"Orthonormal" parameterization of solid sphere?

Question

The standard parameterization of the solid sphere of radius $r$ centered at the origin in $3$-space is $$(\rho\cos\theta\sin\phi,\rho\sin\theta\sin\phi,\rho\cos\phi)\quad\rho\in[0,r],\theta\in[0,2\pi],\phi\in[0,\pi].$$ But this has Jacobian $\rho^2\sin\phi$ - among other things, this means that we cannot randomally pick a point on the sphere in an "unbiased" manner (by choosing random values for the parameters $\rho,\theta,\phi$ in their intervals) because some areas of the sphere (the points near the center) are more likely to be picked than others.

My question is: how can we construct an "orthonormal" parameterization from this one, which is still based on spherical coordinates? In other words, can we change this to an "arc-length parameterization" as is commonly done with one-dimensional curves, so that the Jacobian is everywhere $1$?

Answer 1

You can't! If you could, the sphere would be flat (have $0$ Gaussian curvature).

Answer 2

To choose a point uniformly randomly on the surface of a sphere, see this.

The paramatrization here is not nice at the caps, so differential geometry rules don't apply there. And it may bother you that the caps are not "as likely" to be chosen randomly as other individual points on the sphere. (Their $0\%$ likelihood is different from other points $0\%$ likelihood.) But integrating this distribution over any area of size $A$ will give you the same result.