About: Total curvature

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In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: 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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rdfs:label	Totalkrümmung (de) Total curvature (en) Варіація повороту кривої (uk) 总曲率 (zh)
rdfs:comment	In der Kurventheorie, einem Teilgebiet der Mathematik, wird die Totalkrümmung einer Kurve definiert als das Integral ihrer Krümmung , also als . (de) In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: 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. (en) Варіація повороту кривої — інтеграл кривини кривої за її довжиною. (uk) 在数学中的曲线微分几何的研究中，一个浸入在平面上的曲线的总曲率是曲率的曲线积分：闭曲线的总曲率是 2π 的整数倍，该整数称为曲线的指数或转数。其中转数是单位切向量关于起点的绕数，或者等价的高斯映射的次数。局部不变量曲率和整体拓扑不变量指数的关系是高维黎曼几何的代表性结果，如高斯－博內定理。 (zh)
foaf:depiction
dcterms:subject	Curvature (mathematics) Curves
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Link from a Wikipage to another Wikipage	Torsion of curves Arc length Unit circle Topological invariant Convex curve Mathematics Gauss map Gauss–Bonnet theorem Curvature (mathematics) Simple polygon Fáry–Milnor theorem Curvature Tangent developable Regular homotopy Riemannian geometry Plane curve Curves Winding number Whitney–Graustein theorem Integral Knot (mathematics) Knot invariant Signum function Differential geometry of curves Immersion (mathematics) Total absolute curvature Polygonal chain Tangent vector Turning number
sameAs	Total curvature Total curvature Total curvature Total curvature Total curvature Total curvature Total curvature
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has abstract	In der Kurventheorie, einem Teilgebiet der Mathematik, wird die Totalkrümmung einer Kurve definiert als das Integral ihrer Krümmung , also als . (de) In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: 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. (en) Варіація повороту кривої — інтеграл кривини кривої за її довжиною. (uk) 在数学中的曲线微分几何的研究中，一个浸入在平面上的曲线的总曲率是曲率的曲线积分：闭曲线的总曲率是 2π 的整数倍，该整数称为曲线的指数或转数。其中转数是单位切向量关于起点的绕数，或者等价的高斯映射的次数。局部不变量曲率和整体拓扑不变量指数的关系是高维黎曼几何的代表性结果，如高斯－博內定理。 (zh)
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foaf:isPrimaryTopicOf	wikipedia-en:Total_curvature
is Link from a Wikipage to another Wikipage of	Enneper surface Convex curve Quadrisecant Gauss–Bonnet theorem Fáry–Milnor theorem Regular homotopy Cohn-Vossen's inequality Winding number Knot invariant Cartan–Hadamard conjecture Tangent indicatrix Immersion (mathematics) Total absolute curvature
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