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]

References

1. ^ 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.
2. ^ 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.
3. ^ 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.
4. ^ 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.
5. ^ 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.