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In mathematics, 'curvature' refers to 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,'' 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 differential manifold. This article deals primarily with the first concept.
The primordial example of extrinsic curvature is that of a circle which has curvature equal to the inverse of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature. Further, 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 direction of the bend as well as its sharpness. The curvature of more complex objects (such as surfaces or even curved ''n''-dimensional spaces) are 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.

Contents

Curvature of plane curves

Local expressions

Example

Curvature of space curve

Curvature of 2-dimensional surfaces

Curvature of space

See also

Curvature of plane curves

For a plane curve ''C'', the curvature at a given point ''P'' has a magnitude equal to the ''reciprocal'' of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point, its center shaping the curve's evolute), and is a vector pointing in the direction of that circle's center. The smaller the radius ''r'' of the osculating circle, the larger the magnitude of the curvature (1/''r'') will be; so that where a curve is "nearly straight", the curvature will be close to zero, and where the curve undergoes a tight turn, the curvature will be large in magnitude.
The magnitude of curvature at points on physical curves can be measured in diopters (also spelled dioptre); a diopter has the dimension ''length^-1''.
A straight line has curvature 0 everywhere; a circle of radius ''r'' has curvature 1/''r'' everywhere.

Local expressions

For a plane curve given parametrically as $c(t)\; =\; (x(t),y(t)).$
the curvature is
:
For the less general case of a plane curve given explicitly as $y=f(x)$ the curvature is
:
This form is widely used in engineering, for example; to derive the equations of bending of beams, deriving approximations to the fluid flow around surfaces (in aeronautics) and the free surface boundary conditions in ocean waves. In all such applications, the assumption is made that the slope is small compared with unity, so that the approximation:
:
may be used. This approximation yields a straightforward linear equation describing the phenomenon, which would otherwise remain intractable.
If a curve is defined in polar coordinates as $r(\; heta)$, then its curvature is

ight)^{3/2}}
where here the prime refers to differentiation with respect to $heta$.

Example

Consider the parabola $y\; =\; x^2$. We can parametrize the curve simply as $c(t)\; =\; (t,\; t^2)\; =\; (x,\; y)$,
: $dot\{x\}=\; 1,quadddot\{x\}=0,quad\; dot\{y\}=\; 2t,quadddot\{y\}=2.$
Substituting
:
ight|= {1cdot 2-(2t)(0) over (1+(2t)^2)^{3/2} }={2 over (1+4t^2)^{3/2}}.

Curvature of space curve

:''See Frenet-Serret formulas for a fuller treatment curvature and the related concept of torsion.''
For a parametrically defined space curve its curvature is:
:
Given a function ''r''(''t'') with values in 'R'³, the curvature at a given value of $t$ is
:

Curvature of 2-dimensional surfaces

For a two-dimensional surface embedded in 'R'³, consider the intersection of the surface with a plane containing the normal vector and one of the tangent vectors at a particular point. This intersection is a plane curve and has a curvature. This is the 'normal curvature', and it varies with the choice of the tangent vector. The maximum and minimum values of the normal curvature at a point are called the 'principal curvatures', ''k''₁ and ''k''₂, and the directions of the corresponding tangent vectors are called 'principal directions'.
Here we adopt the convention that a curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, otherwise negative.
The 'Gaussian curvature', named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, ''k''₁''k''₂. It has the dimension of 1/length² and is positive for spheres, negative for one-sheet hyperboloids and zero for planes. It determines whether a surface is locally (when it is positive) or locally saddle (when it is negative).
The above definition of Gaussian curvature is ''extrinsic'' in that it uses the surface's embedding in 'R'³, normal vectors, external planes etc. Gaussian curvature is however in fact an ''intrinsic'' property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss' celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.
An intrinsic definition of the Gaussian curvature at a point ''P'' is the following: imagine an ant which is tied to ''P'' with a short thread of length ''r''. He runs around ''P'' while the thread is completely stretched and measures the length C(''r'') of one complete trip around ''P''. If the surface were flat, he would find C(''r'') = 2π''r''. On curved surfaces, the formula for C(''r'') will be different, and the Gaussian curvature ''K'' at the point ''P'' can be computed as
:$$
K = lim_{r
arr 0} (2 pi r - mbox{C}(r)) cdot rac{3}{pi r^3}.
The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss-Bonnet theorem.
Because curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.
The 'mean curvature' is equal to the sum of the principal curvatures, ''k''₁+''k''₂, over 2. It has the dimension of 1/length. Mean curvature is closely related to the first variation of surface area, in particular a minimal surface like a soap film has mean curvature zero and soap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.

Curvature of space

By extension of the former argument, a space of three or more dimensions can be intrinsically curved; the full mathematical description is described at curvature of Riemannian manifolds. Again, the curved space may or may not be conceived as being embedded in a higher-dimensional space. In recent physics jargon, the embedding space is known as the 'bulk' and the embedded space as a '''p''-brane' where ''p'' is the number of dimensions; thus a surface (membrane) is a 2-brane; normal space is a 3-brane etc.
After the discovery of the intrinsic definition of curvature, which is closely connected with non-Euclidean geometry, many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In the theory of general relativity, which describes gravity and cosmology, the idea is slightly generalised to the "curvature of space-time"; in relativity theory space-time is a pseudo-Riemannian manifold. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the underlying space-time curvature that is physically significant.
Although an arbitrarily-curved space is very complex to describe, the curvature of a space which is locally isotropic and homogeneous is described by a single Gaussian curvature, as for a surface. A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. An example of negatively curved space is hyperbolic geometry. A space or space-time without curvature (formally, with zero curvature) is called 'flat'. For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat space-time. There are other examples of flat geometries in both settings, though. A torus or a cylinder can both be given flat metrics, but differ in their topology. Other topologies are also possible for curved space. See also shape of the universe.

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.

★ Differential geometry of curves for a full treatment of curves embedded in an Euclidean space of arbitrary dimension.

★ Gauss map for more geometric properties of Gauss curvature.

★ Gauss-Bonnet theorem for an elementary application of curvature.

★ Mean curvature at one point on a surface

★ Hertz's principle of least curvature an expression of the Principle of Least Action