Total curvature

In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:

${\displaystyle \int _{a}^{b}k(s)\,ds=2\pi N.}$

The total curvature of a closed curve is always an integer multiple of 2π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces.