sph2cartvec

Convert vector from spherical basis representation to cartesian representation

Syntax

vr = sph2cartvec(vs,az,el)

Description

vr = sph2cartvec(vs,az,el) converts the components of a vector or set of vectors, vs, from a spherical basis representation to a representation in a local cartesian coordinate system, vr. A spherical basis representation is the set of components of a vector projected into the right-handed spherical basis given by $$({\widehat{e}}_{az},{\widehat{e}}_{el},{\widehat{e}}_R)$$. The orientation of a spherical basis depends upon its location on the sphere as determined by azimuth, az, and elevation, az. See Spherical basis representation of vectors.

Each column of vs is a vector in the spherical basis. The third dimension of vs can correspond to different spherical bases if az and el.

Examples

Cartesian Representation of Azimuthal Vector

Start with a vector in a spherical basis located at 45° azimuth, 45° elevation. The vector points along the azimuth direction. Compute the vector components with respect to Cartesian coordinates.

vs = [1;0;0];
vr = sph2cartvec(vs,45,45)

vr = 3×1

   -0.7071
    0.7071
         0

Cartesian Representation of Two Vectors at Two Points

Start with a two vectors in a spherical basis, one pointing along the local z-direction and the other pointing in the local y-direction.

Load the vectors pairs into two different pages. The pages correspond to points (az=20,el=2) and (az=90,el=35). Compute the components of these vectors with respect to the spherical bases at these points.

az = [20 90];
el = [2 35];

Set up page 1 with two vectors.

vs1 = [0 0 1];
vs2 = [0 1 0];
vs(:,:,1) = [vs1' vs2'];

Put the same vectors into page 2.

vs(:,:,2) = [vs1' vs2'];

Transform all vectors to the global cartesian coordinate frame.

vr = sph2cartvec(vs,az,el)

vr = 
vr(:,:,1) =

    0.9391   -0.0328
    0.3418   -0.0119
    0.0349    0.9994


vr(:,:,2) =

         0         0
    0.8192   -0.5736
    0.5736    0.8192

Input Arguments

vs — Vector in spherical basis representation

real-valued 3-by-N matrix | real-valued 3-by-N-by-M MATLAB^® array

Vector in a spherical basis representation, specified as a real-valued 3-by-N matrix or a real-valued 3-by-N-by-M MATLAB array. N is the number of vectors to convert.

• If vs is a 3-by-N matrix, each column represents a vector in one spherical basis defined by a scalar az and el arguments.

• If vs is a 3-by-N-by-M MATLAB array, each column each column represents a vector in a spherical basis. Each column of vs contains the three components of a vector in the right-handed spherical basis. Each page of vs corresponds to different spherical bases. Both the az and el input arguments must be length-M row vectors. Each entry in the az and el vectors corresponds to a page in vs. The m^-th page of vr is converted using the m^-th values of az and el.

Units are in meters.

Example: [45.0;10;20.3]

Data Types: single | double

az — Azimuth angle

scalar | real-valued M-by-1 vector

Azimuth angle, specified as a scalar or real-valued M-by-1 vector in the closed range [–180,180]. M is the number of azimuth angles and must match the size of el. 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, 0° azimuth angle and 0° elevation angle specify a point on the x-axis while an azimuth angle of –90° and an elevation angle of 0° specify a point on the y-axis. Angle units are in degrees.

Example: [10,45]

Data Types: single | double

el — Elevation angle

scalar | real-valued M-by-1 vector

Elevation angle specified as a scalar or real-valued M-by-1 vector in the closed range [–90,90]. M is the number of elevation angles and must match the size of az. To define the elevation angle of a point on the sphere, construct a vector from the origin to the point. The elevation angle is the angle from vector's orthogonal projection into the xy-plane to the vector itself. As examples, 0° elevation angle defines the equator of the sphere and ±90° elevation define the north and south poles, respectively. Angle units are in degrees.

Example: [2,-3]

Data Types: single | double

Output Arguments

vr — Vectors in cartesian basis representation

real-valued 3-by-N matrix | real-valued 3-by-N-M MATLAB array

Cartesian basis representation of output vectors, returned as a real-valued 3-by-N matrix or real-valued MATLAB array. vr has the same size as vs. Each column of vr contains the three components of the vector in a cartesian coordinates.

More About

Spherical basis representation of vectors

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 is a set of three mutually orthogonal unit vectors $$({\widehat{e}}_{az},{\widehat{e}}_{el},{\widehat{e}}_R)$$ defined at a point on the sphere. The first unit vector points along lines of azimuth at constant radius and elevation. The second points along the lines of elevation at constant azimuth and radius. Both are tangent to the surface of the sphere. The third unit vector points radially outward.

The orientation of the basis changes from point to point on the sphere but is independent of R so as you move out along the radius, the basis orientation stays the same. The following figure illustrates the orientation of the spherical basis vectors as a function of azimuth and elevation:

For any point on the sphere specified by az and el, the basis vectors are given by:

$$\begin{array}{ll}{\widehat{e}}_{az}\hfill & =-\sin (az)\widehat{i}+\cos (az)\widehat{j}\hfill \\ {\widehat{e}}_{el}\hfill & =-\sin (el)\cos (az)\widehat{i}-\sin (el)\sin (az)\widehat{j}+\cos (el)\widehat{k}\hfill \\ {\widehat{e}}_R\hfill & =\cos (el)\cos (az)\widehat{i}+\cos (el)\sin (az)\widehat{j}+\sin (el)\widehat{k}\text{.}\hfill \end{array}$$

Any vector can be written in terms of components in this basis as $$v=v_{az}{\widehat{e}}_{az}+v_{el}{\widehat{e}}_{el}+v_R{\widehat{e}}_R$$. The transformations between spherical basis components and Cartesian components take the form

$$\left[\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right]=\left[\begin{array}{ccc}-\sin (az)& -\sin (el)\cos (az)& \cos (el)\cos (az)\\ \cos (az)& -\sin (el)\sin (az)& \cos (el)\sin (az)\\ 0& \cos (el)& \sin (el)\end{array}\right]\left[\begin{array}{c}{v}_{az}\\ v_{el}\\ v_R\end{array}\right]$$

.

and

$$\left[\begin{array}{c}{v}_{az}\\ v_{el}\\ v_R\end{array}\right]=\left[\begin{array}{ccc}-\sin (az)& \cos (az)& 0\\ -\sin (el)\cos (az)& -\sin (el)\sin (az)& \cos (el)\\ \cos (el)\cos (az)& \cos (el)\sin (az)& \sin (el)\end{array}\right]\left[\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right]$$.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Does not support variable-size inputs.

Version History

Introduced in R2013a

R2026a: Multiple spherical coordinate systems

The version accepts vector inputs to support multiple spherical coordinate systems.