Exercise 7.5.8

Proof. Let $P=(a,b,c)\ne (0,0,1)$ be a point of $S_2$, so $a^2+b^2+c^2=1$. Then $c\ne 1$. Write $N=(0,0,1)$ the north pole. Any point $M=(x,y,z)$ lies on the line $(\mathit{NP})$, if and only if $\stackrel{\to }{\mathit{NM}}=\lambda \stackrel{\to }{\mathit{NP}},\lambda \in {ℝ}^{\text{∗}}$, which gives the parametric system of equations

The intersection with the equatorial plane is given by $z=0$, so $\lambda =1∕(1-c)$, which gives $x=a∕(1-c),y=b∕(1-c)$ :

(where the points $(x,y,0)$ are identified with the complex numbers $x+\mathit{iy}$.) □