Antenna
Toolbox™ uses two types of coordinate system: *rectangular coordinate
system* and *spherical coordinate system *.

Antenna
Toolbox uses the *rectangular coordinate system* to visualize
antenna or array geometry. The toolbox uses the *spherical coordinate system
* to visualize antenna radiation patterns.

Visualize the geometry of a default `monopoleTopHat`

antenna from
the antenna library.

m = monopoleTopHat; show(m);

The toolbox displays the top-hat monopole antenna in the
*rectangular* or *Cartesian* coordinate
system.

The *rectangular* coordinate system also called
*Cartesian* coordinate system specifies a position in space
as an ordered 3-tuple of real numbers, `(x,y,z)`

, with respect to
the origin `(0,0,0)`

.

You can view the 3-tuple as a point in space, or equivalently as a vector in three-dimensional Euclidean space. When viewed as a vector in space, the coordinate axes are basis vectors and the vector gives the direction to a point in space from the origin. Every vector in space is uniquely determined by a linear combination of the basis vectors. The most common set of basis vectors for three-dimensional Euclidean space are the standard unit basis vectors:

$$\{[1\text{\hspace{0.05em}}\text{\hspace{0.17em}}0\text{\hspace{0.05em}}\text{\hspace{0.17em}}0],[0\text{\hspace{0.05em}}\text{\hspace{0.17em}}1\text{\hspace{0.05em}}\text{\hspace{0.17em}}0],[0\text{\hspace{0.05em}}\text{\hspace{0.17em}}0\text{\hspace{0.05em}}\text{\hspace{0.17em}}1]\}$$

.

Any three linearly independent vectors define a basis for three-dimensional space. However, the Antenna Toolbox assumes that the basis vectors you use are orthogonal.

The standard distance measure in space is
the *l ^{2}* norm,
or Euclidean norm. The Euclidean norm of a vector [

$$\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$$

The Euclidean norm gives the length of the vector measured from the origin as
the hypotenuse of a right triangle. The distance between two vectors
[*x**0** **y**0** **z**0*]
and
[*x**1** **y**1** **z**1*] is:

$$\sqrt{{({x}_{0}-{x}_{1})}^{2}+{({y}_{0}-{y}_{1})}^{2}+{({z}_{0}-{z}_{1})}^{2}}$$

Visualize the radiation pattern of the default `monopoleTopHat`

antenna.

m = monopoleTopHat; pattern(m,75e6);

The toolbox displays the radiation pattern of the top-hat monopole using
*spherical * coordinate system represented by azimuth and
elevation angles.

The *spherical * coordinate system defines a vector or point in
space with a distance *R* and two angles. You can represent the
angles in this coordinate system:

Azimuth and elevation angles

Phi (Φ) and theta (θ) angles

*u*and*v*coordinates

The *azimuth angle* is the angle from the
positive *x*-axis to the vector's orthogonal
projection onto the *xy* plane, moving in the
direction towards the y-axis. The azimuth angle is in the range –180 and 180
degrees.

The *elevation angle* is the angle from the
vector's orthogonal projection on the *xy* plane
toward the positive *z*-axis, to the vector. The
elevation angle is in the –90 and 90 degrees.

The φ angle is the angle from the
positive *x*-axis to the vector's orthogonal projection
onto the *xy* plane, moving in the direction
towards the y-axis. The azimuth angle is between –180 and 180 degrees.

The θ angle is the angle from the positive *z*-axis to the
vector itself. The θ angle is in the range 0 degrees and 180 degrees.

These angles are an alternative to using azimuth and elevation angles to express the location of point in a unit sphere.

You can
define *u* and *v* in
terms of φ and θ:

$$\begin{array}{l}u=\mathrm{sin}\theta \mathrm{cos}\varphi \\ v=\mathrm{sin}\theta \mathrm{sin}\varphi \end{array}$$

In terms of azimuth and elevation angles, the *u * and
*v* coordinates are:

$$\begin{array}{l}u=\mathrm{cos}el\mathrm{sin}az\\ v=\mathrm{sin}el\end{array}$$

The values of *u * and *v* satisfy the inequalities:

$$\begin{array}{l}-1\le u\le 1\\ -1\le v\le 1\\ {u}^{2}+{v}^{2}\le 1\end{array}$$

The φ and θ angles in terms of *u* and
*v* are:

$$\begin{array}{l}\mathrm{tan}\varphi =u/v\\ \mathrm{sin}\theta =\sqrt{{u}^{2}+{v}^{2}}\end{array}$$

The azimuth and elevation angles in terms of *u* and
*v* are:

$$\begin{array}{l}\mathrm{sin}el=v\\ \mathrm{tan}az=\frac{u}{\sqrt{1-{u}^{2}-{v}^{2}}}\end{array}$$

Convert rectangular coordinates to spherical coordinates (*az*,
*el*, *R*) using:

$$\begin{array}{l}R=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}\\ az={\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)\\ el=\mathrm{tan}{}^{-1}\left(\frac{z}{\sqrt{{x}^{2}+{y}^{2}}}\right)\end{array}$$

Convert spherical coordinates (*az*, *el*,
*R*) to rectangular coordinates using:

$$\begin{array}{l}x=R\mathrm{cos}(el)\mathrm{cos}(az)\\ y=R\mathrm{cos}(el)\mathrm{sin}(az)\\ z=R\mathrm{sin}(el)\end{array}$$

where:

*R*is the distance from the antenna*el*and*az*are the azimuth and elevation angles

[1] Balanis, C.A. *Antenna Theory: Analysis and Design*.
3rd Ed. New York: Wiley, 2005.