"Orthonormal" parameterization of solid sphere? 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 $1$?

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