Total absolute curvature

In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve.^[1]

If the curve is parameterized by its arc length, the total absolute curvature can be expressed by the formula

${\displaystyle \int |\kappa (s)|ds,}$

where s is the arc length parameter and κ is the curvature. This is almost the same as the formula for the total curvature, but differs in using the absolute value instead of the signed curvature.^[2]

Because the total curvature of a simple closed curve in the Euclidean plane is always exactly 2π, the total absolute curvature of a simple closed curve is also always at least 2π. It is exactly 2π for a convex curve, and greater than 2π whenever the curve has any non-convexities.^[2] When a smooth simple closed curve undergoes the curve-shortening flow, its total absolute curvature decreases monotonically until the curve becomes convex, after which its total absolute curvature remains fixed at 2π until the curve collapses to a point.^[3][4]

The total absolute curvature may also be defined for curves in three-dimensional Euclidean space. Again, it is at least 2π (this is Fenchel's theorem), but may be larger. If a space curve is surrounded by a sphere, the total absolute curvature of the sphere equals the expected value of the central projection of the curve onto a plane tangent to a random point of the sphere.^[5] According to the Fáry–Milnor theorem, every nontrivial smooth knot must have total absolute curvature greater than 4π.^[2]