# Why can the mean curvature vector be expressed by a limit of a line integral over closed curves?

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.

• This definitely is not right. Note that the quantity on the left goes to $0$, since the average value of $\nu-\nu_i$ [horrid notation, by the way] goes to $0$. My back-of-envelope calculation shows that you need some constant times $|\gamma_\epsilon|^3$ in the denominator. (Then there's always the question of whether for this author mean curvature is the average of the principal curvatures or their sum.) – Ted Shifrin Apr 28 '17 at 18:01
• @TedShifrin thanks for the comment. Could you give some more explanation for the divergent to $0$. Which inequalities/ properties did you use to get a hint on the denominator? – Joei Apr 29 '17 at 15:06
• @TedShifrin Are you referring to $\frac{1}{arclength} \int_\gamma f(x,y,z) ds =$ average value of $f(x,y,z)$? Does this hold for vector fields as well? – Joei Apr 29 '17 at 15:22
• Sure it does. (Think about it component by component, if you prefer.) I can share the details of my computation, but basically I assumed that the surface was a graph over the $xy$-plane at the origin with horizontal tangent plane at the point, with principal directions along the axes. Then you're looking at $z=\frac12(k_1x^2+k_2y^2)+\dots$ and you can consider $\gamma_\epsilon$ what you get over a circle of radius $\epsilon$ in the plane. Applying Green's Theorem allows you to approximate this and get $\dfrac1{2\pi\epsilon}H\pi\epsilon^3$ in the vertical (normal) direction. – Ted Shifrin Apr 29 '17 at 15:57