Geodesic curves

Grasshopper
1

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.
https://solitaryroad.com/c335.html
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,

Geodesic-and-normal-curvature-of-a-curve-on-a-smooth-surface_W640
https://www.researchgate.net/publication/234790847_Straightest_Geodesics_on_Polyhedral_Surfaces

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

thanks in advance!