mean curvature (plane curve)

Let $\mathrm{\Gamma }$ be a piecewise $C^1$ planar curve.

The total curvature, ${\kappa }_{total}$ , of $\mathrm{\Gamma }$ is defined to be ${\int }_{\mathrm{\Gamma }}|\kappa (s)|𝑑s$ where $\mathrm{\Gamma }$ is parameterized by arclength $s$and $\kappa (s)$ is the curvature (http://planetmath.org/CurvatureOfACurve)of $\mathrm{\Gamma }$.

The mean curvature of $\mathrm{\Gamma }$ is defined to be the ratio of the total curvature to the length of $\mathrm{\Gamma }$ :

$$M(\mathrm{\Gamma })=\frac{{\kappa }_{total}(\mathrm{\Gamma })}{L(\mathrm{\Gamma })}$$