Stenographic projection of $S^{2}$ sphere.

Question

In my notes:

the there exists a line $\imath \left ( p \right )$ that intersects any fixed coordinate N and any point p in $S^{2}$.

Consider a point $p_{1}$ in the region z>0. Indeed, the line $\imath \left ( p_{1} \right )$ intersects $p_{1}$ and N and cuts plane z=0 at some point $q_{1}$ on the plane z=0. However, I fail to understand why for any point p in the region $z>0$, the map of the point p is to the region outside of $S^{2}$.

Any clarification is much appreciated.

Answer 1

Have you tried putting "stereographic projection" into an image search? When I did with Google, this was the second result:

(Unfortunately reverse image search was not able to give me the original source of the image.)

Indeed, if you solve for the point $q$ to get an algebraic expression for it as a function of $p$, you should be able to justify formally why this occurs.