The phi angle (φ) is the angle from the positive y-axis to the vector’s orthogonal projection onto the yz plane. The angle is positive toward the positive z-axis. The phi angle is between 0 and 360 degrees. The theta angle (θ) is the angle from the x-axis to the vector itself. The angle is positive toward the yz plane. The theta angle is between 0 and 180 degrees.

The figure illustrates phi and theta for a vector that appears as a green solid line.

The coordinate transformations between φ/θ and az/el are described by the following equations

The u/v coordinates for the hemisphere x ≥ 0 are derived from the phi and theta angles.

The relations are

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