# uv2azel

Convert u/v coordinates to azimuth/elevation angles

## Syntax

## Description

converts
the `AzEl`

= uv2azel(`UV`

)*u*/*v* space coordinates
to their corresponding azimuth/elevation angle pairs.

## Examples

### Conversion of U/V Coordinates to AzEl

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

azel = uv2azel([0.5; 0])

`azel = `*2×1*
30.0000
0

## Input Arguments

`UV`

— Angle in u/v space

two-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 *u*^{2} + *v*^{2}≤
1.

**Data Types: **`double`

## Output Arguments

`AzEl`

— Azimuth/elevation angle pairs

two-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`

.

## More About

### U/V Space

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.

To convert azimuth and elevation to *u* and *v* use the
transformation

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

which is valid only in the range *abs(az)≤=90*.

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

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

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 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

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

### Azimuth Angle, Elevation Angle

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. By default, the boresight direction of an element or array
is aligned with the positive *x*-axis. The boresight direction is the
direction of the main lobe of an element or array.

**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 shown as a green solid line.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Does not support variable-size inputs.

**Introduced in R2012a**

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