Let $\vec{x}(a,b)$ be a parameterization of an arbitrary unit vector in 3 dimensions:
- $a,b\in\mathbb{R}$
- $||\vec{x}(a,b)||=1$ for all $a,b$.
- For a given unit vector, there exists some finite $a,b$ such that $\vec{x}(a,b)$ is that unit vector.
Is there some $\vec{x}(a,b)$ such that both $\frac{d}{da}\vec{x}(a,b)$ and $\frac{d}{db}\vec{x}(a,b)$ are finite and nonzero at all values of $a,b$? If so, what is one such parameterization?
Note that spherical coordinates $\vec{x}(\theta,\phi)=(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)$ with $r=1$ fails these requirements as $\frac{d}{d\phi}\vec{x}(\theta,\phi)=\vec{0}$ at $\theta=\frac{\pi}{2}$.
It feels like the answer is no, and that I will need to settle on a locally smooth parameterization.