Can geodesics on any surface ( smooth, embedded $\mathbb R ^3)$ be mapped as straight lines on the plane? Please provide examples.
Are there some other global properties of surfaces that may need to be satisfied for a plane straight line representation ?
Stereographic projection of great circles of a sphere clearly do not map to straight lines on the plane. Longitudes of a sphere are obvious maps but may be trivial examples.
Not clear if any isometries need be modified.
Appreciate all helpful suggestions.