Embedding of $\mathbb R^2$ into the unit sphere $S^2$? How to find an embedding of $\mathbb R^2$ into the unit sphere $S^2$?
I know there is such an embedding because $S^2$ is a one-point compactification of $\mathbb R^2$, but I don't know how to write this embedding explicitly.
Thanks!

A well known construction is the so called "stereographic projection". Think of $\def\R{\mathbb R}\R^2$ embedded into $\R^3$ as $\R^2\times\{0\}$ and the sphere embedded as the set of euclidian unit vectors. Now for each $x \in \R^2$, think of the line joining it with $e_3 = (0,0,1)$. It hits the sphere at exactly one other point besides $e_3$. Call this point $\phi(x)$. Then $\phi\colon \R^2 \to S^2$ is an embedding.
To give $\phi$ exactly, note that points on the line are of the form 
$$ tx + (1-t)e_3 = (tx_1, tx_2, 1-t) $$
which is on the sphere iff 
$$ t^2x_1^2 + t^2x_2^2 + (1-t)^2 = 1 \iff (1 + x_1^2+ x_2^2)t^2 - 2t = 0 $$
Hence $t=0$ (which corresponds to $e_3$) or 
$$ t = \frac{2}{1+x_1^2 + x_2^2} \leadsto \phi(x) = \left(\frac{2x_1}{1+x_1^2+x_2^2}, \frac{2x_2}{1+x_1^2+x_2^2}, \frac{-1+x_1^2+x_2^2}{1+x_1^2+x_2^2}\right) $$
