# curvature

Curvature is a measure of the amount by which a curve,
a surface, or any other manifold deviates from a straight line, a plane,
or a hyperplane (the multidimensional equivalent of a plane). For a plane
curve, the curvature at a given point has a magnitude equal to one over
the radius of an osculating circle (a
circle that "kisses," or just touches, the curve at the given point) and
is a vector pointing in the direction of
that circle's center. The smaller the radius *r* of the osculating
circle, the greater the magnitude of the curvature (1/*r*) will be.
A straight line has zero curvature everywhere; a circle of radius r has
a curvature of magnitude 1/*r* everywhere.

Another way to define the curvature at any point *P* on a plane curve
is the difference between the angle *θ* made by the tangent at that point with a fixed straight line, and the angle *θ* + *δθ* made by the tangent at the adjacent point *P'* with the same straight line. The curvature at *P* may therefore be
defined as the rate of change, *k*, or *θ* at that point.
In terms of differential calculus,
for the curve *y* = *f* (*x*)

## Types of curvature

For a two-dimensional surface, there are two kinds of curvature: a **Gaussian
curvature** (or **scalar curvature**) and a **mean
curvature**. To compute these at a given point, consider the intersection
of the surface with a plane containing a fixed normal vector (an arrow sticking
out perpendicularly) at the point. This intersection is a plane and has
a curvature; if the plane is varied, this curvature also changes, and there
are two extreme values – the maximal and the minimal curvature, which
are known as the **main curvatures**, 1/*R*_{1} and 1/*R*_{2}. (By convention, a curvature is taken to be positive
if its vector points in the same direction as the surface's chosen normal,
otherwise it is negative.) The Gaussian curvature is equal to the product
1/*R*_{1}*R*_{2}. It is everywhere positive for
a sphere, everywhere negative for a hyperboloid and pseudosphere, and everywhere zero
for a plane. It determines whether a surface has elliptic (when it is positive)
or hyperbolic (when it is negative) geometry at a point. The integral of
the Gaussian curvature over the whole surface is closely related to the
surface's Euler characteristic.
The mean curvature is equal to the sum of the main curvatures, 1/*R*_{1} + 1/*R*_{2}. A minimal surface, like that of
a soap film, has a mean curvature of zero.

## Curvature tensor

In the case of higher-dimensional manifolds, curvature is defined in terms of a curvature tensor, which describes what happens to a vector that is transported around a small loop of the manifold.

The curvature tensor is the basic mathematical object in Einstein's general theory of relativity. It determines how the tipping of light cones changes from time to time and place to place in the history of the universe.

## Curvature constant

The curvature constant is a constant (*k*) appearing in the Robertson-Walker metric which determines the curvature of the spatial
geometry of the universe. The three standard Friedmann models have: *k* > 0 for positive curvature (spherical geometry), *k* < 0 for negative curvature (hyperbolic geometry), and *k* = 0 for zero
curvature (flat geometry).