 In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry
Curvature facts
 Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as 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 in terms of the lengths of curves within a Riemannian manifold
 This article deals primarily with the extrinsic concept
 The canonical example of extrinsic curvature is that of a circle, which has a curvature equal to the reciprocal of its radius everywhere
 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
 More commonly curvature is a scalar quantity, but one may also define 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 ndimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor
 This article sketches the mathematical framework which describes the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space
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