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 ?
Os raios de curvatura inferiores a 1 mm atendem ao especificado conforme trabalho de Yoshida et al.
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Os outros tres parametros a serem avaliados foram a proporcao aurea interincisivos postratamento ortodontico pelo metodo proposto por Mondelli (2003), a coincidencia da linha media dentaria com a facial e a curvatura incisal.
Horarios de incubacao menores em relacao ao tempo de colonizacao fazem com que a curva de degradacao aumente a sua curvatura aumentando a assintota com consequente diminuicao na taxa de degradacao (DHANOA, 1988), o que pode explicar os valores de degradacao dos alimentos avaliados (Tabela 2 e 3).
Esse aspecto de curvatura da espiga pode trazer grandes restricoes a comercializacao, uma vez que o mercado exige espigas cilindricas.
Enquanto que o metodo de interpolacao Spline e um metodo utilizado para ajuste de uma superficie de curvatura minima por meio dos dados pontuais de entrada, esse metodo e melhor pace superficies que representam tendencias gradativas, mmo o caso das chuvas medias na bacia Nao e apropriado se houver gandes intervalos nas superficies em uma pequena distancia horizontal (Esri, 1999).
Las regiones delgadas y gruesas de la corteza han sido cartografiadas desde la curvatura original cero de los espesores estimados de la corteza.
Inoltre, ogni soluzione matematica al problema di appiattire sulla superficie cartacea la curvatura della Terra era condannata ad una certa inaccuratezza.
O desgaste por erosao deve ser significativamente reduzido com o aumento do diametro das coifas, e a correspondente reducao de velocidade do gas, e com uma possivel aumento no raio de curvatura da regiao de saida.
A altura do dossel forrageiro foi considerada como o comprimento do nivel do solo ate a curvatura da ultima folha completamente expandida.
O termo acamamento de planta refere-se a curvatura do caule em direcao ao solo, causada pela massa de agua acumulada nas espigas maduras, ventos, baixa resistencia do colmo, entre outros fatores.