I have a problem where I want to describe a uniform distrubution on a random 3D surface. Suppose I have given a star-convex surface by means of a radial function \begin{equation} r=f(\theta,\phi) \end{equation} with elevation $\theta$, azimuth $\phi$ and radius $r$. Thus, for every angle pair $[\theta,\phi]^T\in[0,\pi)\times[0,2\pi)$ a radius is given by function f describing a point on a surface in spherical coordinates. Is there an analytical way to describe a uniform distribution of points on the surface imposed by the radial function?
Your help is highly appreciated.
If you take probability density of proportional to the metrics, you will get the uniform distribution. For spherical coordinates:
$$ \omega(\theta,\phi) \propto g = \left|\begin{pmatrix}r\\ 0 \\\partial r/\partial \theta\end{pmatrix}\times \begin{pmatrix}0\\ r\cos\theta \\\partial r/\partial \phi\end{pmatrix}\right|. $$
If you integrate $\omega(\theta,\phi)$ over any region, it will be proportional to the area of the region by design, which is the definition of uniform distribution.