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 ?
Treatment with Xiaflex offers men a medical treatment option for Peyronie's disease that does not prevent them from being candidates for future surgical procedures to correct remaining penile curvature. Whereas surgical procedures will straighten the penis, treatment with Xiaflex offers the potential for some recovery of penile length and girth, while also straightening the curvature.
In a subsequent controlled randomized study on patients with advanced resistant PD, Cavallini and Biagiotti (2002) administered propionyl-L-carnitine (PLC) (1g twice daily) + 10 intraplaque infiltrations (one/week) with verapamil 10 mg for 3 months and compared results with those of the control group (tamoxifen 20 mg twice/day + 10 intraplaque infiltrations once weekly with verapamil 10 mg for 3 months); they demonstrated that the PLC + verapamil association (versus tamoxifen + verapamil control group) was able to significantly reduce both plaque volume (p < 0.01) and penile curvature (p < 0.01) [72].
(15) The clinical presentation of epithelioid sarcoma includes penile nodularity or mass, urethral stenosis, penile curvature, dysuria and pain during erection.
The initial diagnosis of hypospadias is typically made after birth during physical exam, where boys with hypospadias are found to have a ventral skin deficiency with a dorsal hood of foreskin and an abnormally located meatus with varying degrees of ventral penile curvature. The standard classification of hypospadias is based on location of the urethral meatus: distal, midshaft, or proximal (Fig.
The symptomatic (i.e., pain, penile curvature interfering with penetration) incidence of PD has been estimated at 1%, and the asymptomatic (i.e., palpable plaque) prevalence is estimated between 0.4% to 1.0% (Carson, Jordan, & Gelbard, 1999; Gelbard, Dorey, & James, 1990; Greenfield & Levine, 2005; Lindsay et al., 1991).
Chordee or penile curvature is a downward curvature of the penis that typically accompanies the more severe forms of hypospadias.
If approved by the FDA, XIAFLEX is expected to be the first and only biologic therapy indicated for the treatment of Peyronie's disease, an excess of inelastic collagen causing penile curvature deformity, added the company.
They are used to assess the degree of penile curvature resulting from Peyronie's disease before corrective surgery.
The evolution of this concept required numerous experiences of dividing the urethral plate and performing extensive ventral corporeal dissection, only to have significant penile curvature persist with artificial erection.
The company has filed the application for XIAFLEX for the potential treatment of Peyronie's disease, an excess of inelastic collagen causing penile curvature deformity.
The remaining 3 patients (25%) were managed conservatively because of painful erection, painful coitus, erectile dysfunction and penile curvature after 1 year follow-up.
Auxilium said that the trials' co-primary endpoints are the mean percent improvement from baseline in penile curvature compared to placebo and the mean change from baseline in the Peyronie's disease bother domain of the Peyronie's Disease Questionnaire (PDQ) compared to placebo.