Definition of radial projection

Question

In my notes there is a place where "we project radially onto the ball $\partial B(x,r) \subseteq \mathbb{R}^n$", but this map is not precisely defined. I would like to confirm whether I understand it correctly.

Is the radial prjection onto $\partial B(x,r)$ defined as follows $\pi_{x}:\mathbb{R}^n \setminus\{x\} \to \partial B(x,r)$, $\pi_{x}(y)=\frac{(y-x)r}{|y-x|}+x$ ? My slight confusion comes from the fact that we project onto $\partial B(x,r)$ as opposed to the unit sphere $\mathbb{S}^{n-1}$ at the origin. The latter seems to be defined by the map $\pi_{x}(y)=\frac{y-x}{|y-x|}$, so to map it onto $\partial B(x,r)$ I just rescaled by $r$ and translated to $x$. Does this look right?

It seems to be correct. But you should tell us what is projected. — Paul Frost, Aug 11 '22 at 15:27
It was a certain set, lying in $\mathbb{R}^n \setminus \partial B(x,r)$. — Leon5, Aug 11 '22 at 22:01

score 0 · Accepted Answer · answered Aug 11 '22 at 15:31

Yes, it is correct. Radial projection onto the unit sphere would be the map $\pi_0(y) = \frac{y}{|y|}$, while radial projection onto the sphere of radius $r$ centered at the origin would be $\pi(y) = r(\frac{y}{|y|})$. For radial projections onto spheres centered at points $x$ other than $0$, you can either define them as $\pi_1(y)= r(\frac{y-x}{|y-x|})$ or as $\pi_2(y) = x + r(\frac{y-x}{|y-x|})$.

Both definitions are fine since for any set $A\subset \mathbb R^n\setminus\{x\}$, the images of $A$ under $\pi_1$ and $\pi_2$ only differ by a translation by the basepoint $x$. So it's a matter of whether you want to keep track of the basepoint in this way, and it's easy to translate between the two.

Definition of radial projection

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