{\displaystyle {\begin{aligned}\varphi ^{-1}:\mathbb {R} ^{2}&\mapsto S^{2}\setminus \{N\}\\(x,y)&\mapsto ({\frac {2x}{x^{2}+y^{2}+1}},{\frac {2y}{x^{2}+y^{2}+1}},{\frac {x^{2}+y^{2}-1}{x^{2}+y^{2}+1}})\end{aligned}}}