Total curvature

In mathematical study of the differential geometry of curves, the total curvature of a 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 2π, called the index of the curve; it is the winding number of the unit tangent about the origin. 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.

The total curvature of a curve γ in a higher dimensional Euclidean space (equipped with its arclength parameterization) can be obtained by flattening out the tangent developable to γ into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in "n"-dimensional space is

: $int_a^b left|gamma"(s)
ight|sgn kappa_{n-1}(s),ds$

where κ_"n"−1 is last Frenet curvature (the torsion of the curve) and sgn is the signum function.

According to the Whitney-Graustein theorem, the total curvature is invariant under a regular homotopy of a curve.

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