Graph Theory—stereographic projection

Question

Context: I'm in an introductory Graph Theory class, and my professor decided to discuss stereographic projection for some reason unknown to me. It's a topic that's not even mentioned in the textbook we're using.

QUESTION: Express the stereographic projection algebraically given the following definitions:
X : the sphere x$^2$ + y$^2$ + z$^2$ = 1,
N : the north pole (0, 0, 1),
${\rm I\!R}$$^2$ : the plane z = 0,
Y : X — {N}, and
$\phi$ : the function Y$\rightarrow$${\rm I\!R}$$^2$

That is, find $\alpha$(x, y, z) and $\beta$(x, y, z) where $\phi$(P) = ($\alpha$, $\beta$).

He also gave us a hint: Write the equation of the line through N and P (the point on the sphere) and intersect it with the plane z = 0.

Thoughts in my head: What??? This seems a little intimidating since (1) there are several parts, (2) because it isn't even explained in the textbook, and (3) because my professor isn't the best communicator/teacher. So I'm having trouble where to start. I apologize if this is a bad question, but really my brain isn't producing very much help either.

Answer 1

I think this should help show the mapping:

Answer 2

Let $P=(x_0,y_0,z_0)\in Y$, i.e. it's a point on the sphere, distinct to the north pole $N$.

Then $\phi(P)$ is defined as the intersection of line $NP$ with the $x, y$-plane $H$ (which has equation $z=0$).
Note that $H$ is identified with $\Bbb R^2$ in the exercise by regarding $(x,y,0)\in H$ as $(x,y)\in\Bbb R^2$.

Now the line $NP$ consists of points $N+t(P-N)$ with $t\in\Bbb R$, that is, of points $$(0,0,1)+t(x_0,y_0,z_0-1)\ =\ (tx_0,\,ty_0,\,1+t(z_0-1))$$ You can obtain the unique $t$ for which this point is on $H$, by solving $1+t(z_0-1)=0$, and then substitute it into $\phi(P) =(tx_0,ty_0) $.