Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the following:

For any two points $P,Q \in \mathbb{C}$, can we find a curve $\gamma$ connecting $P$ and $Q$, such that $\Pi^{-1}(f(\gamma))$ lies in a great circle on the unit sphere, and that $\Pi^{-1}(f(\gamma))$ cover the circle at most once as points go from $P$ to $Q$ along the curve $\gamma$?

Any ideas or comments are really appreciated!