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Encyclopedia > Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a differential manifold. This article deals primarily with the first concept. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...


The primordial example of extrinsic curvature is that of a circle, which has curvature equal to the inverse of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point. Circle illustration This article is about the shape and mathematical concept of circle. ... This article is about an authentication, authorization, and accounting protocol. ... An osculating circle A circle with 4-point contact at a vertex of a curve In differential geometry, the osculating circle of a curve at a point, is a circle which: Touches the curve at that point Has its unit tangent vector , equal to the unit tangent of the curve...


In a plane, this is a scalar quantity, but in three or more dimensions it is described by a curvature vector that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor. In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... An open surface with X-, Y-, and Z-contours shown. ... This article is about the idea of space. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...


The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space. See the links below for further reading.

Contents

1 dimension in 2 dimensions: Curvature of plane curves

For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point, its center shaping the curve's evolute), and is a vector pointing in the direction of that circle's center. The smaller the radius r of the osculating circle, the larger the magnitude of the curvature (1/r) will be; so that where a curve is "nearly straight", the curvature will be close to zero, and where the curve undergoes a tight turn, the curvature will be large in magnitude. Image File history File links Osculating_circle. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... The reciprocal function: y = 1/x. ... This article is about an authentication, authorization, and accounting protocol. ... An osculating circle A circle with 4-point contact at a vertex of a curve In differential geometry, the osculating circle of a curve at a point, is a circle which: Touches the curve at that point Has its unit tangent vector , equal to the unit tangent of the curve... In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ...


The magnitude of curvature at points on physical curves can be measured in diopters (also spelled dioptre); a diopter has the dimension length-1. A dioptre, or diopter, is a non-SI unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in metres (i. ...


A straight line has curvature 0 everywhere; a circle of radius r has curvature 1/r everywhere.


Local expressions

For a plane curve given parametrically as c(t) = (x(t),y(t)), the curvature is

F[x,y]= frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}

For the less general case of a plane curve given explicitly as y = f(x) the curvature is

kappa=frac{|y''|}{(1+y'^2)^{3/2}}

This quantity is common in physics and engineering; for example, in the equations of bending in beams, the 1D vibration of a tense string, approximations to the fluid flow around surfaces (in aeronautics), and the free surface boundary conditions in ocean waves. In such applications, the assumption is almost always made that the slope is small compared with unity, so that the approximation: A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... This vibrating glass beam may be modeled as a cantilever beam with acceleration, variable linear density, variable section modulus, some kind of dissipation, springy end loading, and possibly a point mass at the free end. ... Figure 1. ... The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ... This article is about the mathematical term. ...

kappa approx frac{d^2y}{dx^2}

may be used. This approximation yields a straightforward linear equation describing the phenomenon, which would otherwise remain intractable.


If a curve is defined in polar coordinates as r(θ), then its curvature is

kappa(theta) = frac{r^2 + 2r'^2 - r r''}{left(r^2+r'^2 right)^{3/2}}

where here the prime refers to differentiation with respect to θ.


Example

Consider the parabola y = x2. We can parametrize the curve simply as c(t) = (t,t2) = (x,y), A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

dot{x}= 1,quadddot{x}=0,quad dot{y}= 2t,quadddot{y}=2

Substituting

kappa(t)= left|frac{dot{x}ddot{y}-dot{y}ddot{x}}{({dot{x}^2+dot{y}^2)}^{3/2}}right|= {1cdot 2-(2t)(0) over (1+(2t)^2)^{3/2} }={2 over (1+4t^2)^{3/2}}

1 dimension in 3 dimensions: Curvature of space curves

See Frenet-Serret formulas for a fuller treatment of curvature and the related concept of torsion.

For a parametrically defined space curve its curvature is: In vector calculus, the Frenet-Serret formulas describe the dynamic properties of a particle which moves along a continuous, differentiable curve in three-dimensional space . ... Differential geometry of curves In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. ...

F[x,y,z]=frac{sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}

Given a function r(t) with values in R3, the curvature at a given value of t is Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

kappa = frac{|dot{r} times ddot{r}|}{|dot{r}|^3}

where dot{r} and ddot{r} correspond to the first and second derivatives of r(t), respectively.


2 dimensions: Curvature of surfaces

In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces have intrinsic curvature, independent of an embedding.


For a two-dimensional surface embedded in R3, consider the intersection of the surface with a plane containing the normal vector and one of the tangent vectors at a particular point. This intersection is a plane curve and has a curvature. This is the normal curvature, and it varies with the choice of the tangent vector. The maximum and minimum values of the normal curvature at a point are called the principal curvatures, k1 and k2, and the directions of the corresponding tangent vectors are called principal directions. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... A normal vector is a vector which is perpendicular to a surface or manifold. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... Principal curvature is the inverse of the radius of the osculating circle. ...


Here we adopt the convention that a curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, otherwise negative. A negative number is a number that is less than zero, such as −3. ...


The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k1k2. It has the dimension of 1/length2 and is positive for spheres, negative for one-sheet hyperboloids and zero for planes. It determines whether a surface is locally convex (when it is positive) or locally saddle (when it is negative). Curvature is the amount by which a geometric object deviates from being flat. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... For other uses, see Sphere (disambiguation). ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, a = b, it is a hyperboloid of revolution. ... In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ...


The above definition of Gaussian curvature is extrinsic in that it uses the surface's embedding in R3, normal vectors, external planes etc. Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss' celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... The Theorema Egregium (Remarkable Theorem) is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces. ...


An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. He runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, he would find C(r) = 2πr. On curved surfaces, the formula for C(r) will be different, and the Gaussian curvature K at the point P can be computed as

 K = lim_{r rarr 0} (2 pi r - mbox{C}(r)) cdot frac{3}{pi r^3}.

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss-Bonnet theorem. This article is about the concept of integrals in calculus. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ... The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ...


The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss-Bonnet theorem is Descartes' theorem on total angular defect. In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ... The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ... In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ...


Because curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...


The mean curvature is equal to the sum of the principal curvatures, k1+k2, over 2. It has the dimension of 1/length. Mean curvature is closely related to the first variation of surface area, in particular a minimal surface like a soap film has mean curvature zero and soap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero. In mathematics, mean curvature of a surface is a notion from differential geometry. ... Area is the measure of how much exposed area any two dimensional object has. ... Verrill Minimal Surface In mathematics, a minimal surface is a surface with a mean curvature of zero. ... A soap film is a physical realization of a minimal surface. ... A soap bubble. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...


3 dimensions: Curvature of space

By extension of the former argument, a space of three or more dimensions can be intrinsically curved; the full mathematical description is described at curvature of Riemannian manifolds. Again, the curved space may or may not be conceived as being embedded in a higher-dimensional space. In recent physics jargon, the embedding space is known as the bulk and the embedded space as a p-brane where p is the number of dimensions; thus a surface (membrane) is a 2-brane; normal space is a 3-brane etc. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ...


After the discovery of the intrinsic definition of curvature, which is closely connected with non-Euclidean geometry, many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In the theory of general relativity, which describes gravity and cosmology, the idea is slightly generalised to the "curvature of space-time"; in relativity theory space-time is a pseudo-Riemannian manifold. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the underlying space-time curvature that is physically significant. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Gravity is a force of attraction that acts between bodies that have mass. ... This article is about the physics subject. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...


Although an arbitrarily-curved space is very complex to describe, the curvature of a space which is locally isotropic and homogeneous is described by a single Gaussian curvature, as for a surface; mathematically these are strong conditions, but they correspond to reasonable physical assumptions (all points and all directions are indistinguishable). A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. An example of negatively curved space is hyperbolic geometry. A space or space-time without curvature (formally, with zero curvature) is called flat. For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat space-time. There are other examples of flat geometries in both settings, though. A torus or a cylinder can both be given flat metrics, but differ in their topology. Other topologies are also possible for curved space. See also shape of the universe. 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ... Lines through a given point P and asymptotic to line l. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In geometry, a torus (pl. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... The shape of the Universe is an informal name for a subject of investigation within physical cosmology. ...


See also

Look up curvature in Wiktionary, the free dictionary.

In differential geometry, the curvature form describes curvature of principal bundle with connection. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G of a space X with a group G. Analogous to the Cartesian product, a principal bundle P is equipped with An action of G on P, analogous to... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measures distribution of mass is curved. It is related to notions of curvature in geometry. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In differential geometry, the geodesic curvature vector is a property of curves in a metric space which reflects the deviance of the curve from following the shortest arc length distance along each infinitesimal segment of its length. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In differential geometry, the Gauss map (named, like so many things, after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere . ... The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ... In mathematics, mean curvature of a surface is a notion from differential geometry. ... The principle of least constraint is another formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829. ... A dioptre, or diopter, is a non-SI unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in metres (i. ... Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ...

External links

References

Coolidge, J.L. "The Unsatisfactory Story of Curvature". The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 375-379


  Results from FactBites:
 
Curvature - definition of Curvature in Encyclopedia (836 words)
For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
Curvature form for the appropriate notion of curvature for vector bundles and principal bundles with connection.
Geodesic curvature vector - encyclopedia article about Geodesic curvature vector. (460 words)
In differential geometry, the geodesic curvature vector is a property of curves in a metric space which reflects the deviance of the curve from following the shortest arc length distance along each infinitesimal segment of its length.
is the curvature vector k of the projection of the curve C onto the tangent plane at P.
A curve for which the geodesic curvature is everywhere vanishing is called a geodesic.
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