azel2uv

Convert azimuth/elevation angles to u/v coordinates

Syntax

UV = azel2uv(AzEl)

Description

UV = azel2uv(AzEl) converts the azimuth/elevation angle pairs to their corresponding coordinates in u/v space.

Examples

Conversion of Azimuth and Elevation to UV

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

uv = azel2uv([30;0])

uv = 2×1

    0.5000
         0

Input Arguments

AzEl — Azimuth/elevation angle pairs

two-row matrix

Azimuth and elevation angles, specified as a two-row matrix. Each column of the matrix represents an angle pair in the form [azimuth;elevation]. Azimuth angles must lie in the range [-90, 90]. Units are in degrees.

Data Types: double

Output Arguments

UV — Angle in u/v space

two-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.

More About

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.

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 these two coordinates systems is

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

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

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

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

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

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

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

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

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

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

Does not support variable-size inputs.

Version History

Introduced in R2012a