[1] If the curve is parameterized by its arc length, the total absolute curvature can be expressed by the formula where s is the arc length parameter and κ is the 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.
[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.
[5] According to the Fáry–Milnor theorem, every nontrivial smooth knot must have total absolute curvature greater than 4π.