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

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]}

**^**Brook, Alexander; Bruckstein, Alfred M.; Kimmel, Ron (2005), "On similarity-invariant fairness measures", in Kimmel, Ron; Sochen, Nir A.; Weickert, Joachim (eds.),*Scale Space and PDE Methods in Computer Vision: 5th International Conference, Scale-Space 2005, Hofgeismar, Germany, April 7-9, 2005, Proceedings*, Lecture Notes in Computer Science, vol. 3459, Springer-Verlag, pp. 456–467, doi: 10.1007/11408031_39.- ^
^{a}^{b}^{c}Chen, Bang-Yen (2000), "Riemannian submanifolds",*Handbook of differential geometry, Vol. I*, North-Holland, Amsterdam, pp. 187–418, doi: 10.1016/S1874-5741(00)80006-0, MR 1736854. See in particular section 21.1, "Rotation index and total curvature of a curve", pp. 359–360. **^**Brakke, Kenneth A. (1978),*The motion of a surface by its mean curvature*(PDF), Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., Appendix B, Proposition 2, p. 230, ISBN 0-691-08204-9, MR 0485012.**^**Chou, Kai-Seng; Zhu, Xi-Ping (2001),*The Curve Shortening Problem*, Boca Raton, Florida: Chapman & Hall/CRC, Lemma 5.5, p. 130, and Section 6.1, pp. 144–147, doi: 10.1201/9781420035704, ISBN 1-58488-213-1, MR 1888641.**^**Banchoff, Thomas F. (1970), "Total central curvature of curves",*Duke Mathematical Journal*,**37**(2): 281–289, doi: 10.1215/S0012-7094-70-03736-1, MR 0259815.