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]
References
edit- ^ 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, ISBN 978-3-540-25547-5.
- ^ a b c Chen, Bang-Yen (2000), "Riemannian submanifolds", Handbook of differential geometry, Vol. I, vol. 1, North-Holland, Amsterdam, pp. 187–418, doi:10.1016/S1874-5741(00)80006-0, ISBN 978-0-444-82240-6, 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.