Curvature
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Curvature is the amount by which a geometric object deviates from being flat. The word flat might have very different meanings depending on the objects considered (for curves it is a straight line and for surfaces it is a Euclidean plane).
In this article we consider the most basic examples: the curvature of a plane curve and the curvature of a surface in Euclidean space. See the links below for further reading.
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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), and is a vector pointing in the direction of that circle's center. The magnitude of curvature at points on physical curves can be measured in diopters (also spelled dioptre); a diopter has the dimension one per meter.
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.
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 <math>c(t) = (x(t),y(t))<math> the curvature is
- <math>\kappa= \frac{\dot{x}\ddot{y}-\dot{y}\ddot{x}}{({\dot{x}^2+\dot{y}^2)}^{3/2}}
<math>
where the dots denote differentiation with respect to t.
For a plane curve given implicitly as <math>f(x,y)=0<math> the curvature is
- <math>\kappa=
\nabla\cdot\left(\frac{\nabla f}{\|\nabla f\|}\right) <math>
that is, the divergence of the direction of the gradient of f. This last formula also gives the mean curvature of an hypersurface in Euclidean space (up to a constant).
Curvature of space curves
A full treatment of curves embedded in an Euclidean space of arbitrary dimension (a space curve) is given in the article on parametric curves.
Curvature of surfaces in 3-space
For two-dimensional surfaces embedded in R3, there are two kinds of curvature: Gaussian curvature and Mean curvature. To compute these at a given point of the surface, consider the intersection of the surface with a plane containing a fixed normal vector at the point. This intersection is a plane curve and has a curvature; if we vary the plane, this curvature will change, and there are two extremal values - the maximal and the minimal curvature, called the principal curvatures, k1 and k2, the extremal directions 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. She 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, she 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
- <math>
K = \lim_{r \rarr 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}.<math>
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 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 like a soap film has mean curvature zero and soap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature 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
In cosmology, the concept of "curvature of space" is considered, which is the curvature of corresponding pseudo-Riemannian manifolds, see curvature of Riemannian manifolds.
A space without curvature is called a "flat space" or Euclidean 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 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.
- Gauss map for more geometric properties of Gauss curvature.
- Gauss-Bonnet theorem for an elementary application of curvature.