(@needham2021visual page 117)
All curves that pass through a point $p$ of a curved surface in the same direction $T$ have the same normal curvature $\kappa_n(T)$ as the normal section in that direction. $\blacksquare$
The inhabitants of the surface can try to draw different curves at $p$ sharing the same $T$ but with different curvatures. They can think that they can control de curvature of the curve, but in certain sense this is false. They only can control the geodesic component of the curvature (see normal and geodesic curvature of a curve), while the normal component is fixed by the surface.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: