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?