The conformal compactification of the prototype Euclidean space $\mathbb R^n$ can be carried out by embedding $\mathbb R^n$ into $\mathbb R^{n+1,1}$ with Minkowski signature. One then considers the null-cone $\mathcal N$ given by the null-set of the obvious quadratic form $Q$ on $\mathbb R^{n+1,1}$, with the origin removed in order to have a proper hypersurface. The final step is to define the conformal compactification as the quotient $$\mathcal N/\sim,$$ where $\sim$ is the equivalence relation on rays, i.e. $x,y\in\mathcal N$ are equivalent (and we write $x\sim y$) if there exists $\lambda\in\mathbb R\smallsetminus\{0\}$ such that $x = \lambda y$.
As a concrete example we can take the real line $\mathbb R$. The null-cone $\mathcal N$, in this case, is just a double cone without vertex in $\mathbb R^3$, and the equivalence relation $\sim$ gives $\mathcal N/\sim\equiv S^1$, i.e. the unit circle. Therefore the circle $S^1$ is the conformal compactification of the real line $\mathbb R$, as one'd expect.
Similarly one obtains that the conformal compactification of $\mathbb R^2$ is the sphere $S^2$, and it turns out that the relation between the two objects is the stereographic projection.
My question is: how can I explicitly and systematically derive the inclusion of, say, $\mathbb R$ into $S^1$, or that of $\mathbb R^2$ into $S^2$, according to the above outlined construction? In the example of the sphere, for instance, such correspondence is, like I've said before, the stereographic projection.
Further hints towards the systematic determination of such inclusion for generic (pseudo-)Riemannian manifolds (e.g. Minkowski 4-spacetime) into their conformal compactification would be greatly appreciated.