Curvature

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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 an 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.

1 Curvature of plane curves
- 1.1 Local expressions
- 1.2 Example
2 Curvature of space curve
3 Curvature of surfaces in 3-space
4 Curvature of space
5 See also
6 External links

[edit] 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 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 one-per-meter.

A straight line has curvature 0 everywhere; a circle of radius r has curvature 1/r everywhere.

[edit] Local expressions

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

$F[x,y]= \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}$

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

$\kappa=\frac{y''}{(1+y'^2)^{3/2}}$

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:

$\kappa\approx\frac{d^2y}{dx^2}$

may be used. This approximation yields a straightforward linear equation describing the phenomenon, which would otherwise remain intractable.

If curve is defined in polar coordinates as $r (θ)$ , its curvature is:

$\kappa(\theta) = \frac{r^2 + 2r'^2 - r r''}{\left(r^2+r'^2 \right)^{3/2}}$

where here the dash refers to differentiation with respect to $θ$ .

[edit] Example

Consider the parabola $y = x 2$ . We can parametrize the curve simply as $c (t) = (t, t 2) = (x, y)$ ,

$\dot{x}= 1,\quad\ddot{x}=0,\quad \dot{y}= 2t,\quad\ddot{y}=2$

Substituting

$\kappa(t)= \left|\frac{\dot{x}\ddot{y}-\dot{y}\ddot{x}}{({\dot{x}^2+\dot{y}^2)}^{3/2}}\right|= {1\cdot 2-(2t)(0) \over (1+(2t)^2)^{3/2} }={2 \over (1+4t^2)^{3/2}}$

[edit] Curvature of space curve

For a parametrically defined space curve its curvature is:

$F[x,y,z]=\frac{\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}$

[edit] Curvature of surfaces in 3-space

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

$K = \lim_{r \rarr 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}.$

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

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 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.

[edit] Curvature of space

In the theory of general relativity, which describes gravity and 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 or space-time without curvature is called flat. See also shape of the universe. 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.

[edit] 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.
Differential geometry of curves for a full treatment of curves embedded in an Euclidean space of arbitrary dimension.
Frenet-Serret formulas for the differential equations characterizing the trajectory of a one dimensional curve embedded in three dimensions.
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