Given the standard equirectangular map of planet earth, say $M((\theta, \phi))$, for $0\leq \theta \leq 2\pi, -\pi/2 \leq \phi \leq \pi/2$.
Now rotate the planet by $90°$ such that a new map $M'((\theta', \phi'))$ is obtained, looking like this.
The task is to derive a formula for the projection $f$ such that $$M'((\theta', \phi')) = M(f(\theta, \phi)).$$
I thought that's easy, but looking at some points or axes and their corresponding location, I don't see a pattern. Any ideas or references appreciated, thanks!
