Curvature
Wiki, Facts

 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 n-dimensional 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