Let $\phi : U \subset \mathbb{R}^k \to \mathbb{R}^n$ be an embedding of a $k$-dimensional surface in $\mathbb{R}^n$. Is there a general prescription for selecting points $p$ in $U$ such that the points $\phi(p)$ will be uniformly distributed in $\phi(U)$?

I first thought about this in the context of uniformly distributing points on the sphere. One has the standard parametrization $\phi : (0,1) \times (0,1) \to \mathbb{R}^3$ where $$\phi(u,v) = (\cos(2\pi u) \sin(\pi v), \sin (2\pi u) \sin (\pi v) , \cos(\pi v)) $$ Uniformly selected points from $(0,1) \times (0,1)$ get mapped to non-uniformly distributed points on the sphere by $\phi$. It seems like instead we have to select points $(x,y)$ uniformly from $(0,1) \times (0,1)$, then map them to points $(u,v) = \psi(x,y) \equiv \left(x, \frac{1}{\pi}\cos^{-1}(2y - 1)\right)$, and then finally use $\phi$ to map those points to the sphere in order to get uniformly distributed points on the sphere.

I'm looking to generalize this treatment of the sphere to embedded $k$-surfaces. That is, how do we select a map $\psi : U \to U$ such that $\phi \circ \psi : U \to \mathbb{R}^n$ takes points uniformly distributed on $U$ to points uniformly distributed on the $k$-dimensional surface $(\phi \circ \psi)(U)$?