This question arises as a possible step in answering this unsolved question on MSE. Given a unit vector $v \in S^{d-1} \subset \mathbb{R}^d$, I'm looking for an explicit formula for the set $$V = \{ u \in S^{d-1} \mid u \cdot v > 0 \} $$ in spherical coordinates. For $d=2$ we can easily take $v$'s polar coordinate $\theta \in (-\pi, \pi]$ to obtain $$V = (\theta-\pi/2, \theta+\pi/2) $$ modulo $2\pi$. Unfortunately this approach runs into a problem for $d \geq 3$. In spherical coordinates $v = (\theta, \phi)$ with $\theta \in (-\pi, \pi]$ and $\phi \in (-\pi/2, \pi/2]$, consider the following examples:
\begin{align} v_1 &= (0, 0) &\implies&&& V_1 = (-\pi/2, \pi/2) \times (-\pi/2, \pi/2) \\ v_2 &= (\pi/2, 0) &\implies&&& V_2 = (0, \pi) \times (-\pi/2, \pi/2) \\ v_3 &= (0, \pi/2) &\implies&&& V_2 = (-\pi, \pi) \times (0, \pi/2) \\ \end{align}
The first two examples suggest we could do as in the two-dimensional case (subtracting and adding $\pi/2$ to each coordinate), but the last one fails in this respect. Subtracting and adding $\pi/2$ to each coordinate does give the correct set but with respect to a different (equivalent) spherical coordinate system, namely that which takes $\theta \in (-\pi/2, \pi/2]$ and $\phi \in (-\pi, \pi]$. This is inconsistent here, and I haven't found a way to remediate this problem. Any help most welcome!
This doesn't seem likely to work. If you think about what you're after in 3D, you want some sort of "nice" description of an arbitrary hemisphere in terms of constraints on latitude and longitude. For very special hemispheres, that's fine, but it's going to involve some messiness no matter what. This is more or less equivalent to describing the corresponding equator ("great circle") of that hemisphere in spherical; see here. You'd want a hyper-spherical generalization. Before proceeding, you should see if the $d=3$ case is of any use to you.
The trouble with spherical coordinates is that, outside of the $d=2$ case, they're not at all symmetric. Sometimes they can be useful if your problem has some recursive structure (e.g. computing the volume of hyperspheres), but otherwise it's a likely to be closer to putting a round peg in a square hole.