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Φ:M⟶R3on U:(x,y)⟼(X,Y,Z)=(2 xx2+y2+1,2 yx2+y2+1,x2+y2−1x2+y2+1)on V:(x′,y′)⟼(X,Y,Z)=(2 x′x′2+y′2+1,2 y′x′2+y′2+1,−x′2+y′2−1x′2+y′2+1)\begin{array}{llcl} \Phi:& M & \longrightarrow & \mathbb{R}^3 \\ \mbox{on}\ U : & \left(x, y\right) & \longmapsto & \left(X, Y, Z\right) = \left(\frac{2 \, x}{x^{2} + y^{2} + 1}, \frac{2 \, y}{x^{2} + y^{2} + 1}, \frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\right) \\ \text{on}\ V : & \left({x'}, {y'}\right) & \longmapsto & \left(X, Y, Z\right) = \left(\frac{2 \, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \frac{2 \, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\right) \end{array}Φ:M⟶R3on U:(x,y)⟼(X,Y,Z)=(2 xx2+y2+1,2 yx2+y2+1,x2+y2−1x2+y2+1)on V:(x′,y′)⟼(X,Y,Z)=(2 x′x′2+y′2+1,2 y′x′2+y′2+1,−x′2+y′2−1x′2+y′2+1)\begin{array}{llcl} \Phi:& M & \longrightarrow & \mathbb{R}^3 \\ \text{on}\ U : & \left(x, y\right) & \longmapsto & \left(X, Y, Z\right) = \left(\frac{2 \, x}{x^{2} + y^{2} + 1}, \frac{2 \, y}{x^{2} + y^{2} + 1}, \frac{x^{2} + y^{2} - 1}{x^{2} + y^{2} + 1}\right) \\ \text{on}\ V : & \left({x'}, {y'}\right) & \longmapsto & \left(X, Y, Z\right) = \left(\frac{2 \, {x'}}{{x'}^{2} + {y'}^{2} + 1}, \frac{2 \, {y'}}{{x'}^{2} + {y'}^{2} + 1}, -\frac{{x'}^{2} + {y'}^{2} - 1}{{x'}^{2} + {y'}^{2} + 1}\right) \end{array}