The Project Tool is very powerful but ironically there's one thing it doesn't do - project a curve onto a surface without trimming or adding. I would like to project a curve onto a surface and just have the resulting curve....
Kevin
+1
Maybe some modes or a couple approaches would be good. Riffing on this:
One approach (I think this is Kevin's initial wish) - Adjust the view to show curve(s) and the target surface. This indicates direction of the projection. Then Project to create a new curve "pressed" onto the surface so that all points and edges are coincident with the surface. It does not trim or clip the surface.
Another approach is to have a new "Working Surface" or Constrain to Surface, similar to working plan