If $f: X \to Y$ is a quasi-isometry between geodesic spaces, and $\gamma: [a,b] \to X$ is a geodesic segment in $X$, is $f\circ \gamma$ a geodesic segment in $Y$? I know that $f \circ \gamma$ is a quasi-geodesic segment since $f$ is a quasi-isometric embedding. But since $f$ is actually a qu...