I am trying to formulate a bijective map $f_n:S^n\setminus (1,0,\dots,0)\to R^n$. I consider $S^n$ in $n$-spherical coordinates, that is, $S^n = \{(r,\phi_1,\dots,\phi_n)\in R^{n+1}\ |\ r = 1\}$, and $\phi_i\in [0,2\pi)$ for all $i$.
It seems natural to map the point $(1,\pi,\phi_2,\dots,\phi_n)$ to the origin of $R^n$, and $(1,\{0,2\pi\},\dots,\phi_n)$ to infinity (in the limits). All other points will be finite in Euclidean $n$-space. In 1 dimension, I view it as setting the circle on top of the real line, removing the north-most point, and stretching the two sides, each to their respective infinities. Similarly in 2 dimensions, place the unit sphere so it sits on the origin of $R^2$, remove the north pole, and stretch the rest out to infinity.
So far I have $f_1(r,\phi_1) = \frac{\phi_1-\pi}{\phi_1(\phi_1-2\pi)}$, which is a bijection, but I can't seem to able to get any further.
Thanks for any help or leads in the right direction.
Spherical coordinates would not be my approach. Even in $\mathbb R^3$ the coordinate system breaks down. The standard approach would be stereographic projection from the north pole. Since you're choosing to omit $(1,0,\dots,0)$, write $x=(x_1,y)$ and let $\pi(x) = \dfrac1{1-x_1}y$.