find f and $d_pf$

Question

Let $$S_1=\{(x,y,z) | x^2+y^2+z^2=1\}-\{N\}$$

$$S_2= \{(x,y,z,0) | x,y\in \Bbb R\}$$

$f:S_1\to S_2$

$f$ is stereographic projection.

,where $\ell$ is a line passing through $N=(0,0,1)$ and $p$

$f(p)= \ell \cap S_2=q$

I want to find $f$ and $d_pf$. Please help me to do this question. I googled and found many results but these are not understandable for me. I have seen such a projection at first time.

Answer 1

$S_2$ should be

$$S_2 = \{ (x, y, 0): x, y \in \mathbb R\} \subset \mathbb R^3$$

To write down $f:S_1 \to S_2$, Parametrize the line joining $N$ and $p = (x, y, z)$ by

$$N + t(N-p) = (0,0,1) + t (-x, -y, 1-z) = (-tx, -ty, 1+t(1-z))$$

$f(p)$ is the point where the $z$ coordinate is $0$, that is $t =1/(z-1)$.Thus

$$f(p) = \bigg(\frac{x}{1-z}, \frac{y}{1-z}, 0\bigg). $$