Geodesic lines on a 3D surface without a discrete ending point
I would like to reproduce the geodesic lines on some sort of surface in 3D, given by a function f(x,y), from some starting points. Reproducing this is what I have in mind with the idea of understanding geodesics a bit better.
On my prior post, it was made clear that fixing starting and ending points makes the problem much more challenging, so I have abandoned that idea. In one of the comments, the thought of using the tangent vectors (or the normal vector to the surface) at every point was floated, and I would like to ask for help exploring that approach.
Each point has an entire tangent vector space associated with it, and therefore, it seems like knowing the normal vector does not determine a specific direction to move the curve forward. Small ("differential") increments in x and y seem intuitively the way to go, but each point on the surface poses the same dilemma.
I'm attaching a file as a toy example of a surface with normal vectors on it.