uv2phitheta

Convert u/v coordinates to phi/theta angles

Syntax

PhiTheta = uv2phitheta(UV)

Description

PhiTheta = uv2phitheta(UV) converts the u/v space coordinates to their corresponding phi/theta angle pairs.

Examples

Conversion of U/V Coordinates

Find the corresponding φ/θ representation for u = 0.5 and v = 0.

PhiTheta = uv2phitheta([0.5; 0])

PhiTheta = 2×1

         0
   30.0000

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 u² + v²≤ 1.

Data Types: double

Output Arguments

PhiTheta — Phi/theta angle pairs

two-row matrix

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

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

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

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

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

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

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 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

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

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

Does not support variable-size inputs.