I'm interested in how we might generalize the Mercator projection from the 2-sphere to the $n$-sphere.
The Mercator projection is normally defined for spherical coordinates given in the form $(r, \theta_1^\text{lat}, \theta_2)$, where $\theta_1^\text{lat} \in [-\pi / 2, \pi / 2]$, $\theta_2 \in [0, 2\pi]$ , and the superscript 'lat' denotes latitude. In this case, the cartesian coordinates $(x_1, x_2)$ of the projected point are given by $$ x_1 = \theta_2 \quad\text{and}\quad x_2 = \ln\left( \tan \left( \frac{\theta_1^\text{lat}}{2} + \frac{\pi}{4} \right) \right). $$
Converting to the usual definition of spherical coordinates, where $\theta_1$ is the polar angle, with $\theta_1^\text{pol} \in [0, \pi]$, gives $$ x_2 = \ln\left( \cot \left( \frac{\theta_1^\text{pol}}{2}\right) \right). $$
Now consider the hyperspherical coordinates $(r, \theta_1^\text{pol}, \theta_2^\text{pol}, \ldots, \theta_{n - 1})$, where $\theta_i^\text{pol} \in [0, \pi]$ and $\theta_{n - 1} \in [0, 2\pi]$. How can we generalize the Mercator projection to this coordinate system? It's clear to me that $x_1 = \theta_{n - 1}$ as before, but I'm not sure if the projections for the other angles generalize in the same way. Is it still true that $$ x_i = \ln\left( \cot \left( \frac{\theta_{i - 1}^\text{pol}}{2}\right) \right), $$ for $i \in [2, n]$? If not, how could we derive the formula?