Another way to get Geodesics on a surface

I have a doubt related to finding geodesic lines on a surface or mesh.
The methods I use is good when the the base surface doesn’t have negative curvature areas , because at this area - by increasing the strength of the goals decreasing the geodesics curvature kg- the geodesics skip this part (the valley) and connects between the 2 positive parts.
It is just a wonder if we can inverse the process; I mean to have originally (Always straight) grid of lines and force them to follow a certain path on the base surface including the negative curvature parts!