Solve d_{1}=R-sqrt{R^2-(c_{1}/2)^2}

$$d _ { 1 } = R - \sqrt { R ^ { 2 } - ( c _ { 1 } / 2 ) ^ { 2 } }$$

Solve for R (complex solution)

$$\left\{\begin{matrix}R=\frac{c_{1}^{2}}{8d_{1}}+\frac{d_{1}}{2}\text{, }&d_{1}\neq 0\text{ and }arg(2R-2d_{1})<\pi \\R\in \mathrm{C}\text{, }&d_{1}=0\text{ and }c_{1}=0\text{ and }arg(R)<\pi \\R=d_{1}\text{, }&d_{1}=\frac{c_{1}}{2}\text{ or }d_{1}=-\frac{c_{1}}{2}\end{matrix}\right.$$

Solve for c_1 (complex solution)

$$c_{1}=-2\sqrt{d_{1}}\sqrt{2R-d_{1}}$$

$$c_{1}=2\sqrt{d_{1}}\sqrt{2R-d_{1}}\text{, }d_{1}=R\text{ or }arg(2d_{1}-2R)\geq \pi$$

Solve for R

$$\left\{\begin{matrix}R=\frac{c_{1}^{2}}{8d_{1}}+\frac{d_{1}}{2}\text{, }&\left(d_{1}\leq -\frac{|c_{1}|}{2}\text{ and }d_{1}<0\right)\text{ or }\left(d_{1}>0\text{ and }d_{1}\leq \frac{|c_{1}|}{2}\right)\\R\geq 0\text{, }&d_{1}=0\text{ and }c_{1}=0\end{matrix}\right.$$

Solve for c_1

$$\left\{\begin{matrix}c_{1}=-2\sqrt{d_{1}\left(2R-d_{1}\right)}\text{, }&\left(R\neq 0\text{ and }d_{1}\geq 0\text{ and }-\left(2d_{1}-2R\right)\geq 0\text{ and }R\leq -\sqrt{d_{1}\left(2R-d_{1}\right)}\text{ and }d_{1}\leq 2R\right)\text{ or }\left(-\left(2d_{1}-2R\right)\geq 0\text{ and }d_{1}>0\right)\text{ or }\left(d_{1}\geq 0\text{ and }-\left(2d_{1}-2R\right)\geq 0\text{ and }d_{1}<2R\right)\text{ or }\left(d_{1}=0\text{ and }R=0\right)\text{ or }\left(R\neq 0\text{ and }d_{1}\geq 2R\text{ and }-\left(2d_{1}-2R\right)\geq 0\text{ and }R\leq -\sqrt{d_{1}\left(2R-d_{1}\right)}\text{ and }d_{1}\leq 0\right)\text{ or }\left(d_{1}=2R\text{ and }R<0\right)\\c_{1}=2\sqrt{d_{1}\left(2R-d_{1}\right)}\text{, }&\left(-\left(2d_{1}-2R\right)\geq 0\text{ and }d_{1}>0\right)\text{ or }\left(R\neq 0\text{ and }d_{1}\geq 0\text{ and }-\left(2d_{1}-2R\right)\geq 0\text{ and }R\leq -\sqrt{d_{1}\left(2R-d_{1}\right)}\text{ and }d_{1}\leq 2R\right)\text{ or }\left(d_{1}\geq 0\text{ and }-\left(2d_{1}-2R\right)\geq 0\text{ and }d_{1}<2R\right)\text{ or }\left(d_{1}\geq 2R\text{ and }-\left(2d_{1}-2R\right)\geq 0\text{ and }R\leq -\sqrt{d_{1}\left(2R-d_{1}\right)}\text{ and }d_{1}\leq 0\right)\text{ or }\left(R\leq 0\text{ and }d_{1}=2R\right)\end{matrix}\right.$$