Parametrization of a sphere such that distances are separable

Question

Question

Is there a parameterization $(u, v)$ of a 2-sphere -- i.e. a map projection -- such that the geodesic (great-circle) distance function $d$ is "separable"? By separable I mean that the distance between two points $\mathbf{r}_1=(u_1,v_1)$ and $\mathbf{r}_2=(u_2,v_2)$ can be expressed as

$$d(\mathbf{r}_1, \mathbf{r}_2) = q(f(u_1, u_2) + g(v_1, v_2)),$$

where $q$, $f$, and $g$ are functions.

Motivation

I am working with a numerical technique that exploits this "separability" property of the Euclidean metric with Cartesian coordinates, which I want to extend to spheres.

Answer 1

\begin{align}d(r_1,r_2)&=\cos^{-1}(r1\cdot r2), \\ &=\cos^{-1}(u_1v_1+u_2v_2), \end{align}