by the parametrization $${\mathbb{P}}^1_{{\mathbb{C}}} \ni (s,t) \; \mapsto \; \big( (s^2 - t^2): i(s^2 + t^2): 2st
\big) \in C$$ and its inverse $$\left\{ 
    \begin{array}{lcll}
        (\alpha: \beta : \gamma) & \mapsto & (\alpha - i \beta : \gamma) &
\mathrm{if\ } (i\alpha +
\beta, \gamma) \neq (0,0) \\
        (\alpha: \beta : \gamma) & \mapsto & (\gamma: - \alpha - i \beta) &
\mathrm{otherwise}
    \end{array}
    \right.$$ The identification between ${\mathbb{P}}^1_{{\mathbb{C}}}$ and $S^2$ by stereographic projection is provided by the following equations: $$\begin{array}{l}
    \left\{
    \begin{array}{lcll}
        (0,0,1) & \mapsto & (0:1) \in {\mathbb{P}}^1_{{\mathbb{C}}} \\ [2mm]
        (\lambda, \mu, \nu) & \mapsto & \left( 1: \frac{\lambda + i \mu}{1 - \nu}
\right) \in {\mathbb{P}}^1_{{\mathbb{C}}} & \mathrm{for\ all\ } (\lambda, \mu, \nu) \in S^2
\setminus \big\{ (0,0,1) \big\}
    \end{array}
    \right. \\ [8mm]
    \left\{
    \begin{array}{lcll}
        (0:1) & \mapsto & (0,0,1) \in S^2 \\ [2mm]
        (1: a + ib) & \mapsto & \left( \frac{2a}{a^2 + b^2 + 1}, \frac{2b}{a^2 +
b^2 + 1}, \frac{a^2 + b^2 - 1}{a^2 + b^2 + 1}  \right) & \mathrm{for\ all\ } a,b
\in {\mathbb{R}}\end{array}
    \right.
\end{array}$$ For example if $v = (1:i:0) \in C$, then the corresponding element of $S^2$ is the South pole $(0,0,-1)$.

In this way it is possible to 