High Quality Content by WIKIPEDIA articles! In mathematical study of the differential geometry of curves, the total curvature of a immersed plane curve is the integral of curvature along a curve taken with respect to arclength: int_a^b k(s),ds. The total curvature of a closed curve is always an integer multiple of 2?, called the index of the curve, or turning number – it is the winding number of the unit tangent about the origin, or equivalently the degree of the Gauss map. This relationship between a local invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher-dimensional Riemannian geometry such as the Gauss–Bonnet theorem.

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