While proving that the quotient space $\overline{B}^n/S^{n-1}$ is homeomorphic to $S^n$, I needed to construct a continuous function $p:\overline{B}^n\to S^n$. I figured that by fixing two points $R = (0, 0,\dots, r)$ and $L = (0, 0,\dots, -r)$ in the closed $n$-ball $\overline{B}^n$ of radius $r$, I could then use the function $p(x) = \begin{cases}(x_1,\dots,x_n, \sqrt{r^2 - ||x||^2}):(x_n - r)^2 \leq (x_n + r)^2\\\ (x_1,\dots,x_n, -\sqrt{r^2 - ||x||^2}):(x_n + r)^2 < (x_n - r)^2\end{cases}$ (where $||x||$ is the standard norm in $\mathbb{R}^n$) to determine the sign of the $(n+1)$th component of the image of $x \in \overline{B}^n$. However, knowing that the closed unit interval can be mapped to the circle by the parametrization $x \mapsto (\cos(2\pi x), \sin(2\pi x))$, I can't help but to wonder whether some similar construction could be made from $\overline{B}^n$ to $S^n$?
That is, do you happen to know some nice/comfortable continuous mappings from a general closed $n$-ball to $n$-sphere, similar to the parametrization of a unit circle by a unit interval?
There is no map that "wraps" the Cartesian space around a sphere analogous to the covering of the unit circle by a line (because the sphere is simply-connected in dimension greater than $1$), but there is the map defined by $$ x \mapsto \biggl(\sin(|x|)\frac{x}{|x|}, \cos(|x|)\biggr), $$ which sends the open ball of radius $\pi$ about the origin diffeomorphically to the unit sphere with the "south pole" $(0, \dots, 0, -1)$ removed, and which collapses the boundary of the ball to the south pole. (This map is closely related to the Riemannian exponential map of a round sphere.)