Formula for Stereographic Projection of Ball of Radius $R$

Question

Let the following be the set of points of radius $R$: $$ B^n_R:=\{x\in\mathbb{R}^{n+1}\,\,| \,\,||x||_2^2=R^2\} $$ What is the formula to stereographically project $B_R^n$ to the $n$-dimensional hyperplane $E^n$? $$ E^n:=\{x\in\mathbb{R}^{n+1}\,\,| \,\,x_{n+1}=0\} $$

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So I'd do a stereographic projection from the north pole $(0,...,0,R)$ onto $E^n$.

Answer 1

In Cartesian coordinates it's $(x_1,\dots,x_{n+1})\to (\frac{Rx_1}{R-x_{n+1}},\dots,\frac {Rx_n}{R-x_{n+1}},0)$.

See here.

Answer 2

The solution is to do the following projections:

1) Projection to unit sphere

2) Stereographic projection

3) Rescaling by radius R