(This answer was written to address the *original* version of the question, before the second paragraph was added.)

As an example, on $S^2$ it's always possible to define a coordinate patch

$$\psi:S^2\rightarrow R^3$$

using typical spherical coordinates $(r,\theta,\phi)$, defined such that $\psi(x)=(1,0,0)$ and $\psi(y)=(1,0,\phi_y)$. The coordinate patch can be defined everywhere on $S^2$ except in some neighborhoods of the poles. Like any coordinate patch, $\psi$ is injective (although it isn't surjective), so $\psi^{-1}$ is defined on $\psi$'s image. Then the simplest possible definition for the map $f$ using those coordinates would be

$$f(\lambda)=\psi^{-1}(1,0,\lambda \phi_y)\ .$$

There are of course many other possible ways to define $f$, which take the image of $f$ along different paths on $S^2$ between $x$ and $y$.

This post imported from StackExchange Mathematics at 2014-10-05 10:04 (UTC), posted by SE-user Red Act