Curvature

• For the meaning in architecture see curvature.
• For use in medicine, " small and large curvature " see stomach.
• See also: (homo) incurvatus in se ( " the crooked man to himself "); Krumm.

Curvature is a term from mathematics, indicated in its simplest sense, the local deviation of a curve from a straight line. The same term also stands for the curvature, which quantitatively indicates for each point of the curve, how strong is this local deviation.

Building on the concept of curvature for the curve can be the curvature of a surface in three-dimensional space described by examining the curves in the curvature of this surface. A certain part of the curvature information of an area, the Gaussian curvature, depends only on the geometry of the inner surface, i.e., the first fundamental form (or the metric tensor ), which specifies how the arc length is calculated from curves.

This intrinsic curvature term can be generalized to manifolds of arbitrary dimension with a metric tensor. On such manifolds of parallel transport along curves is explained and the curvature variables indicate how big is the change in direction of vectors in the parallel transport along closed curves after a round. One application is the theory of general relativity, which describes gravity as a curvature of spacetime. More generally, can be transferred to principal bundles with connection this term. These find application in the gauge theory, in which the curvature quantities (eg the electromagnetic field ) describe the strength of the fundamental interactions.

Curvature of a curve

Of the curvature of a plane curve is understood in the geometry, the change in direction when passing through the curve. The curvature of a straight line is everywhere equal to zero, because its direction does not change. A circle ( arc ) with the same curvature radius everywhere, because its direction changes everywhere equally strong. The smaller the radius of the circle, the greater its curvature. As a measure of curvature of a circle is used, the size, the ratio of central angle and length of a circular arc. The central angle is equal to the angle between the tangent to the circle at the end points of the arc. To define the curvature of a curve in an arbitrary point is considered in accordance with a curved section of length containing the point in question and whose tangents intersect at the ends at an angle. Thus, the curvature at the point is

Defined, if this derivative exists. Is the curvature at a point equal to zero, is defined as the reciprocal of the curvature as the radius of curvature; This is the radius of curvature of the circle through the point, that is the circle that best approximates the curve in this point. The center of this circle is the center of curvature, and can be constructed by the radius of curvature is removed perpendicular to the tangent of the curve, in the direction in which the curved bend.

If the curve is given as a graph of a function, then for the slope angle of the curve, ie with the chain rule. Applies or for the arc length. Thus we obtain for the curvature

Here, the curvature can be positive or negative, depending on whether the slope angle of the curve with increasing abscissa is increasing or decreasing, i.e., whether the function is convex or concave.

Definitions

Is the position vector of a point on the curve as a function of arc length. The curvature of the curve is then defined as

The curvature is given by the amount of discharge of the unit tangent vector of the arc length and thus indicates how quickly changes when passing through the curve, the tangent direction as a function of the arc length. The curvature at a point of the curve is independent of the selected parameterization according to the arc length.

For plane curves, one can define the curvature of related signed an orientation of the normal bundle of the curve. Such an orientation is provided by a continuous unit normal vector field along the curve. It always exists, since every plane curve is orientable. The curvature is non-zero, then the curve is signed by the dot product

Defined. The curvature is thus positive if they ( i.e. if equal to the principal normal unit vector ) in the direction of curves and negative if it curves in the opposite direction ( that is, if applies ). The definition is again independent of the parameterization according to the sheet length, but the sign is dependent on the choice of the longitudinal curve. The amount given above provides the definition of the curvature unsigned. In a left-hand curve is positive and negative in a right turn.

One regular parameterized curve in the plane can be assigned an orientation about the direction of passage. Is an additional orientation of the plane is specified, by an orientation on the normal bundle is induced. This is the unit normal vector, so that the overall base is favorably oriented. Thus the sign of the curvature of a curve parameterized function of the orientation of the plane and the passage sense of the parametric curve.

One curve, which is given as the zero set of a function with regular value, the curvature with respect to the restricted sign on the curve normalized gradient field can be assigned.

Properties

The circle of curvature is the uniquely determined circle whose contact is OK with the curve at the contact point. The curvature at a point is then exactly equal to zero when there is a contact order with the tangent. The evolute of a curve is the locus of their centers of curvature. This gives a center of curvature as the limit of intersections of two normals, the approach to each other. According to Cauchy can thus be defined as the curvature of a plane curve.

The curvature of a curve in space is like the turn of a bewegungsinvariante quantity that describes the local characteristics of the curve. Both sizes come in the Frenet formulas as coefficients.

If the curvature with sign for a parameterized by arc length curve in the plane oriented, then the following equations hold:

Each of the two equations is equivalent to the definition of the curvature with sign for parameterized curves. In Cartesian coordinates, the equations imply that form and a fundamental system of solutions for linear ordinary differential equation whose solution is given by with. From the figure, in turn, the parameterization of the curve is obtained by integration by arc length. The specification of a starting point, a starting direction and the curvature as a function of arc length thus determines the curve clearly. As is given by a rotation of the angle, it follows further that distinguish between two curves with the same curvature function only by an actual motion in the plane. Moreover, it follows from these considerations that the curvature with sign by

Is given, the angle of the tangent vector is a fixed direction and is measured in increasing positive rotation.

If one limits the parameterization of a plane curve in the vicinity of a point on the curve so that it is injective, then you can associate each curve point unambiguously assign the normal vector. This assignment may be considered as mapping of the curve in the unit circle, by tacking the normal vector to the origin of the coordinate system. A curved section of length containing the point, then a part of a curved section of the length on the unit circle. Then for the curvature at the point

This idea can be applied to surfaces in space by conceives a unit normal vector field on the surface as a mapping into the unit sphere. This figure is called the Gauss map. Looking at the ratio of surface areas instead of the arc lengths, providing thereby the patch in the unit sphere with a sign depending on whether the Gauss map preserves or reverses the sense of rotation of the boundary curve, then delivers the original definition of the Gaussian curvature by Gauss. However, the Gaussian curvature is an intrinsic size of the geometry, while a curve has no intrinsic curvature, for each parameterization by arc length is a local isometry between a subset of the real numbers and the curve.

If one considers a normal variation of a parameterized curve on a parameter interval designated by the arc length of the curve segment and is varied, the following applies

The curvature with sign on one point thus indicates how quickly changes the arc length of an infinitesimal curve segment at this point when a normal variation. On land transferred in space, this leads to the notion of mean curvature. The corresponding limit of the ratio of surface areas of normal variation then delivers twice the mean curvature.

This characterization of the curvature explains the following formula for the curvature with sign for a curve, which is given as the zero set of a function. Since the divergence of a vector field in the plane provides the rate of change of the content of an infinitesimal surface element with respect to the vector field for the associated flux divergence of the negative is obtained from the normalized gradient field with the curvature sign:

Where is the Hessian matrix, the trace and the identity matrix. For illustrations, this formula provides twice the mean curvature of surfaces as zero sets in space and is referred to as the formula of Bonnet. Written out and put into another form, the formula in the case of plane curves:

In this case, for example, designated by the partial derivative of the first argument, and the co-factor of. For illustrations of the second expression returns the Gaussian curvature for surfaces as zero sets in space.

Calculation of the curvature for parameterized curves

Is as defined above requires a parameterization of the curve by arc length. By reparametrization, one obtains a formula for arbitrary regular parametrizations. Summing up the first two derivatives of the columns of a matrix together, then the formula is

For plane curves is a square matrix and the formula is simplified by using the product rule for determinants to

, The plane given by the orientation of the standard, then the formula is obtained for the curvature signed by omitting the amount strokes in the numerator.

Level curves

Is the parameterization of the components and functions given, yields the formula for the curvature at the point with the sign of the expression

( The dots denote derivatives with respect to. )

This provides the following special cases:

However, one must note that the sign of the curvature of a curve from its representation, ie they pass depends sense. Since in polar coordinates, the radius by definition must be ≥ 0 always, this can lead to an unnatural curve run and have a different sign behavior of the curvature result. A good example is the lemniscate there - but it occurs even at the hyperbola, since it consists of two branches.

Space curves

For curves in three-dimensional space can be expressed as follows the general formula using the cross product:

Curvature of a surface

One arched regular surface is noticeable in its curvature at a square outwardly increasing deviation of the surface from its tangent plane. A stronger curvature then makes itself felt as a stronger deviation from the plane.

In differential geometry is seen at each point of the curvature radii of the curves of intersection with the built in normal levels (i.e., the surface perpendicular intersecting planes ). In this case, the curvature radii and curvature with respect to the sign of a unit normal vector field on the surface, limited to the plane cross-sectional curve associated. Under these radii of curvature, there is a maximum () and minimum (). The reciprocal values ​​and are called the principal curvatures. The corresponding curvature directions are orthogonal.

The Gaussian curvature and the mean curvature of a regular surface in a point are calculated as follows:

The overall curvature or total curvature of a surface is the integral of the Gaussian curvature of this area:

Curvature in Riemannian geometry

Since Riemannian manifolds are generally embedded in any space, in this branch of differential geometry, a curvature size is needed, which is independent of a surrounding space. For this, the Riemann curvature tensor has been introduced. This measures the extent to which the local geometry of the manifold of the laws of Euclidean geometry is different. Other curvature quantities are derived from the curvature tensor. The main curvature of the Riemannian geometry is the sectional curvature. This derived variable contains all the information contained in the Riemannian curvature tensor. Other simpler derived quantities are the Ricci curvature and the scalar curvature.

A curvature on a Riemannian manifold shown, for example, if one determines the ratio between circumference and radius within the manifold and set to the value that is obtained in a Euclidean space in relationship.

It is worth noting that one can define the surface of a torus, for example, a metric that has no curvature. This can be derived from the fact that you can form a torus as a quotient space of a flat surface.

Application in the theory of relativity

In general relativity, gravity is described by a curvature of space-time caused by the masses of the heavenly bodies. Body and light rays move along the geodesic curvature given by these webs. These tracks give the impression that a force being exerted on the corresponding body.