## Transform Observation Angles

The following equations transform an observation angle directly from one coordinate system to another.

($\theta$,$\phi$) from ($E$,$A$)

\begin{equation} \theta = \cos^{-1} (cos(E) cos(A)) \end{equation} \begin{equation} \phi = \tan^{-1} \biggl(\frac{cos(E) sin(E)}{sin(A)}\biggr) \end{equation}($E$,$A$) from ($\theta$,$\phi$)

\begin{equation} E = \sin^{-1} (sin(\theta) sin(\phi)) \end{equation} \begin{equation} A = \tan^{-1} \biggl(\frac{sin(\theta) cos(\phi)}{cos(\theta)}\biggr) \end{equation}($\theta$,$\phi$) from ($\alpha$,$\epsilon$)

\begin{equation} \theta = \cos^{-1} (cos(\alpha) cos(\epsilon)) \end{equation} \begin{equation} \phi = \tan^{-1} \biggl(\frac{cos(\alpha) sin(\epsilon)}{sin(\alpha)}\biggr) \end{equation}($\alpha$,$\epsilon$) from ($\theta$,$\phi$)

\begin{equation} \alpha = \sin^{-1} (sin(\theta) cos(\phi)) \end{equation} \begin{equation} \epsilon = \tan^{-1} \biggl(\frac{sin(\theta) sin(\phi)}{cos(\theta)}\biggr) \end{equation}## Transform Vector Components

Vector components are transformed indirectly from one coordinate system to another. First, they must be transformed into the intermediate cartesian coordinate system using the following equations.

$E_{\theta}$ and $E_{\phi}$ vector components to $E_x$, $E_y$, and $E_z$

\begin{equation} \begin{bmatrix} E_{x}\\ E_{y}\\ E_{z}\\ \end{bmatrix} = \begin{bmatrix} E_{\theta}cos(\phi)cos(\theta)-E_{\phi}sin(\phi)\\ E_{\theta}sin(\phi)sin(\theta)+E_{\phi}cos(\phi)\\ -E_{\theta}sin(\theta) \\ \end{bmatrix} \end{equation}$E_{\alpha}$ and $E_{\epsilon}$ vector components to $E_x$, $E_y$, and $E_z$

\begin{equation} \begin{bmatrix} E_{x}\\ E_{y}\\ E_{z}\\ \end{bmatrix} = \begin{bmatrix} E_{\alpha}cos(\alpha)\\ -E_{\alpha}sin(\epsilon)sin(\alpha)+E_{\epsilon}cos(\epsilon)\\ -E_{\alpha}cos(\epsilon)sin(\alpha)-E_{\epsilon}sin(\epsilon) \\ \end{bmatrix} \end{equation}$E_{E}$ and $E_{A}$ vector components to $E_x$, $E_y$, and $E_z$

\begin{equation} \begin{bmatrix} E_{x}\\ E_{y}\\ E_{z}\\ \end{bmatrix} = \begin{bmatrix} -E_{E}sin(E)sin(A)+E_{A}cos(A)\\ -E_{E}cos(E)\\ -E_{E}cos(A)sin(E)-E_{A}sin(A) \\ \end{bmatrix} \end{equation}Vector components in the cartesian coordinate system can then be transformed into the final coordinate system using the following equations.

$E_{\theta}$ and $E_{\phi}$ vector components from $E_x$, $E_y$, and $E_z$

\begin{equation} \begin{bmatrix} E_{\theta}\\ E_{\phi}\\ \end{bmatrix} = \begin{bmatrix} E_{x}\, cos(\phi)cos(\theta)+E_{y}\, sin(\phi)cos(\theta)-E_{z}\, sin(\theta)\\ -E_{x}\, sin(\phi)+E_{y}\, cos(\phi)\\ \end{bmatrix} \end{equation}$E_{\alpha}$ and $E_{\epsilon}$ vector components from $E_x$, $E_y$, and $E_z$

\begin{equation} \begin{bmatrix} E_{\alpha}\\ E_{\epsilon}\\ \end{bmatrix} = \begin{bmatrix} E_{x}\, cos(\alpha)-E_{y}\, sin(\epsilon)sin(\alpha)-E_{z}\, cos(\epsilon)sin(\alpha)\\ E_{y}\, cos(\epsilon)-E_{z}\, sin(\epsilon)\\ \end{bmatrix} \end{equation}$E_{E}$ and $E_{A}$ vector components from $E_x$, $E_y$, and $E_z$

\begin{equation} \begin{bmatrix} E_{E}\\ E_{A}\\ \end{bmatrix} = \begin{bmatrix} -E_{x}\, sin(E)sin(A)-E_{y}\, cos(E)-E_{z}\, sin(E)cos(A)\\ E_{x}\, cos(A)-E_{z}\, sin(A)\\ \end{bmatrix} \end{equation}