I stumbled on n-spheres recently, and recalled spherical coordinates.
They are very useful for doing computations on spheres embedded in $\mathbb R^{n+1}$, but oddly uneven, if not to say ugly, when compared to the intrinsic beauty and symmetry of a sphere.
I wonder, if we forget about the embedding Euclidean space, if there are more symmetric ways to parametrize the points of a n-sphere.
For those who are acquainted with spherical coordinates I feel I have to explain what I dislike about those. Among other tings, $x_1$ depends on $\varphi_1$, but $x_2$ depends on $\varphi_1, \varphi_2$. The domain for $\varphi_1$ is not the domain of $\varphi_{i+1}$. If you look at it from a distance, it looks strange, somehow not right.
So if we start from the geometry of the n-sphere, are there more natural coordinates?
What I've tried so far is to state some properties of the spheres. For every point $p$ there is an antipode $\bar p$. There is a metric $d(p,q)$ that corresponds to the length of the arc between the points. (I failed to find an elegant way to proof the triangular equation.) The only point with $d(p,q)=\pi$ is $q=\bar p$. I didn't get far from those basic considerations, and I wonder if all those things already have a name for which I could do a search on the Internet.