Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, 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 in terms of the lengths of curves within a Riemannian manifold.

This article deals primarily with extrinsic curvature. Its canonical example is that of a circle, which has a curvature equal to the reciprocal 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.

Curvature is normally a scalar quantity, but one may also define a curvature vector that takes into account the direction of the bend in addition to its magnitude. 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.

This article sketches the mathematical framework which describes the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.

A migrating wild-type Dictyostelium discoideum cell whose boundary is colored by curvature. Scale bar: 5 µm.

Curvature of plane curves

Augustin-Louis Cauchy defined the center of curvature of a curve C as the intersection point of two infinitely close normals to the curve, the radius of curvature as the distance from the point to C, and the curvature itself as the inverse of the radius of curvature.[1]

Let C be a plane curve (the precise technical assumptions are given below). The curvature of C at a point is a measure of how sensitive its tangent line is to moving the point to other nearby points. There are many equivalent ways that this idea can be made precise.

Osculating circle

One way is geometrical. It is natural to define the curvature of a straight line to be constantly zero. The curvature of a circle of radius r should be large if r is small and small if r is large. Thus the curvature of a circle is defined to be the reciprocal of the radius:[2]

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line. The radius of curvature is defined as the reciprocal of the curvature.

Another way to understand the curvature is physical. Suppose that a particle moves along the curve with unit speed. Taking the time s as the parameter for C, this provides a natural parametrization for the curve. The unit tangent vector T (which is also the velocity vector, since the particle is moving with unit speed) also depends on time. The curvature is then the magnitude of the rate of change of T. Symbolically,

FrenetTN
The vectors T and N at two points on a plane curve, a translated version of the second frame (dotted), and the change in T: δT. δs is the distance between the points. In the limit dT/ds will be in the direction N and the curvature describes the speed of rotation of the frame.

This is the magnitude of the acceleration of the particle and the vector dT/ds is the acceleration vector. Geometrically, the curvature κ measures how fast the unit tangent vector to the curve rotates.[3] 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.

These two approaches to the curvature are related geometrically by the following observation. In the first definition, the curvature of a circle is equal to the ratio of the angle of an arc to its length. Likewise, the curvature of a plane curve at any point is the limiting ratio of , an infinitesimal angle (in radians) between tangents to that curve at the ends of an infinitesimal segment of the curve, to the length of that segment ds, i.e., /ds.[4] If the tangents at the ends of the segment are represented by unit vectors, it is easy to show that in this limit, the magnitude of the difference vector is equal to , which leads to the given expression in the second definition of curvature.

Precise definition

Suppose that C is a twice continuously differentiable immersed plane curve, which here means that there exists a parametric representation of C by a pair of functions γ(t) = (x(t), y(t)) such that the first and second derivatives of x and y both exist and are continuous, and

throughout the domain. For such a plane curve, there exists a reparametrization of C with respect to arc length s such that

[5]

The velocity vector T(s) is the unit tangent vector. The unit normal vector N(s), the curvature κ(s), the oriented or signed curvature k(s), and the radius of curvature R(s) are given by

Expressions for calculating the curvature in arbitrary coordinate systems are given below.

Animations of the signed curvature and the acceleration vector T′(s)

Lissajous-Curve nebeneinander animated
Lemniscate nebeneinander animated

Signed curvature

The sign of the signed curvature k indicates the direction in which the unit tangent vector rotates as a function of the parameter along the curve. If the unit tangent rotates counterclockwise, then k > 0. If it rotates clockwise, then k < 0. So, for example, the sign of the curvature of the graph of a function is the same as the sign of the second derivative (see below).

The signed curvature depends on the particular parametrization chosen for a curve. For example, the unit circle can be parametrised by (cos θ, sin θ) (counterclockwise, with k > 0), or by (cos(−θ), sin(−θ)) (clockwise, with k < 0). More precisely, the signed curvature depends only on the choice of orientation of an immersed curve. Every immersed curve in the plane admits two possible orientations.

Local expressions

For a plane curve given parametrically in Cartesian coordinates as γ(t) = (x(t),y(t)), the curvature is

where primes refer to derivatives d/dt with respect to the parameter t. The signed curvature k is

The expression reflects the geometric meaning discussed above, that the curvature is influenced by the change amount of the tangent vector in the direction of the normal vector, as

These can be expressed in a coordinate-independent manner via

Curvature of the graph of a function

For the less general case of a plane curve given explicitly as the graph of a function y = f(x), and using primes instead of d/dx for derivatives, the curvature is

and the signed curvature is

This quantity is common in physics and engineering; for example, in the equations of bending in beams, the one-dimensional 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:

may be used. This approximation yields a straightforward linear equation describing the phenomenon.

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

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

Implicit curve

For implicit curves defined by a function with partial derivatives , , , , the curvature is given by[6]

Examples

Consider the parabola y = x2. We can parametrize the curve simply as γ(t) = (t,t2) = (x,y). If we use primes for derivatives with respect to the parameter t, then

Substituting and dropping unnecessary absolute values, get

And the same result may be obtained immediately from the above formula of the curvature of a graph, without parametrizing.

A Lissajous curve with a 3:2 ratio of frequencies can be parametrized in this way:

Applying the formula it turns out to have signed curvature:

(For more details on this example see osculating circle.)

Curvature of space curves

Torus-Knot uebereinander animated
Animation of the curvature and the acceleration vector T′(s)

As in the case of curves in two dimensions, the curvature of a regular space curve C in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus if γ(s) is the arc-length parametrization of C then the unit tangent vector T(s) is given by

and the curvature is the magnitude of the acceleration:

The direction of the acceleration is the unit normal vector N(s), which is defined by

The plane containing the two vectors T(s) and N(s) is called the osculating plane to the curve at γ(s). The curvature has the following geometrical interpretation. There exists a circle in the osculating plane tangent to γ(s) whose Taylor series to second order at the point of contact agrees with that of γ(s). This is the osculating circle to the curve. The radius of the circle R(s) is called the radius of curvature, and the curvature is the reciprocal of the radius of curvature:

The tangent, curvature, and normal vector together describe the second-order behavior of a curve near a point. In three-dimensions, the third order behavior of a curve is described by a related notion of torsion, which measures the extent to which a curve tends to move in a helical path in space. The torsion and curvature are related by the Frenet–Serret formulas (in three dimensions) and their generalization (in higher dimensions).

Local expressions

For a parametrically-defined space curve in three dimensions given in Cartesian coordinates by γ(t) = (x(t), y(t), z(t)), the curvature is

where the prime denotes differentiation with respect to the parameter t. This can be expressed independently of the coordinate system by means of the formula

where × is the vector cross product. Equivalently,

Here the T denotes the matrix transpose. This last formula is also valid for the curvature of curves in a Euclidean space of any dimension.

Curvature from arc and chord length

Given two points P and Q on C, let s(P,Q) be the arc length of the portion of the curve between P and Q and let d(P,Q) denote the length of the line segment from P to Q. The curvature of C at P is given by the limit

where the limit is taken as the point Q approaches P on C. The denominator can equally well be taken to be d(P,Q)3. The formula is valid in any dimension. Furthermore, by considering the limit independently on either side of P, this definition of the curvature can sometimes accommodate a singularity at P. The formula follows by verifying it for the osculating circle.

Curvature of curves on surfaces

When a one-dimensional curve lies on a two dimensional surface embedded in three dimensions 3, further measures of curvature are available, which take the surface's unit normal vector, u into account. These are the normal curvature, geodesic curvature and geodesic torsion. Any non-singular curve on a smooth surface will have its tangent vector T lying in the tangent plane of the surface orthogonal to the normal vector. The normal curvature, kn, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, kg, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τr, measures the rate of change of the surface normal around the curve's tangent.

Let the curve be a unit speed curve and let t = u × T so that T, u, t form an orthonormal basis: the Darboux frame. The above quantities are related by:

Principal curvature

Minimal surface curvature planes-en
Saddle surface with normal planes in directions of principal curvatures

All curves with the same tangent vector will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing T and u. Taking all possible tangent vectors, 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 normal directions.

Curvature of surfaces

Normal sections

Curvature can be evaluated along surface normal sections, similar to § Curves on surfaces above (see, e.g., Earth radius of curvature).

Gaussian curvature

In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k1k2. It has a dimension of length−2 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-shaped (when it is negative).

Gaussian curvature is 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. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.

Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

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. It 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, she 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 by the Bertrand–Diguet–Puiseux theorem as

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 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.

Because (Gaussian) 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.

Mean curvature

The mean curvature is an extrinsic measure of curvature equal to half the sum of the principal curvatures, k1 + k2/2. It has a dimension of length−1. 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. 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.

Second fundamental form

The intrinsic and extrinsic curvature of a surface can be combined in the second fundamental form. This is a quadratic form in the tangent plane to the surface at a point whose value at a particular tangent vector X to the surface is the normal component of the acceleration of a curve along the surface tangent to X; that is, it is the normal curvature to a curve tangent to X (see above). Symbolically,

where N is the unit normal to the surface. For unit tangent vectors X, the second fundamental form assumes the maximum value k1 and minimum value k2, which occur in the principal directions u1 and u2, respectively. Thus, by the principal axis theorem, the second fundamental form is

Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures.

Shape operator

A related notion of curvature is the shape operator, which is a linear operator from the tangent plane to itself. When applied to a tangent vector X to the surface, the shape operator is the tangential component of the rate of change of the normal vector when moved along a curve on the surface tangent to X. The principal curvatures are the eigenvalues of the shape operator, and in fact the shape operator and second fundamental form have the same matrix representation with respect to a pair of orthonormal vectors of the tangent plane. The Gauss curvature is thus the determinant of the shape tensor and the mean curvature is half its trace.

Curvature of space

By extension of the former argument, a space of three or more dimensions can be intrinsically curved. The curvature is intrinsic in the sense that it is a property defined at every point in the space, rather than a property defined with respect to a larger space that contains it. In general, a curved space may or may not be conceived as being embedded in a higher-dimensional ambient space; if not then its curvature can only be defined intrinsically.

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 spacetime"; in relativity theory spacetime 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 spacetime 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). 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 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 spacetime. 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.

Generalizations

Parallel Transport
Parallel transporting a vector from ANBA yields a different vector. This failure to return to the initial vector is measured by the holonomy of the surface.

The mathematical notion of curvature is also defined in much more general contexts.[7] Many of these generalizations emphasize different aspects of the curvature as it is understood in lower dimensions.

One such generalization is kinematic. The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve; analogously, curvature in higher dimensions can be regarded as a kind of tidal force (this is one way of thinking of the sectional curvature). This generalization of curvature depends on how nearby test particles diverge or converge when they are allowed to move freely in the space; see Jacobi field.

Another broad generalization of curvature comes from the study of parallel transport on a surface. For instance, if a vector is moved around a loop on the surface of a sphere keeping parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known as holonomy.[8] Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; see curvature form. A closely related notion of curvature comes from gauge theory in physics, where the curvature represents a field and a vector potential for the field is a quantity that is in general path-dependent: it may change if an observer moves around a loop.

Two more generalizations of curvature are the scalar curvature and Ricci curvature. In a curved surface such as the sphere, the area of a disc on the surface differs from the area of a disc of the same radius in flat space. This difference (in a suitable limit) is measured by the scalar curvature. The difference in area of a sector of the disc is measured by the Ricci curvature. Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions. They are particularly important in relativity theory, where they both appear on the side of Einstein's field equations that represents the geometry of spacetime (the other side of which represents the presence of matter and energy). These generalizations of curvature underlie, for instance, the notion that curvature can be a property of a measure; see curvature of a measure.

Another generalization of curvature relies on the ability to compare a curved space with another space that has constant curvature. Often this is done with triangles in the spaces. The notion of a triangle makes senses in metric spaces, and this gives rise to CAT(k) spaces.

See also

Notes

  1. ^ *Borovik, Alexandre; Katz, Mikhail G. (2011), "Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus", Foundations of Science, 17 (3): 245–276, arXiv:1108.2885, Bibcode:2011arXiv1108.2885B, doi:10.1007/s10699-011-9235-x
  2. ^ Kline, Morris. Calculus: An Intuitive and Physical Approach (2nd ed.). p. 458.
  3. ^ Pressley, Andrew. Elementary Differential Geometry (1st ed.). p. 29.
  4. ^ Pogorelov, A. V. Differential Geometry (1st ed.). p. 49.
  5. ^ Kennedy, John (2011). "The Arc Length Parametrization of a Curve".
  6. ^ Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces". Computer Aided Geometric Design. 22 (7): 632–658. CiteSeerX 10.1.1.413.3008. doi:10.1016/j.cagd.2005.06.005.
  7. ^ Kobayashi, S.; Nomizu, K. Foundations of Differential Geometry. Wiley Interscience. vol. 1 ch. 2–3.
  8. ^ Henderson; Taimina. Experiencing Geometry (3rd ed.). pp. 98–99.

References

External links

Arc (geometry)

In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve. A common example in the plane (a two-dimensional manifold), is a segment of a circle called a circular arc. In space, if the arc is part of a great circle (or great ellipse), it is called a great arc.

Every pair of distinct points on a circle determines two arcs. If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than π radians (180 degrees), and the other arc, the major arc, will subtend an angle greater than π radians.

Curvature form

In differential geometry, the curvature form describes the curvature of a connection on a principal bundle. It can be considered as an alternative to or a generalization of the curvature tensor in Riemannian geometry.

Curvatures of the stomach

The curvatures of the stomach refer to the greater and lesser curvatures. The greater curvature of the stomach is four or five times as long as the lesser curvature.

Earth radius

Earth radius is the distance from a selected center of Earth to a point on its surface, which is often chosen to be sea level, or more commonly, the surface of an idealized ellipsoid representing the shape of Earth. Because Earth is not a perfect sphere, the determination of Earth's radius can have several values, depending on how it is measured; from its equatorial radius of about 6,378 kilometres (3,963 miles) to its polar radius of about 6,357 kilometres (3,950 miles).

When only one radius is stated, the International Astronomical Union (IAU) prefers that it be Earth's equatorial radius.The International Union of Geodesy and Geophysics (IUGG) gives three global average radii, the arithmetic mean of the radii of the ellipsoid (R1), the radius of a sphere with the same surface area as the ellipsoid or authalic radius (R2), and the radius of a sphere with the same volume as the ellipsoid (R3). All three IUGG average radii are about 6,371 kilometres (3,959 mi). A fourth global average radius not mentioned by the IUGG is the rectifying radius, the radius of a sphere with a circumference equal to the perimeter of the polar cross section of the ellipsoid, about 6,367 kilometres (3,956 mi). The radius of curvature at any point on the surface of the ellipsoid depends on its coordinates and its azimuth, north-south (meridional), east-west (prime vertical), or somewhere in between.

Figure of the Earth

The figure of the Earth is the size and shape of the Earth in geodesy. Its specific meaning depends on the way it is used and the precision with which the Earth's size and shape is to be defined. While the sphere is a close approximation of the true figure of the Earth and satisfactory for many purposes, geodesists have developed several models that more closely approximate the shape of the Earth so that coordinate systems can serve the precise needs of navigation, surveying, cadastre, land use, and various other concerns.

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point:

For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the Theorema egregium.

Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.

General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay. The predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.

Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes; for example, microquasars and active galactic nuclei result from the presence of stellar black holes and supermassive black holes, respectively. The bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.

Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.

Helix

A helix (), plural helixes or helices (), is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helices are coil springs and the handrails of spiral staircases. A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word ἕλιξ, "twisted, curved".

Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially.

Minimum railway curve radius

The minimum railway curve radius is the shortest allowable design radius for the center line of railway tracks under a particular set of conditions. It has an important bearing on constructions costs and operating costs and, in combination with superelevation (difference in elevation of the two rails) in the case of train tracks, determines the maximum safe speed of a curve. Minimum radius of curve is one parameter in the design of railway vehicles as well as trams. Monorails and guideways are also subject to minimum radii.

Radius of curvature

In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold (Besse 1987, p. 43).In relativity theory, the Ricci tensor is the part of the curvature of spacetime that determines the degree to which matter will tend to converge or diverge in time (via the Raychaudhuri equation). It is related to the matter content of the universe by means of the Einstein field equation. In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an Einstein manifold, which have been extensively studied (cf. Besse 1987). In this connection, the Ricci flow equation governs the evolution of a given metric to an Einstein metric; the precise manner in which this occurs ultimately leads to the solution of the Poincaré conjecture.

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.

The curvature tensor is given in terms of the Levi-Civita connection by the following formula:

where [u,v] is the Lie bracket of vector fields. For each pair of tangent vectors u, v, R(u,v) is a linear transformation of the tangent space of the manifold. It is linear in u and v, and so defines a tensor. Occasionally, the curvature tensor is defined with the opposite sign.

If and are coordinate vector fields then and therefore the formula simplifies to

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). The linear transformation is also called the curvature transformation or endomorphism.

The curvature formula can also be expressed in terms of the second covariant derivative defined as:

which is linear in u and v. Then:

Thus in the general case of non-coordinate vectors u and v, the curvature tensor measures the noncommutativity of the second covariant derivative.

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

Scalar curvature

In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action. The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics. The scalar curvature of an n-manifold is defined as the trace of the Ricci tensor, and it can be defined as n(n − 1) times the average of the sectional curvatures at a point.

At first sight, the scalar curvature in dimension at least 3 seems to be a weak invariant with little influence on the global geometry of a manifold, but in fact some deep theorems show the power of scalar curvature. One such result is the positive mass theorem of Schoen, Yau and Witten. Related results give an almost complete understanding of which manifolds have a Riemannian metric with positive scalar curvature.

Scoliosis

Scoliosis is a medical condition in which a person's spine has a sideways curve. The curve is usually "S"- or "C"-shaped. In some, the degree of curve is stable, while in others, it increases over time. Mild scoliosis does not typically cause problems, while severe cases can interfere with breathing. Typically, no pain is present.The cause of most cases is unknown, but is believed to involve a combination of genetic and environmental factors. Risk factors include other affected family members. It can also occur due to another condition such as muscles spasms, cerebral palsy, Marfan syndrome, and tumors such as neurofibromatosis. Diagnosis is confirmed with X-rays. Scoliosis is typically classified as either structural in which the curve is fixed, or functional in which the underlying spine is normal.Treatment depends on the degree of curve, location, and cause. Minor curves may simply be watched periodically. Treatments may include bracing or surgery. The brace must be fitted to the person and used daily until growing stops. Evidence that chiropractic manipulation, dietary supplements, or exercises can prevent the condition from worsening is lacking. However, exercise is still recommended due to its other health benefits.Scoliosis occurs in about 3% of people. It most commonly occurs between the ages of 10 and 20. Girls typically are more severely affected than boys. The term is from Ancient Greek: σκολίωσις, translit. skoliosis which means "a bending".

Shape of the universe

The shape of the universe is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes general global properties of its shape as of a continuous object. The shape of the universe is related to general relativity, which describes how spacetime is curved and bent by mass and energy.

Cosmologists distinguish between the observable universe and the global universe. The observable universe consists of the part of the universe that can, in principle, be observed by light reaching Earth within the age of the universe. It encompasses a region of space that currently forms a ball centered at Earth of estimated radius 46.5 billion light-years (4.40×1026 m). This does not mean the universe is 46.5 billion years old; in fact, the universe is 13.8 billion years old, but space itself has also expanded, causing the size of the observable universe to be as stated. (However, it is possible to observe these distant areas only in their very distant past, when the distance light had to travel was much less). Assuming an isotropic nature, the observable universe is similar for all contemporary vantage points.

The global shape of the universe can be described with three attributes:

Finite or infinite

Flat (no curvature), open (negative curvature), or closed (positive curvature)

Connectivity, how the universe is put together, i.e., simply connected space or multiply connected.There are certain logical connections among these properties. For example, a universe with positive curvature is necessarily finite. Although it is usually assumed in the literature that a flat or negatively curved universe is infinite, this need not be the case if the topology is not the trivial one.The exact shape is still a matter of debate in physical cosmology, but experimental data from various independent sources (WMAP, BOOMERanG, and Planck for example) confirm that the observable universe is flat with only a 0.4% margin of error. Theorists have been trying to construct a formal mathematical model of the shape of the universe. In formal terms, this is a 3-manifold model corresponding to the spatial section (in comoving coordinates) of the 4-dimensional spacetime of the universe. The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker (FLRW) model. Arguments have been put forward that the observational data best fit with the conclusion that the shape of the global universe is infinite and flat, but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by 2-dimensional lattice).

Vertebral column

The vertebral column, also known as the backbone or spine, is part of the axial skeleton. The vertebral column is the defining characteristic of a vertebrate in which the notochord (a flexible rod of uniform composition) found in all chordates has been replaced by a segmented series of bone: vertebrae separated by intervertebral discs. The vertebral column houses the spinal canal, a cavity that encloses and protects the spinal cord.

There are about 50,000 species of animals that have a vertebral column. The human vertebral column is one of the most-studied examples.

Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace-free: metric contraction on any pair of indices yields zero.

In general relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves through regions of space devoid of matter. More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold.In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is locally conformally flat: there exists a local coordinate system in which the metric tensor is proportional to a constant tensor. This fact was a key component of Nordström's theory of gravitation, which was a precursor of general relativity.

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