Parametrization of a unit 2 sphere

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Here is the parametrization for a unit 2 sphere locating at the center of a Euclidean 3 dimensional space:

$$x=x(u,v)= \cos u\sin v,\ \ y=y(u,v)=\sin u\sin v,\ \ z=z(u,v)=\cos v, $$

where $0\leq u<2\pi,0\leq v<\pi$. I'm not sure if the inverse $u=u(x,y,z),v=v(x,y,z)$ exists. If it does, then how can one find out its exact form? In addition, it's obvious that expression such as: $e_{x}=\frac{\partial u(x,y,z)}{\partial x}e_u+\frac {\partial v(x,y,z)}{\partial x}e_v$ is meaningless at at least some point on the unit sphere if we can draw a picture. But how can one show this fact in a formal way?

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2 Answers

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$$\frac{y}{x}=\tan u\qquad u=\arctan\frac{y}{x},\qquad v=\arccos z$$ Warning: $\arctan\frac{y}{x}$ is actually atan2.

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$$x=x(u,v)= \cos u\sin v; \ y=y(u,v)=\sin u\sin v; \ z=z(v)=\cos v. $$

Mapping is bijective.

For given latitude and longitude (u,v) respectively a single point on sphere (x,y,z) is defined.

For given point on sphere (x,y,z) a single latitude and a single longitude (u,v) are defined thus:

$$ u=\tan^{-1}\frac {y}{ x}; v=\cos^{-1} z. $$

With $ \tan^{-1}$ function proper quadrant has to be chosen.

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