Curvature  

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In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold. This article deals primarily with the first concept.

The canonical example of extrinsic curvature is that of a circle, which everywhere has curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.

In a plane, this is a scalar quantity, but in three or more dimensions it is described by a curvature vector that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor.

The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.

See also

• Curvature form for the appropriate notion of curvature for vector bundles and principal bundles with connection
• Curvature of a measure for a notion of curvature in measure theory
• Curvature of Riemannian manifolds for generalizations of Gauss curvature to higher-dimensional Riemannian manifolds
• Curvature vector and geodesic curvature for appropriate notions of curvature of curves in Riemannian manifolds, of any dimension
• Curve
• Degree of curvature
• Differential geometry of curves for a full treatment of curves embedded in a Euclidean space of arbitrary dimension
• Dioptre a measurement of curvature used in optics
• Gauss-Bonnet theorem for an elementary application of curvature
• Gauss map for more geometric properties of Gauss curvature
• Hertz's principle of least curvature an expression of the Principle of Least Action
• Mean curvature at one point on a surface
• Minimum railway curve radius
• Radius of curvature
• Second fundamental form for the extrinsic curvature of hypersurfaces in general
• Torsion of a curve