TITLE: Properties of the n-dimensional Stereographic Projection
QUESTION [5 upvotes]: Hello,
I'm looking for an argument that the n-dimensional stereographic projection maps circles (intersections of affine two-dimensional subspaces with S^n) to circles in R^n. I've looked around and the only argument I saw for the n-dimensional case is a generalization of the geometric proof for n = 2 (with the tangent cone) which I don't really feel comfortable with, even when n = 2. Is it possible to reduce it to the n = 2 case somehow or give a "direct", algebraic, proof?

REPLY [6 votes]: I don't know what is the "tangent cone" argument you mention, but anyway here is my favorite  proof of the fact.
The stereographic projection is the restriction of an inversion $I$ from $\mathbb R^{n+1}\setminus \{p\}$ to itself, where $p$ is the pole from which you project (or maybe $I$ is a map from $\mathbb R^{n+1}\cup\{\infty\}$ to itself if you prefer it this way). So it suffices to prove that $I$ sends circles to circles and lines.
First observe that $I$ sends $n$-spheres and hyperplanes to $n$-spheres and hyperplanes. (This follows from the planar case with circles and lines and rotation symmetry. The planar case is elementary). In $\mathbb R^3$, every circle is an intersection of two spheres. Hence its image is an intersection of two sets each of which is a sphere or a plane. Such set is a circle again.
In higher dimensions, either restrict to a suitable 3-dimensional subspace, as Agol said in his answer, or represent a circle as an intersection of several spheres.