phitheta2uv

Convert phi/theta angles to u/v coordinates

Syntax

UV = phitheta2uv(PhiTheta)

Description

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

Examples

Conversion of Phi-Theta Pair

Find the corresponding u-v representation for φ = 30° and φ = 0°.

uv = phitheta2uv([30; 0])

uv = 2×1

     0
     0

Input Arguments

PhiTheta — Phi/theta angle pairs

two-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

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 PhiTheta.

More About

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

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

U/V Space

The u/v coordinates for the hemisphere x ≥ 0 are derived from the phi and theta angles.

The relations are

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

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:

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

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