How do you define the curvature

curvature

This article covers the mathematical term. For the concept of architecture see curvature. For the curvature of the stomach, see stomach.

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

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

This intrinsic curvature concept can be generalized to manifolds of any dimension with a metric tensor. The parallel transport along curves is explained on such manifolds, and the curvature quantities indicate how large the change in direction of vectors is after one revolution during parallel transport along closed curves. One application is the general theory of relativity, which describes gravity as a curvature of space-time. This term can be applied even more generally to main fiber bundles with a connection. These are used in calibration theory, in which the curvature quantities describe the strength of the fundamental interactions (e.g. the electromagnetic field).

Curvature of a curve

Curvature of the circle: $ \ kappa = \ tfrac {1} {r} = \ tfrac {\ varphi} {s} $
Curve and its circle of curvature at the point of the curve P

In geometry, the curvature of a plane curve is the change in direction when moving through the curve. The curvature of a straight line is zero everywhere because its direction does not change. A circle (arc) with the radius $ r $ has the same curvature everywhere, because its direction changes equally strongly everywhere. The smaller the radius of the circle, the greater its curvature. The measure of the curvature of a circle is $ \ tfrac 1r = \ tfrac {\ Delta \ varphi} {\ Delta s} $, the ratio of central angle and length of a circular arc. The central angle is equal to the outer angle between the circular tangents in the end points of the circular arc. In order to define the curvature of any curve at a point, one considers a curve segment of length $ \ Delta s $, which contains the point in question and whose tangents intersect at the end points at the angle $ \ Delta \ varphi $. So the curvature $ \ kappa $ in the point becomes through

$ \ kappa: = \ lim _ {\ Delta s \ rightarrow 0} \ frac {\ Delta \ varphi} {\ Delta s} = \ frac {\ mathrm {d} \ varphi} {\ mathrm {d} s} $

defined, if this differential quotient exists. If the curvature at a point is not equal to zero, then the reciprocal of the curvature is called the radius of curvature; this is the radius of the circle of curvature through this point, i.e. the circle that best approximates the curve at this point. The center of this circle is called the center of curvature and can be constructed by plotting the radius of curvature perpendicular to the tangent of the curve in the direction in which the curve bends.

If the curve is given as a graph of a function $ f \ colon \ R \ to \ R, \, y = f (x) $, then the slope angle $ \ varphi $ of the curve $ \ tfrac {\ mathrm {d} y applies } {\ mathrm {d} x} = \ tan \ varphi $, i.e. with the chain rule $ \ tfrac {\ mathrm {d} ^ 2y} {\ mathrm {d} x ^ 2} = (1+ \ tan ^ 2 \ varphi) \ tfrac {\ mathrm {d} \ varphi} {\ mathrm {d} x} = \ left (1+ \ left (\ tfrac {\ mathrm {d} y} {\ mathrm {d} x} \ right) ^ 2 \ right) \ tfrac {\ mathrm {d} \ varphi} {\ mathrm {d} x} $. For the arc length $ s $, $ \ mathrm {d} s ^ 2 = \ mathrm {d} x ^ 2 + \ mathrm {d} y ^ 2 $ or $ \ tfrac {\ mathrm {d} s} {\ mathrm {d} x} = \ sqrt {1+ \ left (\ tfrac {\ mathrm {d} y} {\ mathrm {d} x} \ right) ^ 2} $. This gives for the curvature

$ \ kappa = \ frac {\ mathrm {d} \ varphi} {\ mathrm {d} s} = \ frac {\ frac {\ mathrm {d} \ varphi} {\ mathrm {d} x}} {\ frac {\ mathrm {d} s} {\ mathrm {d} x}} = \ frac {\ frac {\ mathrm {d} ^ 2y} {\ mathrm {d} x ^ 2}} {\ left (1+ \ left (\ frac {\ mathrm {d} y} {\ mathrm {d} x} \ right) ^ 2 \ right) ^ {\ frac {3} {2}}}. $

The curvature can be positive or negative, depending on whether the angle of rise $ \ varphi $ of the curve is increasing or decreasing with increasing abscissa $ x $, i.e. H. whether the function is convex or concave.

Definitions

Animations of the curvature and the "acceleration vector" $ \ mathrm {d} ^ 2 \ vec {r} / \ mathrm {d} s ^ 2 $

$ \ vec {r} (s) \ in \ R ^ p $ is the position vector of a point on the curve as a function of the arc length $ s $. The curvature $ {\ kappa \,} $ of the curve is then defined as

$ \ kappa = \ left | \ frac {\ mathrm {d} ^ 2 \ vec {r}} {\ mathrm {d} s ^ 2} \ right |. $

The curvature is given by the amount of the derivative of the unit tangent vector $ \ vec t = \ frac {\ mathrm {d} \ vec {r}} {\ mathrm {d} s} $ according to the arc length and thus indicates how fast the tangent direction changes depending on the arc length when running through the curve. The curvature at one point of the curve is independent of the selected parameterization according to the arc length.

You can use the Signed curvature define with respect to an orientation of the normal bundle of the curve. Such an orientation is given by a continuous unit normal vector field $ \ vec N $ along the curve. It always exists because every flat curve can be oriented. If the curvature is not zero, then the curvature is signed by the scalar product

$ \ kappa = \ vec N \ cdot \ frac {\ mathrm {d} \ vec {t}} {\ mathrm {d} s} $

Are defined. The curvature is therefore positive if it bends in the direction of $ \ vec N $ (i.e. if $ \ vec N $ equals the main normal unit vector $ \ vec n = \ frac {\ vec t \, '} {| \ vec t \ , '|} $ with $ \ vec t \,': = \ frac {\ mathrm {d} \ vec t} {\ mathrm {d} s} $) and negative if it curves in the opposite direction ( ie if $ \ vec N = - \ vec n $ applies). The definition is again independent of the parameterization according to the arc length, but the sign depends on the choice of $ \ vec N $ along the curve. The amount $ | \ kappa | $ returns the unsigned definition of curvature given above.

An orientation can be assigned to a regularly parameterized curve in the plane via the direction of travel. If an orientation of the plane is also specified, this induces an orientation on the normal bundle. For this, let $ \ vec N (s) $ be the unit normal vector, so that the ordered basis $ (\ vec t (s), \ vec N (s)) $ is positively oriented. This means that the sign of the curvature of a parameterized curve is dependent on the orientation of the plane and the direction of passage of the parameterized curve. In a left turn $ \ kappa $ is positive and in a right turn it is negative.

A curve $ C = f ^ {- 1} (0) $, which is given as a set of zeros of a function $ f \ colon \ R ^ 2 \ to \ R $ with regular value $ 0 \ in \ R $, can have the curvature with the sign with respect to the normalized gradient field $ \ vec N = \ left. \ frac {\ nabla f} {| \ nabla f |} \ right | _C $.

properties

The circle of curvature is the uniquely determined circle whose order of contact with the curve at the point of contact is $ \ geq 2 $. The curvature in a point is zero if and only if the contact order with the tangent $ \ geq 2 $ is there. The evolution of a curve is the locus of its centers of curvature. A center of curvature is obtained as the limit value of the intersection points of two normals which approach each other. According to Cauchy, this can be used to define the curvature of a plane curve.[1]

The curvature of a space curve, like the winding, is a movement-invariant quantity that describes the local course of a curve. Both quantities appear as coefficients in the Frenet formulas.

If $ \ kappa $ is the signed curvature for a curve parameterized according to the arc length in the oriented plane, then the following equations apply:

$ \ begin {align} \ frac {\ mathrm {d} \ vec {t}} {\ mathrm {d} s} & = & \ kappa \ vec N \ \ frac {\ mathrm {d} \ vec {N }} {\ mathrm {d} s} & = - \ kappa \ vec t & \ end {align} $

Either of the two equations is equivalent to defining signed curvature for parameterized curves. In Cartesian coordinates, the equations mean that $ \ vec t $ and $ \ vec N $ are a fundamental system of solutions to the linear ordinary differential equation

$ \ vec x '(s) = \ kappa (s) \ begin {pmatrix} 0 & -1 \ 1 & 0 \ end {pmatrix} \ cdot \ vec x (s) $

form their solution by

$ \ vec x (s) = \ begin {pmatrix} \ cos \ varphi (s) & - \ sin \ varphi (s) \ sin \ varphi (s) & \ cos \ varphi (s) \ end {pmatrix } \ cdot \ vec x (s_0) $

With

$ \ varphi (s) = \ int_ {s_0} ^ s \ kappa (\ bar s) d \ bar s $

given is. From the mapping $ s \ mapsto \ vec t (s) $ one obtains the parameterization $ s \ mapsto \ vec r (s) $ of the curve according to the arc length by integration. The specification of a starting point $ \ vec r (s_0) $, a starting direction $ \ vec t (s_0) $ and the curvature $ s \ mapsto \ vec \ kappa (s) $ as a function of the arc length clearly determines the curve. Since $ \ vec t (s) $ is given by a rotation of $ \ vec t (s_0) $ by the angle $ \ varphi (s) $, it also follows that two curves with the same curvature function can only be separated by an actual movement in the level differ. In addition, it follows from these considerations that the curvature is signed by

$ \ kappa = \ frac {\ mathrm {d} \ varphi} {\ mathrm {d} s} $

is given, where $ \ varphi \ in \ R $ is the angle of the tangent vector to a fixed direction and is measured increasing in the positive direction of rotation.

If one restricts the parameterization of a plane curve in the vicinity of a curve point $ p $ so that it is injective, then one can uniquely assign the normal vector $ \ vec N (s) $ to each curve point $ q = \ vec r (s) $ . This assignment can be understood as mapping the curve into the unit circle by attaching the normal vector to the origin of the coordinate system. A curve segment of length $ \ Delta s $ that contains the point $ p $ then has a curve segment on the unit circle of length $ \ Delta \ tilde s $. The following then applies to the curvature at point $ p $

$ \ kappa = \ lim _ {\ Delta s \ rightarrow 0} \ frac {\ Delta \ tilde s} {\ Delta s}. $

This idea can be transferred to surfaces in space by understanding a unit normal vector field on the surface as a mapping into the unit sphere. This mapping is called a Gaussian map. If one looks at the ratio of areas instead of the arc lengths and gives the area in the unit sphere a sign, depending on whether the Gaussian map preserves or reverses the direction of rotation of the boundary curve, then this provides the original Gaussian definition of the Gaussian curvature. However, the Gaussian curvature is a quantity of the intrinsic geometry, while a curve has no intrinsic curvature, because every parameterization according to the arc length is a local isometry between a subset of the real numbers and the curve.

If one considers a normal variation $ \ vec r (t) + \ varepsilon \ vec N (t) $ of a parameterized curve with $ \ varepsilon \ in \ R $ on a parameter interval $ \ Delta t $, denoted by $ \ Delta s $ the arc length of the varied curve segment and sets $ \ Delta s_0 = \ Delta s | _ {\ varepsilon = 0} $, then applies

$ \ kappa = - \ lim _ {\ Delta t \ rightarrow 0} \ frac {1} {\ Delta s_0} \ left. \ frac {\ mathrm {d} \ Delta s} {\ mathrm {d} \ varepsilon} \ right | _ {\ varepsilon = 0}. $

The signed curvature at a point indicates how quickly the arc length of an infinitesimal curve segment changes at this point with a normal variation. Transferred to surfaces in space, this leads to the concept of mean curvature. The corresponding limit value of the ratio of areas with normal variation then provides twice the mean curvature.

This characterization of the curvature explains the following formula for the signed curvature for a curve which is the set of zeros $ f ^ {- 1} (0) $ of a function $ f \ colon \ R ^ 2 \ to \ R, \; (x , y) \ mapsto f (x, y) $ is given. Since the divergence of a vector field in the plane provides the rate of change of the content of an infinitesimal area with respect to the flow belonging to the vector field, the negative divergence of the normalized gradient field is used to obtain the curvature with sign:

$ \ kappa = - \ nabla \ cdot \ vec N = - \ nabla \ cdot \ frac {\ nabla f} {| \ nabla f |} = - \ vec N ^ T \ cdot (\ tilde H_f- \ operatorname {sp } (\ tilde H_f) E) \ cdot \ vec N, $

where $ \ tilde H_f: = - \ tfrac {H_f} {| \ nabla f |} $ with the Hesse matrix $ H_f $, $ \ operatorname {sp} (\ tilde H_f) $ the trace and $ E $ the identity matrix is. For mappings $ f \ colon \ R ^ 3 \ to \ R $ this formula delivers twice the mean curvature of surfaces as sets of zeros in space and is called the Bonnet formula. In the case of plane curves, the formula is written out and given a different form:

$ \ kappa = - \ frac {f_y ^ 2f_ {xx} -2f_xf_yf_ {xy} + f_x ^ 2f_ {yy}} {(f_x ^ 2 + f_y ^ 2) ^ {3/2}} = \ vec N ^ T \ cdot \ operatorname {adj} (\ tilde H_f) \ cdot \ vec N. $

Here, z. B. $ f_x $ the partial derivative of $ f $ according to the first argument and $ \ operatorname {adj} (\ tilde H_f) $ the adjuncts of $ \ tilde H_f $. For mappings $ f \ colon \ R ^ 3 \ to \ R $, the second expression gives the Gaussian curvature for surfaces as sets of zeros in space.

Calculation of curvature for parameterized curves

The definition given above presupposes a parameterization of the curve according to the arc length. By re-parameterization one obtains a formula for any regular parameterization $ t \ mapsto \ vec r (t) \ in \ R ^ p $. If one summarizes the first two derivatives of $ \ vec r $ as columns of a matrix $ A (t) = (\ vec r '(t), \ vec r' '(t)) $, then the formula is

$ \ kappa = \ frac {\ sqrt {\ det (A ^ T \ cdot A)}} {| \ vec r '| ^ 3} $.

For plane curves, $ A (t) $ is a square matrix and the formula is simplified to using the product rule for determinants

$ \ kappa = \ frac {| \ det A |} {| \ vec r '| ^ 3} $.

If the plane is given by the $ \ R ^ 2 $ with the standard orientation, then the formula for the curvature with sign is obtained by omitting the absolute lines in the numerator.

Flat curves

If the parameterization is given by the component functions $ x $ and $ y $, then the formula for the signed curvature in the point $ \ vec r (t) = (x (t), y (t)) $ provides the expression

$ \ kappa (t) = \ frac {\ dot x (t) \ ddot y (t) - \ ddot x (t) \ dot y (t)} {\ big (\ dot x (t) ^ 2 + \ dot y (t) ^ 2 \ big) ^ {3/2}} $.

(The points denote derivatives according to $ t $.)

This provides the following special cases:

case 1
The curve is the graph of a function $ f $. The curvature at the point $ \ vec r (x) = \ left (x, f (x) \ right) $ results from
$ \ kappa (x) = \ frac {f '' (x)} {\ left (1 + f '(x) ^ 2 \ right) ^ {3/2}} $.
Case 2
The curve is given in polar coordinates, i.e. by an equation $ r = f (\ varphi) $. In this case one obtains the formula for the curvature in the point $ \ vec r (\ varphi) = f (\ varphi) \ cdot (\ cos \ varphi, \ sin \ varphi) $
$ \ kappa (\ varphi) = \ frac {(f (\ varphi)) ^ 2 + 2 (f '(\ varphi)) ^ 2 - f (\ varphi) f' '(\ varphi)} {\ left [ (f (\ varphi)) ^ 2 + (f '(\ varphi)) ^ 2 \ right] ^ {3/2}} $.

Space curves

For curves in the three-dimensional space $ \ mathbb {R} ^ 3 $, the general formula can be expressed using the cross product as follows:

$ \ kappa (t) = \ frac {| \ vec {r} \, '(t) \ times \ vec {r} \,' '(t) |} {| \ vec {r} \,' (t ) | ^ 3} $

Curvature of a surface

The curvature of a curved regular surface is noticeable by an outwardly quadratically increasing deviation of the surface from its tangential plane. A increased curvature then becomes noticeable as a greater deviation from the plane.

In differential geometry, the radii of curvature of the intersection curves with the normal planes established in $ p $ (i.e.the surface of perpendicular intersecting planes). The sign of a unit normal vector field on the surface, restricted to the plane intersection curve, is assigned to the radii of curvature and curvatures. Among these radii of curvature there is a maximum ($ R_1 $) and a minimum ($ R_2 $). The reciprocal values ​​$ k_1 = \ tfrac1 {R_1} $ and $ k_2 = \ tfrac1 {R_2} $ are called main curvatures. The corresponding directions of curvature are perpendicular to one another.

The Gaussian curvature $ K $ and the mean curvature $ H $ of a regular surface at a point $ p $ are calculated as follows:

$ K = \ frac {1} {R_1} \ cdot \ frac {1} {R_2} = k_ {1} \ cdot k_ {2} $
$ H = \ frac {1} {2} \ left (\ frac {1} {R_1} + \ frac {1} {R_2} \ right) = \ frac {1} {2} (k_ {1} + k_ {2}) $

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

$ C = \ int K \, dA = \ int k_ {1} k_ {2} \, dA $

Curvature in Riemannian geometry

Since Riemannian manifolds are generally not embedded in any space, a curvature quantity is needed in this sub-area of ​​differential geometry that is independent of a surrounding space. For this purpose, the Riemann curvature tensor was introduced. This measures the extent to which the local geometry of the manifold deviates from the laws of Euclidean geometry. Further curvature quantities are derived from the curvature tensor. The most important curvature of Riemannian geometry is the cutting curvature. This derived quantity contains all information that is also contained in the Riemann curvature tensor. Other simpler derived quantities are the Ricci curvature and the scalar curvature.

A curvature on a Riemannian manifold can be seen, for example, if the ratio between the circumference and the radius within the manifold is determined and compared to the value $ 2 \ pi $ that is obtained in a Euclidean space.

It is noteworthy that, for example, a metric can be defined on the surface of a torus that has no curvature. This can be deduced from the fact that a torus can be formed as a quotient space from a flat surface.

Application in the theory of relativity

In the general theory of relativity, gravity is described by a curvature of space-time, which is caused by the masses of the celestial bodies. Bodies and rays of light move on the geodetic paths determined by this curvature. These tracks give the impression that a force is being exerted on the corresponding body.

literature

  • Wolfgang Walter: Analysis II. Springer, 1991, 2nd edition, ISBN 3-540-54566-2, pp. 171-174.
  • Konrad Köngisberger: Analysis 1. 2nd edition, Springer, 1992, ISBN 3-540-55116-6, pp. 238-41, 257.
  • Ilja Nikolajewitsch Bronstein, Konstantin Adolfowitsch Semendjajew, Gerhard Musiol, Heiner Mühlig: Paperback of Mathematics. Verlag Harri Deutsch, 7th edition 2008, ISBN 978-3-8171-2007-9, p. 251 ff (excerpt from the English edition (Google))
  • Matthias Richter: Basic knowledge of mathematics for engineers. Vieweg + Teubner 2001, 2nd edition 2008, ISBN 978-3-8348-0729-8, p. 230 (excerpt (Google))
  • A. Albert Klaf: Calculus Refresher. Dover 1956, ISBN 978-0-486-20370-6, pp. 151-168 (excerpt (Google))
  • James Casey: Exploring curvature. Vieweg + Teubner, 1996, ISBN 978-3-528-06475-4.

Web links

Individual evidence

  1. ↑ Alexandre Borovik, Mikhail G Katz: Who gave you the Cauchy-Weierstrass tale? The dual history of rigorous calculus. In: Foundations of Science. 2011, ISSN 1233-1821, pp. 1–32, doi: 10.1007 / s10699-011-9235-x, arxiv: 1108.2885.