Curvature

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 the links below for further reading.

Read more about Curvature:  Curvature of Plane Curves, Curvature of Space Curves, Curves On Surfaces, Higher Dimensions: Curvature of Space, Generalizations

Other articles related to "curvature":

Lipid Bilayer Mechanics - Bending Modulus
... Bending modulus is defined as the energy required to deform a membrane from its intrinsic curvature to some other curvature ... For an ideal bilayer the intrinsic curvature is zero, so this expression is somewhat simplified ... each face must deform to accommodate a given curvature (see bending moment) ...
Basic Results of Small Cancellation Theory - More General Curvature
... More generally, it is possible to define various sorts of local "curvature" on any van Kampen diagram to be - very roughly - the average excess of ...
Lipid Bilayer Mechanics - Curvature
... primary factor governing which structure a given lipid forms is its shape (i.e.- its intrinsic curvature) ... Intrinsic curvature is defined by the ratio of the diameter of the head group to that of the tail group ... PC lipids, this ratio is nearly one so the intrinsic curvature is nearly zero ...
Curvature - Generalizations
... The mathematical notion of curvature is also defined in much more general contexts ... of these generalizations emphasize different aspects of the curvature as it is understood in lower dimensions ... The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve analogously, curvature in higher dimensions can be ...
Schur's Lemma (from Riemannian Geometry)
... Then if the sectional curvature is pointwise constant, that is, there exists some function such that for all and all then is constant, and the manifold has ...