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

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

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

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

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

$$\begin{array}{l}\sin el=\sin \varphi \sin \theta \\ \tan az=\cos \varphi \tan \theta \\ \cos \theta =\cos el\cos az\\ \tan \varphi =\tan el/\sin az\end{array}$$