Parametrizing a Sphere

Find a parametrization for a sphere \(S\) of radius \(2\) centered at the origin.

Solution The equation of the sphere, \(x^{2}+y^{2}+z^{2}=4,\) in rectangular coordinates cannot be expressed explicitly as a function of two of its variables. However, in spherical coordinates, the sphere has the explicit equation \(\rho =2.\) So if we use the spherical coordinates \(\theta \) and \(\phi \) as parameters, then parametric equations of the sphere are given by

and a parametrization of \(S\) is \[ \begin{equation*} \mathbf{r}\left( \theta ,\phi \right) =2\cos \theta \sin \phi \,\mathbf{i} +2\sin \theta \sin \phi \,\mathbf{j}+2\cos \phi \,\mathbf{k} \end{equation*} \]

where \(0\leq \theta \leq 2\pi \) and \(0\leq \phi \leq \pi . \)