# Far Field Coordinate Systems (FF card)

The coordinate systems for the FF card have specific definitions/conventions. Use the spherical coordinate system in general and the Cartesian coordinate system when appropriate for the application.

## Spherical coordinates $\vartheta$ , $\phi$

The far field pattern of an antenna is assumed to be independent of the distance from the antenna. Specify the $\vartheta$ and $\phi$ angles of the direction to the observation point $r$ .

Unit vectors of the coordinate system:

(1)

## Cartesian coordinates u, v

The far field pattern of an antenna is assumed to be independent of the distance from the antenna. Specify the first two coordinates (u, v) of the unit vector to the observation point $r$ .

Observation point on the unit sphere:

(2) $\stackrel{^}{r}=\left[\begin{array}{c}u\\ v\\ \sqrt{1-{u}^{2}-{v}^{2}}\end{array}\right]$

Convert between spherical and Cartesian coordinates:

(3) $\begin{array}{l}u=\mathrm{sin}\vartheta \mathrm{cos}\phi \\ v=\mathrm{sin}\vartheta \mathrm{sin}\phi \\ \vartheta =\mathrm{arccos}\left(\sqrt{1-{u}^{2}-{v}^{2}}\right)\\ \phi =\mathrm{atan}2\left(v,u\right)\end{array}$