Let $\Bbb S^2 =${$(x,y,z) \in \Bbb R^3 : x^2+y^2+z^2=1$}. Find a bijection $T$ from $H=$ {$(x,y,z) \in \Bbb S^2 :z>0$} to $\Bbb R^2$ such that the segments of halfcircels in $H$ will be mapped on segments of straight lines in $\Bbb R^2$.
I was thinking about putting a plane on the northpole of $H$ and imagining some sort of light beams coming out of $(0,0,0)$ who are going to that plane. But I'm not sure how to find a real bijection who does that.