My brother, who is in hospital atm and cannot verify by himself asked me to post the following question, thank you in advance, and sorry if the topic has already been covered, i do not have the math knowledge to look for it in the forums.
He wants to map, on the unit sphere with polar coordinates, the square with vertices $(0,1)$, $(1,0)$. $(-1,0)$, $(0,-1)$ with centre $(0,0)$ having $(0,1)$ and $(0,-1)$ as poles.
He wants to know if the map $(a,b)\mapsto (a\pi , \arcsin(b))$ is adequate or at least it works.
He would also like to know if there's a more clever way to do that.
Thank you.