Say I have a unit circle. And I want to map a point on such circle to the face of a unit hemisphere. (Imagine mapping points on a shadow of a hemisphere to the hemisphere).
in the diagram below, $A$ represents the very left-hand side of the hemisphere. $B$ represents the center of the hemisphere's face (not the 3D center). And $C$ is some arbitrary point where $C_z$ is in between $A_z$ and $B_z$
Given coordinates $x, y$ how would you get the angles $\theta, \phi$ on the hemisphere?
On
$x^2 + y^2 +z^2 = r^2\\ z = \sqrt {r^2 - x^2 - y^2}$
And that would be the most simple minded, mapping from the disk to the hemisphere.
If you want this in spherical coordinates
$x = r \cos \theta \sin \phi\\ y = r\sin \theta \sin \phi\\ z = r \cos \phi$
$\theta = \tan^{-1} \frac {y}{x}\\ \phi = \tan^{-1} \frac {\sqrt {x^2+y^2}}{z}$
I suggest you first map the point $(x,y)$ from the disk to the point $(x,y,\sqrt{1-(x^2+y^2)})$ in the hemisphere and then transform into spherical coordinates.