I like to visualize some typical curved lines on surfaces in $ \mathbb R^3 $ where $\int KdA$ and $\int k_g\,ds$ (constituents of the Gauß-Bonnet thm ) when each of these scalars is held constant. Simplest among them is the small/latitude circle on a sphere.
How are the parametrizations implemented?
I think such understanding is basic to classical Gaussian geometric ideation. Please correct me, if I sound wrong.