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 $E^n$? $$ E^n:=\{x\in\mathbb{R}^{n+1}\,\,| \,\,x_{n+1}=0\} $$

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

score 2 · Accepted Answer · 2019-07-11T02:14:08.847

2

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.

edited Jul 11 '19 at 02:14

answered Jul 11 '19 at 01:17

@DavidK I took your advice. – Jul 11 '19 at 02:15

score 0 · Answer 2 · answered Jul 11 '19 at 00:07

0

The solution is to do the following projections:

1) Projection to unit sphere

2) Stereographic projection

3) Rescaling by radius R

answered Jul 11 '19 at 00:07

ndrizza

1,348

Formula for Stereographic Projection of Ball of Radius $R$

2 Answers2