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# 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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### Conversion of U/V Coordinates

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

`PhiTheta = uv2phitheta([0.5; 0]);`

## Input Arguments

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### UV — Angle in u/v spacetwo-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

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### PhiTheta — Phi/theta angle pairstwo-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.

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### U/V Space

The u/v coordinates for the hemisphere x ≥ 0 are derived from the phi and theta angles, as follows:

u = sin(θ) cos(φ)

v = sin(θ) sin(φ)

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

u = cos(el) sin(az)

v = sin(el)

The values of u and v satisfy the inequalities

–1 ≤ u ≤ 1

–1 ≤ v ≤ 1

u2 + v2 ≤ 1

Conversely, the phi and theta angles can be written in terms of u and v

tan(φ) = v/u

sin(θ) = sqrt(u2 + v2)

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

sin(el) = v

tan(az) = u/sqrt(1 – u2 – v2)

### 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