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There must be some error in my formulas for the stereographic projection of the sphere and the inverse projection. However, I can't find the error. Here's what I have:

Let $S_K^n$ be the sphere with sectional curvature $K$, then the stereographic projection of $S_K^n$ to th $n$-dimensional hyperplane is:

$$ x=(x_1,...,x_{n+1}) \mapsto \frac{1}{1-\sqrt{K}x_{n+1}}(x_1,...,x_n,0) $$

and the inverse of the projection is: $$ x=(x_1,...,x_n,0)\mapsto \frac{2}{K||x||_2^2+1}\left(x_1,...,x_n,\frac{K||x||_2^2-1}{2\sqrt{K}}\right) $$

What I want to do is the stereographic projection from the north pole, where the sphere is centered at 0 with radius $R=\frac{1}{\sqrt{K}}$.

ndrizza
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    What makes you think that there is some error? When you check your formulas, remember that the one for "from sphere to plane" only needs to hold in case that the point is already on the sphere. So there is a constraint that must be fulfilled. – Reinhard Meier Jul 15 '19 at 16:01

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Your formulas are correct (as shown in Riemannian Manifolds: An Introduction to Curvature by Lee 1997, Formula 3.9).

Remember that to prove that $\sigma^{-1}(\sigma(x))=x$ for $x \in S^n_R$ you need the fact that $R^2 = ||x_{1:n}||^2 + x_{n+1}^2$, where $R=1/\sqrt{K}$ in your formulas.

Ondrej Skopek
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