Accelerating the pace of engineering and science

# phitheta2uv

Convert phi/theta angles to u/v coordinates

## Description

example

UV = phitheta2uv(PhiTheta) converts the phi/theta angle pairs to their corresponding u/v space coordinates.

## Examples

expand all

### Conversion of Phi/Theta Pair

Find the corresponding u/v representation for φ = 30  degrees and θ = 0 degrees.

`UV = phitheta2uv([30; 0]);`

## Input Arguments

expand all

### PhiTheta — Phi/theta angle pairstwo-row matrix

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

Data Types: double

## Output Arguments

expand all

### 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 PhiTheta.

expand all

### 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

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

### 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

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