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":

... 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) ...

**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 ...

**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 ...

... 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 ...