uv2phitheta

Convert u/v coordinates to phi/theta angles

Description

example

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

Examples

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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

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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

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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.

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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

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