curvature

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curvature

1. any normal or abnormal curving of a bodily part
2. Geometry the change in inclination of a tangent to a curve over unit length of arc. For a circle or sphere it is the reciprocal of the radius

curvature

(ker -vă-cher) See radius of curvature.

Curvature

 

(mathematics), a quantity characterizing the deviation of a curve or surface from a line or plane. The deviation of the arc MN (see Figure 1) of the curve L from the tangent MP at the point M may be characterized by the average curvature Kav of this arc, which is equal to the quotient of the angle a between the tangents at the points M and N to the length Δs of the arc MN:

The average curvature of the arc of a circle is equal to the reciprocal of the circle’s radius and thus gives an intuitive characterization of the degree of curvature of a circle: the arc’s curvature increases with decreasing radius.

Figure 1

The limit of the average curvature as the point N approaches the point M, that is, as s → 0, is called the curvature k of the curve L at the point M:

The quantity R, the reciprocal of the curvature, is usually called the radius of curvature of the curve L at M.

If the curve L is the graph of a function y =f(x), then the curvature k of this curve may be calculated from the formula

The curvature k of the curve L is, generally speaking, a function of the arc length s, measured from a certain point M on the curve. If L1 and L2 are curves whose curvatures, as functions of their respective arc lengths, are the same, then L1 and L2 are congruent, that is, they may be superimposed by a motion. For this reason, the definition of the curvature of a plane curve as a function of the arc length is usually called the natural (intrinsic) equation of this curve.

In order to characterize the deviation of a space curve L from the plane, we introduce the concept of torsion. The torsion σ at the point M of the curve is defined as the limit of the ratio of the angle β between the osculating planes to the curve at the points M and N to the length Δs of the arc MN as N approaches M:

The angle β is taken to be positive if the rotation of the osculating plane at N as N approaches M is counterclockwise when viewed from M. The curvature and torsion, specified as functions of the arc length, define the curve L to within its position in space.

The deviation of a surface from a plane may be measured as follows. All possible planes are drawn through the normal at a given point M on the surface. The sections of the surface by these planes are called normal sections, and the curvatures of the normal sections at M are called the normal curvatures of the surface at this point. The maximum and minimum of the normal curvatures at the specified point M are called the principal curvatures. If k1 and k2 are the principal curvatures, then the quantities K = k1k2 and H = ½(k1 + K2) are called, respectively, the total curvature (or Gaussian curvature) and the average curvature of the surface at the point M. These curvatures of the surface determine the normal curvatures and may therefore be viewed as measures of the deviation of the surface from the plane. In particular, if K = 0 and H = 0 at all points of the surface, then the surface is a plane.

The total curvature does not change under bending (that is, deformations of the surface that do not change the length of curves on the surface). If, for example, the total curvature is equal to zero at all points of the surface, then each sufficiently small piece of the surface is applicable to a plane. The total curvature without reference to the surrounding space constitutes the central concept of the intrinsic geometry of the surface. The average curvature is related to the external shape of the surface.

The concept of curvature may be generalized to objects of a more general nature. For example, the concept of curvature arises in Riemann spaces and is a measure of the deviation of these spaces from Euclidean spaces.

REFERENCES

Blaschke, V. Differentsial’naia geometriia i geometricheskie osnovy teorii otnositel’nosti Einshteina, vol. 1. Moscow-Leningrad, 1935. (Translated from German.)
Rashevskii, P. K. Kurs differentsial’noigeometrii, 4th ed. Moscow, 1956.
Pogorelov, A. V. Differentsial’naia geometriia, 5th ed. Moscow, 1969.

E. G. POZNIAK


Curvature

 

a scarcely noticeable entasis, or convex swelling, given certain parts of a building. Curvature lends considerable sculptural expressiveness to a building and eliminates optical distortion when straight-sided forms of a building are viewed from a distance or when appreciable foreshortening occurs. It is found primarily in architecture employing the classical orders.

curvature

[′kər·və·chər]
(mathematics)
The reciprocal of the radius of the circle which most nearly approximates a curve at a given point; the rate of change of the unit tangent vector to a curve with respect to arc length of the curve.
References in periodicals archive ?
Respecto a la parte inelastica, la ecuacion resultante tiene en cuenta directamente los valores obtenidos del diagrama momento curvatura en el sentido que considera que la curvatura plastica es la diferencia entre la curvatura ultima y la curvatura de fluencia.
Dimensoes dos fios antes e apos a laminacao em mm Diametro inicial Retangular obtido Raio de curvatura 0,49 0,408 x 0,402 0,778 0,65 0,479 x 0,64l 0,79l Tabela 3.
Tambien se advierte la curvatura en que el querer uno u otro bien no termina en el como su correlato, a diferencia del simple deseo, sino que se prolonga en la obra realizada por el mismo yo que la quiere: el querer es inseparablemente querer-hacer (en un sentido amplio de hacer, que se extiende a los actos inmanentes ejercidos voluntariamente).
Todos los arcos son iguales porque coinciden con la curvatura del nervio diagonal.
La curvatura del fruto central de la fila externa, con excepcion del mayor valor (p < 0,0300) que se observo para este con la BEH en la tercera y cuarta mano del experimento 5, no vario entre tratamientos con o sin BEH en las restantes (p > 0,1424) ni en las otras manos (p > 0,0934) de los demas experimentos (Cuadro 2).
Una vez tomadas las radiografias previas se midio el angulo de la curvatura segun la tecnica de Schneider (9) (figura 2), incluyendo en la muestra solo aquellos canales que presentaron angulacion entre 25 y 35[grados].
Os metodos utilizados foram o inverso quadratico da distancia, krigagem, minima curvatura e triangulacao, esses metodos sao descritos a seguir.
m] = curvaturas da secao referentes ao estadio I, ao estadio II e a um valor medio entre esses dois estadios, respectivamente.
La perturbacion en la curvatura [zeta] es la cantidad mas importante en cosmologia; es mediante ella que podemos describir la estadistica de las estructuras a gran escala, y es precisamente ella la que nos permitira definir adecuadamente el principio cosmologico.
De la observacion del primer molar inferior se puede destacar como a pesar de la configuracion aplanada de la raiz distal hasta en un 86,11 % de las ocasiones, la presencia de un conducto adicional en dicha raiz solo llega hasta un 11 % de las muestras analizadas; por otro lado se observa una ausencia total de fusion radicular y una curvatura hacia distal en un 93,36 % de los casos (Tabla II).
El analisis de la curvatura tanto en plano como en perfil de la cuenca se ve reflejado en la Figura 9, donde se evidencia el comportamiento de las laderas de vertiente a lo largo y ancho de la cuenca como resultado del relieve derivado por hipsometria y la pendiente, estimulado por la dinamica de un balance erosivo y de las fuerzas tectonicas presentes.
Femur anterior recto, con suave curvatura en el primer tercio, metaesterno con debil cresta sobre la linea media que se desarrolla posteriormente.