# Dictionary Definition

curvature

### Noun

1 (medicine) a curving or bending; often
abnormal; "curvature of the spine"

2 the rate of change (at a point) of the angle
between a curve and a tangent to the curve

3 the property possessed by the curving of a line
or surface [syn: curve]

# User Contributed Dictionary

## English

### Noun

- The shape of something curved.

## Italian

### Noun

curvature- Plural of curvatura

# Extensive Definition

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, 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. 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.

## One dimension in two dimensions: Curvature of plane curves

For a plane curve
C, the mathematical definition of curvature uses a parametric
representation of C with respect to the arc length
parametrization. It can be computed given any regular
parametrization by a more complicated formula given below. Let
γ(s) be a regular
parametric curve, where s is the arc length, or
natural parameter. This determines the unit tangent vector T, the
unit normal vector N, the curvature κ(s), the signed
curvature k(s) and the radius of curvature at each point:

- T(s)=\gamma'(s),\quad T'(s)=k(s)N(s),\quad \kappa(s) = \|\gamma(s)\| = \left|k(s)\right|, \quad R(s)=\frac.

The curvature of a straight
line is identically zero. The curvature of a circle of radius R'' is constant,
i.e. it does not depend on the point and is equal to the reciprocal
of the radius:

- \kappa = \frac.

Thus for a circle, the radius of curvature is
simply its radius. Straight lines and circles are the only plane
curves whose curvature is constant. Given any curve C and a point P
on it where the curvature is non-zero, there is a unique circle
which most closely approximates the curve near P, the osculating
circle at P. The radius of the osculating circle is the radius
of curvature of C at this point.

### The meaning of curvature

Suppose that a particle moves on the plane with
unit speed. Then the trajectory of the particle will trace out a
curve C in the plane. Moreover, taking the time as the parameter,
this provides a natural parametrization for C. The instanteneous
direction of motion is given by the unit tangent vector T and the
curvature measures how fast this vector rotates. If a curve keeps
close to the same direction, the unit tangent vector changes very
little and the curvature is small; where the curve undergoes a
tight turn, the curvature is large.

The magnitude of curvature at points on physical
curves can be measured in diopters (also spelled dioptre)
— this is the convention in optics. A diopter has the
dimension \scriptstyle[.

### Local expressions

For a plane curve given parametrically as c(t) =
(x(t),y(t)), the curvature is

- F[x,y]= \frac.

For the less general case of a plane curve given
explicitly as y=f(x) the curvature is

- \kappa=\frac.

This quantity is common in physics and engineering; for example, in
the equations of
bending in beams, the 1D
vibration
of a tense string, approximations to the fluid flow around surfaces
(in aeronautics), and the free surface boundary conditions in ocean
waves. In such applications, the assumption is almost always made
that the slope is small
compared with unity, so that the approximation:

- \kappa \approx \frac

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(\theta), then its curvature is

- \kappa(\theta) = \frac

where here the prime refers to differentiation
with respect to \theta.

### Example

Consider the parabola y = x^2. We can parametrize the curve simply as c(t) = (t, t^2) = (x, y),- \dot= 1,\quad\ddot=0,\quad \dot= 2t,\quad\ddot=2

- \kappa(t)= \left|\frac\right|= =

## One dimension in three dimensions: Curvature of space curves

- See Frenet-Serret formulas for a fuller treatment of curvature and the related concept of torsion.

For a parametrically defined space curve its
curvature is:

- F[x,y,z]=\frac

- \kappa = \frac

## Two dimensions: Curvature of surfaces

In contrast to curves, which do not have
intrinsic curvature, but do have extrinsic curvature (they only
have a curvature given an embedding), surfaces have intrinsic
curvature, independent of an embedding.

For a two-dimensional surface embedded in
R3,
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, k1 and k2, 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, k1k2. 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 convex (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 R3, 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) = 2πr. 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_ (2 \pi r - \mbox(r)) \cdot \frac.

The integral of the Gaussian
curvature over the whole surface is closely related to the
surface's Euler
characteristic; see the Gauss-Bonnet
theorem.

The discrete analog of curvature, corresponding
to curvature being concentrated at a point and particularly useful
for polyhedra, is the
(angular)
defect; the analog for the Gauss-Bonnet
theorem is
Descartes' theorem on total angular defect.

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,
k1+k2, 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 such as a soap film, has
mean curvature zero and a 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.

## Three dimensions: 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;
mathematically these are strong conditions, but they correspond to
reasonable physical assumptions (all points and all directions are
indistinguishable). 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 a 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
- Dioptre a measurement of curvature used in optics.

## External links

- 3D-XplorMath Java applets Applets for space curves with osculating circles.
- The History of Curvature

## References

Coolidge, J.L. "The Unsatisfactory Story of Curvature". The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 375-379curvature in Catalan: Curvatura

curvature in Czech: Křivost křivky

curvature in German: Krümmung

curvature in Spanish: Curvatura

curvature in French: Courbure

curvature in Korean: 곡률

curvature in Italian: Curvatura

curvature in Lithuanian: Kreivumas

curvature in Hungarian: Görbület

curvature in Japanese: 曲率

curvature in Polish: Krzywizna krzywej

curvature in Portuguese: Curvatura

curvature in Russian: Кривизна

curvature in Finnish: Kaarevuus

curvature in Swedish: Krökning

curvature in Ukrainian: Кривина
(математика)

curvature in Chinese: 曲率