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

$p(x)=\{\begin{array}{l}({x}_{1},\dots ,{x}_{n},\sqrt{{r}^{2}-||x|{|}^{2}}):({x}_{n}-r{)}^{2}\le ({x}_{n}+r{)}^{2}\\ \text{}({x}_{1},\dots ,{x}_{n},-\sqrt{{r}^{2}-||x|{|}^{2}}):({x}_{n}+r{)}^{2}({x}_{n}-r{)}^{2}\end{array}$

(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 $x\mapsto (\mathrm{cos}(2\pi x),\mathrm{sin}(2\pi x))$, 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?