Draw Line on Arbitrary Surface

I’m working on a 3D modeling tool and we use data from OpenLayers to construct a 3D mesh of some land. I want this behavior: you click on the mesh and it puts a point where you clicked on the mesh. Then, after the first point has been made, if you hover over the mesh with your mouse, a helper line will appear between the first point you made and where you mouse is currently. This helper line is drawn on the mesh, i.e. conforms to it’s surface. This sounded like simple functionality but is proving quite complex.

I’ve looked into geodesics, and shortest path algorithms but they seem really complex. I’m still fairly new to three.js. I’ve also read about UV mapping, or maybe casting a shadow to create this line.

Does any one have any pointers for this? Or things to look into? Thank you!