Invariance
According to the Whitney–Graustein theorem, the total curvature is invariant under a regular homotopy of a curve: it is the degree of the Gauss map. However, it is not invariant under homotopy: passing through a kink (cusp) changes the turning number by 1.
By contrast, winding number about a point is invariant under homotopies that do not pass through the point, and changes by 1 if one passes through the point.
Read more about this topic: Total Curvature
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