I came across a section (4.5) in the paper of Gabriel Taubin "A Signal Processing Approach To Fair Surface Design" which I would like to understand better:
"It is known that, for a curvature continuous surface, if the curve $\gamma$ shrinks to the point $v_i$, the (line) integral converges to the mean curvature $\kappa(v_i)$ of the surface at the point $v_i$ times the normal vector $N_i$ at the same point " $$ \lim_{\epsilon \rightarrow 0} \frac{1}{|\gamma_\epsilon|} \int_{v\in \gamma_\epsilon} (v-v_i)dl(v) = \kappa(v_i) N_i$$
Where $\gamma_\epsilon$ is a closed curve embedded in the surface which encircles the vertex $v_i$ and $|\gamma_\epsilon|$ is the length of the curve.
How can one prove this statement?
M.Do Carmo. "Differential Geometry of Curves and Surfaces" is cited but I could not find anything which helped me prove it.
He is using this to define a normal vector for polyhedral surfaces via the discrete Laplacian, which can be understood as an approximation of the above line integral.