How can we build a topological atlas for the $n$-dimensional disk $D^n=\{ x\in \mathbb{R}^n \mid \lVert x \rVert \leqslant 1 \}$ as a manifold with boundary ? Specifically, how to construct the maps of charts that cover boundary points ?
Thanks in advance !! Greetings.
One way is to pick a point $p\in\partial D^n$ and perform the inversion in unit sphere centered at $p$: namely, $$x\mapsto p+\frac{x-p}{\|x-p\|^2} \tag1$$ This maps $D^n\setminus\{p\}$ onto a half-space. Then do the same with another boundary point, and you have $D^n$ covered.
The usual choice is $p=(0,\dots,0,\pm 1)$, which is very much related to stereographic projection.