Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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.

The primordial example of extrinsic curvature is that of a circle, which has curvature equal to the inverse of its radius everywhere. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers 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. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Differential geometry of curves, the osculating circle of a sufficiently smooth plane Curve at a given point on the curve is the Circle whose center

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. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian 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 Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another Linear algebra is the branch of Mathematics concerned with In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express

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.

1 One dimension in two dimensions: Curvature of plane curves
2 One dimension in three dimensions: Curvature of space curves
3 Two dimensions: Curvature of surfaces
4 Three dimensions: Curvature of space
5 See also
6 External links
7 References

One dimension in two dimensions: Curvature of plane curves

For a plane curve C, the mathematical definition of curvature uses a parametric representation of C with respect to the arc length parametrization. In mathematics a plane curve is a Curve in a Euclidian plane (cf In Mathematics, parametric equations are a method of defining a curve It can be computed given any regular parametrization by a more complicated formula given below. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian Let γ(s) be a regular parametric curve, where s is the arc length, or natural parameter. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult This determines the unit tangent vector T, the unit normal vector N, the curvature κ(s), the signed curvature k(s) and the radius of curvature at each point:

$T(s)=\gamma'(s),\quad T'(s)=k(s)N(s),\quad \kappa(s) = \|\gamma''(s)\| = \left|k(s)\right|, \quad R(s)=\frac{1}{\kappa(s)}.$

The curvature of a straight line is identically zero. The curvature of a circle of radius R is constant, i. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the e. it does not depend on the point and is equal to the reciprocal of the radius:

$\kappa = \frac{1}{R}.$

Thus for a circle, the radius of curvature is simply its radius. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which Straight lines and circles are the only plane curves whose curvature is constant. Given any curve C and a point P on it where the curvature is non-zero, there is a unique circle which most closely approximates the curve near P, the osculating circle at P. In Differential geometry of curves, the osculating circle of a sufficiently smooth plane Curve at a given point on the curve is the Circle whose center The radius of the osculating circle is the radius of curvature of C at this point.

The meaning of curvature

Suppose that a particle moves on the plane with unit speed. Then the trajectory of the particle will trace out a curve C in the plane. Moreover, taking the time as the parameter, this provides a natural parametrization for C. The instanteneous direction of motion is given by the unit tangent vector T and the curvature measures how fast this vector rotates. If a curve keeps close to the same direction, the unit tangent vector changes very little and the curvature is small; where the curve undergoes a tight turn, the curvature is large.

The magnitude of curvature at points on physical curves can be measured in diopters (also spelled dioptre) — this is the convention in optics. A dioptre, or diopter, is a Unit of measurement of the Optical power of a lens or curved Mirror, which is equal to the reciprocal A diopter has the dimension $\scriptstyle[{Length^{-1}]}.$

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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and This article is about structural behavior For other meanings see Bending (disambiguation. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves In such applications, the assumption is almost always made that the slope is small compared with unity, so that the approximation:

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

may be used. Slope is used to describe the steepness incline gradient or grade of a straight line. 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 $y = x 2$ . In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular We can parametrize the curve simply as $c (t) = (t, t 2) = (x, y)$ ,

$\dot{x}= 1,\quad\ddot{x}=0,\quad \dot{y}= 2t,\quad\ddot{y}=2$

Substituting

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

One dimension in three dimensions: Curvature of space curves

See Frenet-Serret formulas for a fuller treatment of curvature and the related concept of torsion. "Binormal" redirects here For the category-theoretic meaning of this word see Normal morphism.

For a parametrically defined space curve its curvature is:

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

Given a function r(t) with values in R³, the curvature at a given value of $t$ is

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

where $\dot{r}$ and $\ddot{r}$ correspond to the first and second derivatives of r(t), respectively. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

Two 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. In Differential geometry, the two principal curvatures at a given point of a Surface measure how the surface bends by different amounts in different directions 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. In Differential geometry, the two principal curvatures at a given point of a Surface measure how the surface bends by different amounts in different directions

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 −2

The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k₁k₂. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German It has the dimension of 1/length² and is positive for spheres, negative for one-sheet hyperboloids and zero for planes. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, a hyperboloid is a Quadric, a type of surface in three Dimensions described by the equation {x^2 \over a^2} + It determines whether a surface is locally convex (when it is positive) or locally saddle (when it is negative). In Mathematics, a phenomenon is sometimes said to occur locally if roughly speaking it occurs on sufficiently small or arbitrarily small Neighborhoods

The above definition of Gaussian curvature is extrinsic in that it uses the surface's embedding in R³, normal vectors, external planes etc. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group 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. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group Formally, Gaussian curvature only depends on the Riemannian metric of the surface. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M This is Gauss' celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Gauss's Theorema Egregium (Latin "Remarkable Theorem" is a foundational result in Differential geometry proved by Carl Friedrich Gauss that concerns the

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. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant The Gauss–Bonnet theorem or Gauss–Bonnet formula in Differential geometry is an important statement about Surfaces which connects their geometry (in

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 Geometry, the defect (or deficit 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 or Gauss–Bonnet formula in Differential geometry is an important statement about Surfaces which connects their geometry (in In Geometry, the defect (or deficit 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 ( M, g) (with Riemannian metric g) is a real Differentiable manifold M

The mean curvature is equal to the sum of the principal curvatures, k₁+k₂, over 2. In Mathematics, the mean curvature H of a Surface S is an extrinsic measure of Curvature that comes from Differential It has the dimension of 1/length. Mean curvature is closely related to the first variation of surface area, in particular a minimal surface such as a soap film, has mean curvature zero and a soap bubble has constant mean curvature. Surface area is the measure of how much exposed Area an object has In Mathematics, a Minimal surface is a surface with a Mean curvature of zero A soap bubble is a very thin film of Soap water that forms a Sphere with an iridescent Surface. A soap bubble is a very thin film of Soap water that forms a Sphere with an iridescent Surface. 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. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis For the Mechanical engineering and Architecture usage see Isometric projection.

Three 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. In Mathematics, specifically Differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described 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.

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 mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry 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. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Gravitation is a natural Phenomenon by which objects with Mass attract one another Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a 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.

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). Isotropy is uniformity in all directions Precise definitions depend on the subject area In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. An example of negatively curved space is hyperbolic geometry. In 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. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity 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. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Other topologies are also possible for curved space. See also shape of the universe. The shape of the Universe is an informal name for a subject of investigation within Physical cosmology which describes the Geometry of the Universe

External links

3D-XplorMath Java applets Applets for space curves with osculating circles.
The History of Curvature

References

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

Dictionary

-noun

The shape of something curved.
(mathematics) The extent to which a subspace is curved within a metric space.
(differential geometry) The extent to which a Riemannian manifold is intrinsically curved.

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