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# uv2azel

Convert u/v coordinates to azimuth/elevation angles

## Description

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AzEl = uv2azel(UV) converts the u/v space coordinates to their corresponding azimuth/elevation angle pairs.

## Examples

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### Conversion of U/V Coordinates

Find the corresponding azimuth/elevation representation for u = 0.5  and v = 0.

`AzEl = uv2azel([0.5; 0]);`

## Input Arguments

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### UV — Angle in u/v spacetwo-row matrix

Angle in u/v space, specified as a two-row matrix. Each column of the matrix represents a pair of coordinates in the form [u; v]. Each coordinate is between –1 and 1, inclusive. Also, each pair must satisfy u2 + v2≤ 1.

Data Types: double

## Output Arguments

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### AzEl — Azimuth/elevation angle pairstwo-row matrix

Azimuth and elevation angles, returned as a two-row matrix. Each column of the matrix represents an angle in degrees, in the form [azimuth; elevation]. The matrix dimensions of AzEl are the same as those of UV.

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### U/V Space

The u/v coordinates for the hemisphere x ≥ 0 are derived from the phi and theta angles, as follows:

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

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

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

### Phi Angle, Theta Angle

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

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

### Azimuth Angle, Elevation Angle

The azimuth angle is the angle from the positive x-axis toward the positive y-axis, to the vector's orthogonal projection onto the xy plane. The azimuth angle is between –180 and 180 degrees. The elevation angle is the angle from the vector's orthogonal projection onto the xy plane toward the positive z-axis, to the vector. The elevation angle is between –90 and 90 degrees. These definitions assume the boresight direction is the positive x-axis.

 Note:   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 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.