We can do something similar for the 3-sphere $\mathbb{S}^3$: taking a point in $\mathbb{S}^3$, we take its gnomonic projection to $\mathbb{R}^3$, and we interpret it as the stereographic projection of a point in $\mathbb{S}^3$: this gives a map $\mathbb{S}^3 \to \mathbb{S}^3$ which just doubles the spherical distance from the center of projection. The latter fact can be seen as a geometric consequence of the fact that the the inscribed angle in a circle is half the central angle (Euclid, Elements, III.20 😉). But I'm not sure what plays the role of $\mathrm{Herm}(2)$ in the spherical case.
↧