$p=-\sqrt{-q^{2}-1+\frac{1}{z^{2}}}$
$p=\sqrt{-q^{2}-1+\frac{1}{z^{2}}}\text{, }z\neq 0$

$q=-\sqrt{-p^{2}-1+\frac{1}{z^{2}}}$
$q=\sqrt{-p^{2}-1+\frac{1}{z^{2}}}\text{, }z\neq 0$

$p=\frac{\sqrt{1-\left(qz\right)^{2}-z^{2}}}{|z|}$
$p=-\frac{\sqrt{1-\left(qz\right)^{2}-z^{2}}}{|z|}\text{, }z\neq 0\text{ and }|z|\leq \frac{1}{\sqrt{q^{2}+1}}$

$q=\frac{\sqrt{1-\left(pz\right)^{2}-z^{2}}}{|z|}$
$q=-\frac{\sqrt{1-\left(pz\right)^{2}-z^{2}}}{|z|}\text{, }z\neq 0\text{ and }|z|\leq \frac{1}{\sqrt{p^{2}+1}}$