The *u/v* coordinates for
the positive hemisphere *x* ≥ 0 can be derived
from the phi
and theta angles.

The relation between these two coordinates systems is

In these expressions, φ and θ are the phi and theta
angles, respectively.

In terms of azimuth and elevation, the *u* and *v* coordinates
are

The values of *u* and *v* satisfy
the inequalities

Conversely, the phi and theta angles can be written in terms
of *u* and *v* using

The azimuth and elevation angles can also be written in terms
of *u* and *v*