Mean curvature vector approximated for the discrete Laplace Beltrami Operator

Question

"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 " $$\lim_{\epsilon \to 0} \frac{1}{|\gamma_\epsilon|}\int_{v\in\gamma_\epsilon} (v-v_i)dl(v) = \kappa(v_i)N_i $$ 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. Especially $|\gamma_\epsilon|\to0$. From Gabriel Taubin "A Signal Processing Approach To Fair Surface Design" .

It is states as a common fact from differential geometry. But it is not clear to me. How would one prove this statement?

Does it has something to do with $$\kappa(v_i) N_i = \lim_{r(A)\to 0} \frac{3\nabla(A)}{2A} $$ where $r(A)$ is the diameter of a small region around $v_i$. $\nabla$ is the gradient with respect $x,y,z$ the coordinates.

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. Also the second expression is used to approximate the Laplace-Beltrami operator for triangulated surfaces.

Answer 1

In both papers you mention, the curve integral is introduced as a motivation only. Let $S\subset\mathbb R^3$ be a surface and $p\in S$. We may assume that $p=0$, and that the tangent plane to $S$ at $p$ is the $x$-$y$-plane. We also assume that the $x$- and $y$-axes are the principal curvature directions of $S$ at $p$. Then $S$ can be approximated to second order by the graph of $$(x,y)\mapsto\kappa_1\frac{x^2}2+\kappa_2\frac{y^2}2\;.$$ Let $\gamma_\varepsilon$ be the image of a circle of radius $\varepsilon>0$ around $0$. Then we consider the following integral (EDIT This computation was wrong. Here is a better one, I hope). \begin{multline*}\lim_{\varepsilon\to 0}\frac1{\varepsilon^2|\gamma_\varepsilon|}\int_{\gamma_\varepsilon}v\,d\ell(v)\\=\lim_{\varepsilon\to 0}\frac1{2\pi\varepsilon^3}\int_0^{2\pi}\begin{pmatrix}\varepsilon\cos\theta\\\varepsilon\sin\theta\\\frac{\varepsilon^2}2(\kappa_1\cos^2\theta+\kappa_2\sin^2\theta)\end{pmatrix}\,\varepsilon\,d\theta=\frac H2\,N\;,\end{multline*} where $N$ is the normal vector at $p$ (here, the unit vector in $z$-direction), and $H$ is the mean curvature with respect to $N$. Because of the appearance of $\varepsilon^3$ in the denominator, it might be better to consider the following slightly more stable expression instead: $$\lim_{\varepsilon\to 0}\frac1{\varepsilon^2|\gamma_\varepsilon|}\int_{\gamma_\varepsilon}\langle v,N\rangle\,d\ell(v)=\frac H2\;,$$ where $N$ still is the normal vector at $p$.

If $\gamma_\varepsilon$ has different shapes, then other numbers can occur as limiting values of the integral. For this reason, both papers allows weights $w_{ij}$ in their discretisations, see (6) in Taubin's paper and the formulas near the end of section 2.1 in Sorkine's, approximating the integral above by $$\sum_{j\in N(i)}w_{ij}(v_j-v_i)\;.$$ Sorkine describes how one may choose $w_{ij}$ depending on the local geometry of the mesh used for the discretisation.