Spherical basis vectors in 3-by-3 matrix form
A = azelaxes(az,el)
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 = 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.
az— Azimuth angle
Azimuth angle specified as a scalar in the closed range [–180,180]. Angle units are in degrees. To define the azimuth angle of a point on a sphere, construct a vector from the origin to the point. The azimuth angle is the angle in the xy-plane from the positive x-axis to the vector's orthogonal projection into the xy-plane. As examples, zero azimuth angle and zero elevation angle specify a point on the x-axis while an azimuth angle of 90° and an elevation angle of zero specify a point on the y-axis.
el— Elevation angle
Elevation angle specified as a scalar in the closed range [–90,90]. Angle units are in degrees. To define the elevation of a point on the sphere, construct a vector from the origin to the point. The elevation angle is the angle from its orthogonal projection into the xy-plane to the vector itself. As examples, zero elevation angle defines the equator of the sphere and ±90° elevation define the north and south poles, 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)];