# Geometric intuition for the Weingarten map

Parameterize a hypersurface $M$ by $r: \Omega \rightarrow \mathbb{R}^n$, and let $T_p M$ denote the tangent space at $p = r(u)$. We define the Weingarten map to be the linear map $L_p : T_p M \rightarrow T_p M$ given by $$L_p(v) = -\partial_v N$$ where $N$ is the unit normal vector field of the hypersurface.

What I'm looking for is perhaps some geometric explanation, pictures or software that will let me sort of see what the Weingarten map looks like for 2d surfaces, and why we might expect that critical points for normal curvature occurs at its eigenvectors. (It could also be the case that this is just due to the magic of linear algebra, but some deeper explanation would be nice).

-