Spherical basis vectors in 3-by-3 matrix form
At the point located at 45° azimuth, 45° elevation, compute the 3-by-3 matrix containing the components of the spherical basis.
A = azelaxes(45,45)
A = 3×3 0.5000 -0.7071 -0.5000 0.5000 0.7071 -0.5000 0.7071 0 0.7071
The first column of
A contains the radial basis vector
[0.5000; 0.5000; 0.7071]. The second and third columns are the azimuth and elevation basis vectors, respectively.
A— Spherical basis vectors
Spherical basis vectors returned as a 3-by-3 matrix. The columns contain the unit vectors in the radial, azimuthal, and elevation directions, respectively. Symbolically we can write the matrix as
where each component represents a column vector.
Spherical basis vectors are a local set of basis vectors which point along the radial and angular directions at any point in space.
The spherical basis vectors at the point (az,el) can be expressed in terms of the Cartesian unit vectors by
This set of basis vectors can be derived from the local Cartesian basis by two consecutive rotations: first by rotating the Cartesian vectors around the y-axis by the negative elevation angle, -el, followed by a rotation around the z-axis by the azimuth angle, az. Symbolically, we can write
The following figure shows the relationship between the spherical basis and the local Cartesian unit vectors.
MATLAB® computes the matrix
A = [cosd(el)*cosd(az), -sind(az), -sind(el)*cosd(az); ... cosd(el)*sind(az), cosd(az), -sind(el)*sind(az); ... sind(el), 0, cosd(el)];
Usage notes and limitations:
Does not support variable-size inputs.