Is there a $1-1$ map of the interior of the $n$-simplex to $\mathbb{R}^n$?

Question

I'm looking for a simple $1-1$ map from the interior of the standard $n$-simplex that covers $\mathbb{R}^n$. Any suggestions?

Answer 1

I can tell you how to construct one. Whether it is simple enough is for you to determine. Position the simplex with the origin of $\Bbb R^n$ at its center. Now scale every open radial segment in the simplex to $(0,\infty)$ by your favorite bijection. In whatever direction you choose, let the distance from the origin to the simplex be $d$. Take the point that is $r$ from the origin in the simplex and map it to the point that is$\frac 1{1-\frac rd}$ from the origin in $\Bbb R^n$. The complicated part is coming up with $d$ given an angle of interest.