TITLE: Is there map $\mathbb{S}^2 \to \mathbb{R}^2$ which maps great circles to lines?
QUESTION [7 upvotes]: Let $U \subseteq \mathbb{S}^2$ be some small open set.
Is there a smooth map $f:U \to \mathbb{R}^2$ which preserves the geodesics? 
(i.e if $\alpha$ is (part of) a great circle, then I require $f \circ \alpha$ to be a straight line).
(There is no such conformal map, since every conformal geodesics-preserving map is a scaled isometry. In particular the stereographic projection is excluded from the search).

REPLY [2 votes]: As commented by kimchi lover,
the gnomonic projection  maps great circles to straight lines. (It can be defined on a subset of the hemisphere).