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

:<math>\int_a^b k(s)\,ds = 2\pi N.</math>

The total curvature of a closed curve is always an integer multiple of 2{{pi}}, where ''N'' is called the ''index of the curve'' or ''turning number'' – it is the [[Winding number|winding number]] of the unit [[Tangent vector|tangent vector]] about the origin, or equivalently the degree of the map to the [[Unit circle|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.

==Comparison to surfaces==
{{See|Gaussian_curvature#Total_curvature}}
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|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|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|polygonal chain]], with total curvature&nbsp;2{{pi}}.]]
A finite generalization is that the exterior angles of a triangle, or more generally any [[Simple polygon|simple polygon]], add up to 360°&nbsp;=&nbsp;2{{pi}} radians, corresponding to a turning number of 1. More generally, [[Polygonal chain|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 absolute curvature|total absolute curvature]] of a curve is defined in almost the same way as the total curvature, but using the absolute value of the curvature instead of the signed curvature.
It is 2{{pi}} for [[Convex curve|convex curve]]s in the plane, and larger for non-convex curves.<ref>{{citation
 | last = Chen | first = Bang-Yen
 | contribution = Riemannian submanifolds
 | doi = 10.1016/S1874-5741(00)80006-0
 | mr = 1736854
 | pages = 187–418
 | publisher = North-Holland, Amsterdam
 | title = Handbook of differential geometry, Vol. I
 | year = 2000}}. See in particular section 21.1, "Rotation index and total curvature of a curve", [https://books.google.com/books?id=U2BIz8NNksYC&pg=PA359 pp.&nbsp;359–360].</ref> It can also be generalized to curves in higher dimensional spaces by flattening out the [[Tangent developable|tangent developable]] to {{mvar|γ}} into a plane, and computing the total curvature of the resulting curve.  That is, the total curvature of a curve in {{mvar|n}}-dimensional space is

:<math>\int_a^b \left|\gamma''(s)\right|\sgn \kappa_{n-1}(s)\,ds</math>

where {{math|''κ''<sub>''n''&minus;1</sub>}} is last Frenet curvature (the torsion of the curve) and {{math|sgn}} is the signum function.

The minimum total absolute curvature of any three-dimensional curve representing a given knot is an [[Knot invariant|invariant]] of the knot. This invariant has the value 2{{pi}} for the unknot, but by the [[Fáry–Milnor theorem]] it is at least 4{{pi}} for any other knot.<ref>{{citation|title=On the Total Curvature of Knots|first=John W.|last=Milnor|journal=Annals of Mathematics | series = Second Series|volume=52|year=1950|pages=248–257|jstor=1969467|doi=10.2307/1969467|issue=2}}</ref>

==References==
{{Reflist}}

==Further reading==
*{{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
 | arxiv = math/0606007
 | contribution = Curves of finite total curvature
 | doi = 10.1007/978-3-7643-8621-4_7
 | mr = 2405664
 | pages = 137–161
 | publisher = Birkhäuser, Basel
 | series = Oberwolfach Semin.
 | title = Discrete differential geometry
 | volume = 38
 | year = 2008| s2cid = 117955587 }}

[[Category:Curves]]
[[Category:Curvature (mathematics)]]

{{Sourceattribution|Total curvature}}