Total Curvature - Generalizations

Generalizations

A finite generalization is that the exterior angles of a triangle, or more generally any simple polygon, add up to 360° = 2π radians, corresponding to a turning number of 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.

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

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

The minimum total curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2π for the unknot, but by the Fary–Milnor theorem it is at least 4π for any other knot.