I am looking for examples of $f:\mathbb{S}^{n-1}\to\mathbb{S}^{n-1}$.
The obvious one is:
$f(\mathbf{x}) = \mathbf{Wx}$ where $\mathbf{W}$ is orthogonal and $\mathbf{x}\in\mathbb{S}^{n-1}$.
Are there other functions which have an explicit analytic form that map points from unit sphere to unit sphere?