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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…mehr

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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.