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

The relation between the two coordinates is

$$\begin{array}{l}u=\mathrm{sin}\theta \mathrm{cos}\varphi \\ v=\mathrm{sin}\theta \mathrm{sin}\varphi \end{array}$$

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

$$\begin{array}{l}u=\mathrm{cos}el\mathrm{sin}az\\ v=\mathrm{sin}el\end{array}$$

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

$$\begin{array}{l}-1\le u\le 1\\ -1\le v\le 1\\ {u}^{2}+{v}^{2}\le 1\end{array}$$

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

$$\begin{array}{l}\mathrm{tan}\varphi =u/v\\ \mathrm{sin}\theta =\sqrt{{u}^{2}+{v}^{2}}\end{array}$$

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

$$\begin{array}{l}\mathrm{sin}el=v\\ \mathrm{tan}az=\frac{u}{\sqrt{1-{u}^{2}-{v}^{2}}}\end{array}$$

The φ angle is the angle from the positive *y*-axis
toward the positive *z*-axis, to the vector's
orthogonal projection onto the *yz* plane. The φ
angle is between 0 and 360 degrees. The θ angle is the angle
from the *x*-axis toward the *yz* plane,
to the vector itself. The θ angle is between 0 and 180 degrees.

The figure illustrates φ and θ for a vector that
appears as a green solid line. The coordinate system is relative to
the center of a uniform linear array, whose elements appear as blue
circles.

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

$$\begin{array}{l}\mathrm{sin}(\text{el})=\mathrm{sin}\varphi \mathrm{sin}\theta \hfill \\ \mathrm{tan}(\text{az})=\mathrm{cos}\varphi \mathrm{tan}\theta \hfill \\ \hfill \\ \mathrm{cos}\theta =\mathrm{cos}(\text{el})\mathrm{cos}(\text{az})\hfill \\ \mathrm{tan}\varphi =\mathrm{tan}(\text{el})/\mathrm{sin}(\text{az})\hfill \end{array}$$