant cannot possibly be perpendicular under any diffeomorphic map. But, any $W_{\alpha}$ for which $\alpha < \pi$ is diffeomorphic to $W_{\frac{\pi}{2}}$, although I'm not sure of the reasoning. But what I'm struggling with is how to write all of this into a formal proof. I'm not sure how to rigorously argue about this. Any tips?