Convert u/v coordinates to phi/theta angles
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)
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