# sph2cartvec

Convert vector from spherical basis components to Cartesian components

## Description

converts the components of a vector or set of vectors, `vr`

= sph2cartvec(`vs`

,`az`

,`el`

)`vs`

, from their
*spherical basis representation* to their representation in a local
Cartesian coordinate system. 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,
`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

## Input Arguments

`vs`

— Vector in spherical basis representation

3-by-1 column vector | 3-by-*N* matrix

Vector in spherical basis representation specified as a 3-by-1 column vector or
3-by-*N* matrix. Each column of `vs`

contains the
three components of a vector in the right-handed spherical basis $$({\widehat{e}}_{az},{\widehat{e}}_{el},{\widehat{e}}_{R})$$.

**Example: **`[4.0; -3.5; 6.3]`

**Data Types: **`double`

**Complex Number Support: **Yes

`az`

— Azimuth angle in degrees

scalar in range [–180, 180]

Azimuth angle in degrees, specified as a scalar in the closed range [–180, 180]. 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.

**Example: **`45`

**Data Types: **`double`

`el`

— Elevation angle in degrees

scalar in range [–90, 90]

Elevation angle in degrees, specified as a scalar in the closed range [–90, 90]. 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.

**Example: **`30`

**Data Types: **`double`

## Output Arguments

`vr`

— Vector in Cartesian representation

3-by-1 column vector | 3-by-*N* matrix

Cartesian vector returned as a 3-by-1 column vector or 3-by-*N* matrix having
the same dimensions as `vs`

. Each column of `vr`

contains the three components of the vector in the right-handed *x,y,z*
basis.

## 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:

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

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

$$\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]=\left[\begin{array}{ccc}-\mathrm{sin}(az)& -\mathrm{sin}(el)\mathrm{cos}(az)& \mathrm{cos}(el)\mathrm{cos}(az)\\ \mathrm{cos}(az)& -\mathrm{sin}(el)\mathrm{sin}(az)& \mathrm{cos}(el)\mathrm{sin}(az)\\ 0& \mathrm{cos}(el)& \mathrm{sin}(el)\end{array}\right]\left[\begin{array}{c}{v}_{az}\\ {v}_{el}\\ {v}_{R}\end{array}\right]$$

.

and

$$\left[\begin{array}{c}{v}_{az}\\ {v}_{el}\\ {v}_{R}\end{array}\right]=\left[\begin{array}{ccc}-\mathrm{sin}(az)& \mathrm{cos}(az)& 0\\ -\mathrm{sin}(el)\mathrm{cos}(az)& -\mathrm{sin}(el)\mathrm{sin}(az)& \mathrm{cos}(el)\\ \mathrm{cos}(el)\mathrm{cos}(az)& \mathrm{cos}(el)\mathrm{sin}(az)& \mathrm{sin}(el)\end{array}\right]\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]$$.

## 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**

## See Also

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