If a space has a differential volume element defined by:
$d\Omega=\sin^2(\alpha)\sin(\theta)d\alpha d\theta d\phi$
And $\alpha \in [0,\pi/2]$, $\theta \in [0,\pi]$, and $\phi \in [0,2\pi]$
How can I make a regular grid of points over the space that is uniform with respect to the the differential volume element?
I imagine a solution where I reparameterize the space in terms of, e.g., $u,v,w$, which are sampled uniformly. Then to get $\alpha,\theta,\phi$ I apply some transformation:
$\alpha = f(u)$
$\theta = g(v)$
$\phi = h(w)$
or something like that. How would I do this (i.e. find f, g, and h)?