Convert azimuth/elevation angles to u/v coordinates

converts
the azimuth/elevation
angle pairs to their corresponding coordinates in `UV`

= azel2uv(`AzEl`

)*u*/*v* space.

Find the corresponding *uv* representation for 30° azimuth and 0° elevation.

uv = azel2uv([30;0])

`uv = `*2×1*
0.5000
0

`AzEl`

— Azimuth/elevation angle pairstwo-row matrix

Azimuth and elevation angles, specified as a two-row matrix. Each column of the matrix represents an angle in degrees, in the form [azimuth; elevation].

**Data Types: **`double`

`UV`

— Angle in u/v spacetwo-row matrix

Angle in *u*/*v* space, returned
as a two-row matrix. Each column of the matrix represents an angle
in the form [*u*; *v*]. The matrix
dimensions of `UV`

are the same as those of `AzEl`

.

The *azimuth angle* of a vector is the angle between
the *x*-axis and the orthogonal projection of the vector onto the
*xy* plane. The angle is positive in going from the
*x* axis toward the *y* axis. Azimuth angles lie
between –180 and 180 degrees. The *elevation angle* is the angle
between the vector and its orthogonal projection onto the *xy*-plane. The
angle is positive when going toward the positive *z*-axis from the
*xy* plane. These definitions assume the boresight direction is the
positive *x*-axis.

The elevation angle is sometimes defined in the literature as the angle a vector makes
with the positive *z*-axis. The MATLAB^{®} and Phased Array
System Toolbox™ products do not use this definition.

This figure illustrates the azimuth angle and elevation angle for a vector shown as a green solid line. The coordinate system is relative to the center of a uniform linear array, whose elements appear as blue disks.

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

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

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

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

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Usage notes and limitations:

Does not support variable-size inputs.

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