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

\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 2 $π$ , the total absolute curvature is also always at least 2 $π$ . It is exactly 2 $π$ for a convex curve, and greater than 2 $π$ 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 2 $π$ 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 2 $π$ , 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 Fary–Milnor theorem, every nontrivial smooth knot must have total absolute curvature greater than 4 $π$ .^[2]

References

↑ Brook, Alexander; Bruckstein, Alfred M.; Kimmel, Ron (2005), "On similarity-invariant fairness measures", in Kimmel, Ron; Sochen, Nir A.; Weickert, Joachim, 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, 3459, Springer-Verlag, pp. 456–467, doi:10.1007/11408031_39 .
1 2 3 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, 20, Princeton University Press, Princeton, N.J., Appendix B, Proposition 2, p. 230, ISBN 0-691-08204-9, MR 485012 .
↑ Chou, Kai-Seng; Zhu, Xi-Ping (2001), The Curve Shortening Problem, Boca Raton, FL: 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: 281–289, doi:10.1215/S0012-7094-70-03736-1, MR 0259815 .

Various notions of curvature defined in differential geometry

Differential geometry of curves	Curvature Torsion of a curve Frenet–Serret formulas Radius of curvature (applications) Affine curvature Total curvature Total absolute curvature

Differential geometry of surfaces	Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss–Codazzi equations First fundamental form Second fundamental form Third fundamental form

Riemannian geometry	Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature

Curvature of connections	Curvature form Torsion tensor Cocurvature Holonomy

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