Solve R=rsqrt{f+P/f-P}

$$R = r \sqrt { \frac { f + P } { f - P } }$$

Solve for P (complex solution)

$$\left\{\begin{matrix}P=-\frac{f\left(r^{2}-R^{2}\right)}{r^{2}+R^{2}}\text{, }&\left(R=0\text{ or }R\neq -ir\right)\text{ and }\left(R=0\text{ or }R\neq ir\right)\text{ and }\left(R=0\text{ or }|arg(\sqrt{\frac{R^{2}}{r^{2}}}r)-arg(R)|<\pi \right)\text{ and }f\neq 0\text{ and }r\neq 0\\P\neq f\text{, }&\left(R=0\text{ and }r=0\right)\text{ or }\left(f=0\text{ and }R=ir\text{ and }r\neq 0\right)\text{ or }\left(f=0\text{ and }R=-ir\text{ and }|arg(ir)-arg(-ir)|<\pi \text{ and }r\neq 0\right)\end{matrix}\right.$$

Solve for R (complex solution)

$$R=\sqrt{\frac{P+f}{f-P}}r$$

$$f\neq P$$

Solve for P

$$\left\{\begin{matrix}P=-\frac{f(r^{2}-R^{2})}{r^{2}+R^{2}}\text{, }&(r<0\text{ or }R\geq 0)\text{ and }(r>0\text{ or }R\leq 0)\text{ and }f\neq 0\text{ and }r\neq 0\\P\in [-|f|,f)\cup (f,|f|]\text{, }&R=0\text{ and }r=0\end{matrix}\right.$$

Solve for R

$$R=\sqrt{\frac{P+f}{f-P}}r$$

$$\left(f\geq -P\text{ and }f>P\text{ and }|f|\geq |P|\right)\text{ or }\left(f\leq -P\text{ and }f<P\text{ and }|f|\geq |P|\right)\text{ or }\left(P\neq 0\text{ and }f=-P\right)$$