When can you and when can you not combine partial derivatives

On traditional maps, earth is represented in a flat plane, or by euclidean geometry. however, a globe is a more accurate model that comes from elliptical geometry. how does a globe represent the fact that there are no parallel lines in elliptical geometry? the equator is not parallel to any other latitudinal lines. the north and south poles are never connected by a geodesic. the geodesics connecting the north and south poles never intersect. the geodesics connecting the north and south poles intersect at both of the poles.