Share 'curve on surface'

hello, i am trying to make a curve on a surface, using the endpoints, centre and endpoint of the edge of a subsurface, and points along it, to interpolate a curve. but even when i add more points, when the base surface is highly curved, the lines will not follow the surface closely enough. Is there another way to achieve a curve that more closely follows the surface? Orshould i change the appro…