Total curvature

This relationship between a local geometric 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.

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.

More generally, polygonal chains that do not go back on themselves (no 180° angles) have well-defined total curvature, interpreting the curvature as point masses at the angles.

[1] It can also be generalized to curves in higher dimensional spaces by flattening out the tangent developable to γ into a plane, and computing the total curvature of the resulting curve.

This curve has total curvature 6 π , and index/turning number 3, though it only has winding number 2 about p .
A closed polygonal chain , with total curvature 2 π .