Is there map $\mathbb{S}^2 \to \mathbb{R}^2$ which maps great circles to lines?

Question

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).

Answer 1

As commented by kimchi lover,

the gnomonic projection maps great circles to straight lines. (It can be defined on a subset of the hemisphere).