I need atlases for working with $n$-spheres, specifically $S^1$, $S^3$, and $S^5$. Stereographic projection charts seem like the most straightforward and economical way to make these. However, I want to work only in terms of hyper-spherical coordinates on the spheres, rather than the usual Cartesian coordinates in the $n+1$ space, since I need to be free of any reference to that higher dimensional embedding space.
Not finding such equations anywhere I attempted to derive these charts. I take the n-spheres as centered at the origin of the hyper-spherical coordinates, and project from the "north" and "south" poles onto the equatorial hyper-plane. I define the generalized azimuthal angles over the domain $[\pi/2, -\pi/2]$ rather than the more common convention of $[0, \pi]$. Then substituting the correct coordinate transformation equations from $n+1$ Cartesian to hyper-spherical coordinates into the Cartesian stereographic projection equations and calculating the radius coordinate in the projection hyperplane one gets, for any $n > 1$,
$r_1 = \frac{\cos t}{1-\sin t}$ "north" pole projection, $t \in (\pi/2, -\pi/2]$
$r_2 = \frac{\cos t}{1+\sin t}$ "south pole" $t\in [\pi/2, -\pi/2)$
where $t$ = the highest (or last) generalized azimuthal angle added for that dimension. All the remaining $n-1$ angle coordinates of that hyper-spherical coordinate set carry over to the projection hyperplane unchanged, for both charts, resulting in the requisite number of n coordinates for the n-dimensional hyperplane.
$n = 1$ is a slightly special case since there is only one angle on the 1-sphere, the "latitude", so the domain of $t$ is $[0, 2\pi)$ in this case, with the same equations for $r_1$ and $r_2$ mapping a point on the circle to a point on the real line thru the circle's "equator". This completes the three atlases needed.
However, I have not derived these results completely rigorously, but had to rely on intuition and inference at some steps in the higher dimensions, so have some reservations about their general validity. If anyone has or knows of a correct derivation of atlases meeting these requirements I would greatly appreciate seeing it or a reference to it.