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[[File:Winding Number Around Point.svg|thumb|300px|This curve has total curvature 6''π'', and index/turning number 3, though it only has [[winding number]] 2 about ''p''.]]
In [[mathematics|mathematical]] study of the [[differential geometry of curves]], the '''total curvature''' of an [[immersion (mathematics)|immersed]] [[plane curve]] is the [[integral]] of [[curvature]] along a curve taken with respect to [[arc length]]:


:<math>\int_a^b k(s)\,ds.</math>


The total curvature of a closed curve is always an integer multiple of 2π, called the '''index''' of the curve, or '''[[turning number]]''' – it is the [[winding number]] of the unit [[tangent vector]] about the origin, or equivalently the degree of the map to the [[unit circle]] assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the [[Gauss map]] for surfaces.
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==Comparison to surfaces==
 
This relationship between a local geometric invariant, the curvature, and a global [[topological invariant]], the index, is characteristic of results in higher-dimensional [[Riemannian geometry]] such as the [[Gauss–Bonnet theorem]].
 
== Invariance ==
According to the [[Whitney–Graustein theorem]], the total curvature is invariant under a [[regular homotopy]] of a curve: it is the degree of the [[Gauss map]]. However, it is not invariant under homotopy: passing through a kink (cusp) changes the turning number by 1.
 
By contrast, [[winding number]] about a point is invariant under homotopies that do not pass through the point, and changes by 1 if one passes through the point.
 
== Generalizations ==
[[File:Closed polygonal line.svg|thumb|A closed [[polygonal chain]], with total curvature&nbsp;2''π''.]]
A finite generalization is that the exterior angles of a triangle, or more generally any [[simple polygon]], add up to 360°&nbsp;=&nbsp;2''π'' radians, corresponding to a turning number of 1. More generally, [[polygonal chain]]s that do not go back on themselves (no 180° angles) have well-defined total curvature, interpreting the curvature as point masses at the angles.
 
The total curvature of a curve γ in a higher dimensional [[Euclidean space]] (equipped with its arclength parameterization) can be obtained by flattening out the [[tangent 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
 
:<math>\int_a^b \left|\gamma''(s)\right|\sgn \kappa_{n-1}(s)\,ds</math>
 
where κ<sub>''n''&minus;1</sub> is last Frenet curvature (the [[torsion of curves|torsion]] of the curve) and sgn is the [[signum function]].
 
The minimum total curvature of any three-dimensional curve representing a given [[Knot (mathematics)|knot]] is an [[knot invariant|invariant]] of the knot. This invariant has the value 2π for the unknot, but by the [[Fary–Milnor theorem]] it is at least 4π for any other knot.
 
==References==
*{{citation|first= Wolfgang|last=Kuhnel|title=Differential Geometry: Curves - Surfaces - Manifolds|publisher=American Mathematical Society|year=2005|edition=2nd|isbn=978-0-8218-3988-1}} (translated by Bruce Hunt)
*{{citation|title=On the Total Curvature of Knots|first=John W.|last=Milnor|authorlink=John Milnor|journal=Annals of Mathematics | series = Second Series|volume=52|year=1950|pages=248–257|jstor=1969467|doi=10.2307/1969467|issue=2}}
*{{cite arxiv|first=John M.|last=Sullivan|title=Curves of finite total curvature|year=2007|eprint=math/0606007|class=math.GT }}.
 
[[Category:Curves]]
[[Category:Curvature (mathematics)]]

Latest revision as of 20:11, 27 October 2014


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