Geodesic curves

hi all,

I have a geometrical question about geodesic curves on surfaces, specifcally about how they are traced by using grasshopper.
I mean that from my point of view there is a mismatch between the surface properties and the curve ones.
From the theoretical point of view the normal to the surface and the normal to the curve should be the same and I see that the have no coincidence at all,


somebody have dealt with this question?
somebody can shed some light on the way geodesic curves work in GH?

thanks in advance!