- Total curvature
In mathematical study of the
differential geometry of curves , the total curvature of a plane curve is theintegral ofcurvature along a curve taken with respect to arclength::int_a^b k(s),ds.
The total curvature of a closed curve is always an integer multiple of 2π, called the index of the curve; it is the
winding number of the unit tangent about the origin. This relationship between a local invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher-dimensionalRiemannian geometry such as theGauss-Bonnet theorem .The total curvature of a curve γ in a higher dimensional
Euclidean space (equipped with its arclength parameterization) can be obtained by flattening out thetangent developable to γ into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in "n"-dimensional space is:int_a^b left|gamma"(s) ight|sgn kappa_{n-1}(s),ds
where κ"n"−1 is last Frenet curvature (the torsion of the curve) and sgn is the
signum function .According to the
Whitney-Graustein theorem , the total curvature is invariant under aregular homotopy of a curve.References
*citation|first= Wolfgang|last=Kuhnel|title=Differential Geometry: Curves - Surfaces - Manifolds|publisher=American Mathematical Society|year=2005|edition=2nd|isbn=978-0821839881 (translated by Bruce Hunt)
*citation|title=On the Total Curvature of Knots|first=John W.|last=Milnor|authorlink=John Milnor|journal=The Annals of Mathematics, Second Series|volume=52|number=2|year=1950|pages=248-257|url=http://www.jstor.org/stable/1969467
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