Also found in: Dictionary, Thesaurus, Medical, Acronyms, Wikipedia.
Related to curvature: radius of 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
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005


(ker -vă-cher) See radius of curvature.
Collins Dictionary of Astronomy © Market House Books Ltd, 2006
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.



(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.


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.




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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.


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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
Correcting the curvature with surgery is one of the most common methods while some people with mild scoliosis prefer wearing braces," he added.
Simultaneously, three different area ratios with three different characteristics are modeled in order to observe the effects of area ratio and centerline curvature in order to separate each geometry formed by coupling different diffusion and centerline characteristics, a naming methodology was used such that A1, A2, and A3 indicate area distributions with high diffusion rate around intake throat, high diffusion rate around engine face and modest area distribution respectively.
Thus, for N(k)-manifold with Ricci soliton the quasi-conformal like curvature tensor W takes the form
(iii) The comparison results of lane-change paths have implications for curve selection of parallel parking in constrained steering space studied in [35], since parallel parking maneuver bears similarity to lane-change maneuver in terms of interpolating boundary conditions (position, heading, and curvature at the endpoints).
Killing submersions are determined by two functions on M, its Gaussian curvature and the so-called bundle curvature (for more details, see [23, 24] and Section 2).
In [6], the term (4) is used in a geometry-driven diffusion algorithm that takes into account the curvature of the surface represented by the image, and the curvature controls the amount of diffusion in each region.
The corneal curvature as expressed by simulated K in diopters was recorded before ortho-k treatment and after discontinuation of lens wear.
Do any other readers see landmarks that shouldn't be vivible and can anyone explain why the earth seems to have no curvature across the channel and on to Lundy?
Curvature is a provider of new and pre-owned network hardware and IT infrastructure services.
One point on the moment curvature diagram correspond to a prescribed value of the concrete compressive strain at the outer compressed point of the concrete section (i.e.
We shall here focus on one of the scenarios for the quantum-gravity realm which can motivate such studies, the scenario such that the characteristic scale of quantum gravity (usually assumed to be of the order of the Planck scale) plays the role of scale of curvature of momentum space [8-11].