In differential geometry , the term curvaturetensor may refer to the Riemann curvaturetensor of a Riemannian manifold &mdash see also Curvature of Riemannian manifolds the curvature of an affine connection or covariant derivative on tensors the curvature form of an Ehresmann connection see Ehresmann connection , connection principal bundle or connection vector bundle . See also Tensor disambiguation mathdab zh ... more details
General relativity In the mathematical field of differential geometry , the Riemann curvaturetensor , or Riemann Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel , is the most standard way to express curvature of Riemannian manifolds . It associates 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 a Euclidean space. The curvaturetensor can also be defined for any pseudo ... 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 curvaturetensor ... precise by the Jacobi field Jacobi equation . The curvaturetensor is given in terms of the Levi Civita ... and v , and so defines a tensor. Occasionally, the curvaturetensor is defined with the opposite ... v nabla u w . math The curvaturetensor measures noncommutativity of the covariant derivative , and as such is the integrability ... original position. The Riemann curvaturetensor directly measures the failure of this in a general ... X Z R X,Y Z math where R is the Riemann curvaturetensor. Coordinate expression In local coordinates math x mu math the Riemann curvaturetensor is given by math R rho sigma mu nu dx rho R partial ... nu . math Symmetries and identities The Riemann curvaturetensor has the following symmetries math ... list of symmetries of the curvaturetensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvaturetensor at some point. Simple calculations ... the Gaussian curvature and a , b , c and d take values either 1 or 2. The Riemann tensor ... curvaturetensor of the surface is simply given by math operatorname Ric ab Kg ab . , math Space ... curvaturetensor References citation first A.L. last Besse title Einstein manifolds publisher Springer ... Tensor Category Riemannian geometry Category Tensors in general relativity Category Curvature mathematics ... more details
Riemann curvaturetensor . 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. Curvature of plane curves Let C be a plane curve the precise technical assumptions are given below . The curvature ... curvature is thus the determinant of the shape tensor and the mean curvature is half its trace linear ...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 ... on the context. There is a key distinction between extrinsic curvature , which is defined for objects embedded in another space usually an Euclidean space in a way that relates to the radius of curvature of circles that touch the object, and Curvature of Riemannian manifolds intrinsic curvature , which ... concept. The canonical example of extrinsic curvature is that of a circle , which everywhere has curvature ..., and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its ... 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 surface s or even ... circle.svg float right 250px One way is geometrical. It is natural to define the curvature of a straight line to be identically 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 ... 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 ... more details
curvaturetensor . ref name Kline cite book title Mathematical thought from ancient to modern ... geometry are quadratic form s such as metric tensor s, and the Riemann curvaturetensor . The exterior ... scalar curvature Euclidean vector vector bivector , e.g. inverse metric tensor m 1 covector , linear ... tensor m 3 e.g. 3 form e.g. Riemann curvaturetensor ... m N e.g. N form i.e. determinant , volume ... of tensor theory in engineering Curvature Diffusion MRI Mathematical foundation tensors Diffusion ...About a modern but abstract treatment Tensor intrinsic definition other uses Tensor disambiguation Dablink Note that in common usage, the term tensor is also used to refer to a tensor field . Image Components stress tensor cartesian.svg 300px right thumb Stress, a second order tensor. The tensor s components ... theorem stress tensor stress tensor T takes a direction v as input and produces the stress T sup v sup ... independent of a particular choice of coordinate system . It is possible to represent a tensor by examining ... of numerical values. The coordinate independence of a tensor then takes the form of a covariant transformation ... to that computed in another one. The order or degree of a tensor is the dimensionality of the array ..., a 0th order tensor. A coordinate vector , or 1 dimensional array, can represent a vector, a 1st order tensor. A 2 dimensional array, or square matrix mathematics matrix , is then needed to represent a 2nd order tensor. In general, an order k tensor can be represented as a k dimensional array of components. The order of a tensor is the number of indices necessary to refer unambiguously to an individual component of a tensor. History The concepts of later tensor analysis arose from the work of Carl ... des Tensorkalk ls 1994 . ref The word tensor itself was introduced in 1846 by William Rowan ... of Quaternions ref to describe something different from what is now meant by a tensor. ref Namely ... Eigenschaften der Krystalle in elementarer Darstellung Leipzig, 1898 ref Tensor calculus was developed ... more details
are the Christoffel symbols of the metric. Unlike the Riemann curvaturetensor or the Ricci tensor , which ... The Riemann tensor of an n dimensional Euclidean space vanishes identically, so the scalar curvature ... R to represent three different things the Riemann curvaturetensor math R ijk l math or math R abcd math the Ricci tensor math R ij math the scalar curvature R These three are then distinguished from ... R for the full Riemann curvaturetensor. See also Basic introduction to the mathematics of curved ...Unreferenced date December 2009 In Riemannian geometry , the scalar curvature or Ricci scalar is the simplest curvature invariant of a Riemannian manifold . To each point on a Riemannian manifold, it assigns ..., the scalar curvature represents the amount by which the volume of a geodesic ball in a curved ..., 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 ... metrics are known as Einstein manifold Einstein metrics . The scalar curvature is defined as the trace of the Ricci tensor , and it can be characterized as a multiple of the average of the sectional curvature s at a point. Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult. One of the few is the positive ... , which seeks extremal metrics in a given conformal class for which the scalar curvature is constant. Definition The scalar curvature is usually denoted by S other notations are Sc , R . It is defined as the Trace linear algebra trace of the Ricci curvaturetensor with respect to the metric tensor metric math S mbox tr g , operatorname Ric . math The trace depends on the metric since the Ricci tensor is a 0,2 valent tensor one must first raising and lowering indices raise an index to obtain a 1,1 ... more details
A curvature collineation often abbreviated to CC is vector field which preserves the Riemann tensor in the sense that, math mathcal L X R a bcd 0 math where math R a bcd math are the components of the Riemann tensor. The Set mathematics set of all smooth function smooth curvature collineations forms a Lie algebra under the Lie bracket operation if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra . The Lie algebra is denoted by math CC M math and may be infinity infinite dimension al. Every affine vector field is a curvature collineation. See also Conformal vector field Homothetic vector field Killing vector field Matter collineation Spacetime symmetries relativity stub Category Mathematical methods in general relativity ... more details
of gravitation such as general relativity , curvature scalars play an important role in telling distinct spacetimes apart. Two of the most basic curvature invariants in general relativity are the Curvature ... familiar quadratic invariants of the electromagnetic field tensor in classical electromagnetism ... Syzygy syzygies for the zero th order invariants of the Riemann tensor. They have limitations because ... from Minkowski spacetime using any number of curvature invariants of any order . See also Cartan Karlhede algorithm Carminati McLenaghan invariants Curvature invariant general relativity Ricci decomposition ..., Cambridge Univ. Press 2003 Curvature invariants are studied in Chapter 9 ref references DEFAULTSORT Curvature Invariant Category Riemannian geometry relativity stub Geometry stub ... more details
In differential geometry , the Ricci curvaturetensor , named after Gregorio Ricci Curbastro , represents ... curvature space form . If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold ... R zeta, eta xi math where R is the Riemann curvaturetensor . In local coordinates using the Einstein ... ij R k ikj . math In terms of the Riemann curvaturetensor and the Christoffel symbols , one has math ... tangent vectors is often simply called the Ricci curvature , since knowing it is equivalent to knowing the Ricci curvaturetensor. The Ricci curvature is determined by the sectional curvature s of a Riemannian ... the full curvaturetensor. A notable exception is when the manifold is given a priori as a hypersurface ... g . The Ricci curvature is usefully thought of as a multiple of the Laplacian of the metric tensor ... curvature. Trace free Ricci tensor In Riemannian geometry and general relativity , the trace free ... curvature , math g math is the metric tensor , and math n math is the dimension of math M math ... from that of ordinary Euclidean n space. More generally, the Ricci tensor is defined on any pseudo Riemannian manifold . Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold. The Ricci curvature is broadly applicable to modern Riemannian geometry ... with the former, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global ... at p , the Ricci tensor math mathrm Ric xi , eta math evaluated at math xi, eta math is defined ... serve to define a Ricci tensor associated to any affine connection , not just the Levi Civita connection ... , the Ricci tensor of a Riemannian manifold is symmetric tensor symmetric , in the sense that math ... tensor associated to any torsion free affine connection for which there exists a parallel volume form. It thus follows that the Ricci tensor is completely determined by knowing the quantity math operatorname ... of the sectional curvature, taken over all the 2 planes containing math xi math . There is an n &minus ... more details
In differential geometry , the curvature form describes curvature of a connection form connection on a principal bundle . It can be considered as an alternative to or generalization of curvaturetensor in Riemannian geometry . Definition Let G be a Lie group with Lie algebra math mathfrak g math , and P B be a principal bundle principal G bundle . Let be an Ehresmann connection on P which is a math mathfrak g math valued Differential form one form on P . Then the curvature form is the math mathfrak g math valued 2 form on P defined by math Omega d omega 1 over 2 omega, omega D omega. math Here math d math stands for exterior derivative , math cdot, cdot math is defined by math alpha otimes X, beta otimes Y alpha wedge beta otimes X, Y mathfrak g math and D denotes the exterior covariant derivative . In other terms, math , Omega X,Y d omega X,Y omega X , omega Y . math Curvature form in a vector bundle If E B is a vector bundle. then one can also think of as a matrix of 1 forms and the above formula becomes the structure equation math , Omega d omega omega wedge omega, math where math wedge math is the Exterior power wedge product . More precisely, if math omega i j math and math Omega i j math denote components of and correspondingly, so each math omega i j math is a usual 1 form and each math Omega i j math is a usual 2 form then math Omega i j d omega i j sum k omega i k wedge omega k j . math For example, for the tangent bundle of a Riemannian manifold , the structure group is O n and is a 2 form with values in o n , the skew symmetric matrix antisymmetric matrices . In this case the form is an alternative description of the curvaturetensor , i.e. math ,R X,Y Omega X,Y , math using the standard notation for the Riemannian curvaturetensor, Bianchi identities ... to the mathematics of curved spacetime Chern Simons form Curvature of Riemannian manifolds Gauge theory curvature Category Differential geometry Category Curvature mathematics es forma de ... more details
the exponential map at p . The sectional curvature is a smooth real valued function on the 2 Grassmannian fiber bundle bundle over the manifold. The sectional curvature determines the Riemann curvaturetensorcurvaturetensor completely. Definition Given a Riemannian manifold and two linearly independent ... over langle u,u rangle langle v,v rangle langle u,v rangle 2 math Here R is the Riemann curvaturetensor ... curvaturetensorcurvature of Riemannian manifolds curvaturecurvature Category Riemannian geometry ...In Riemannian geometry , the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds . The sectional curvature K &sigma sub p sub depends on a two dimensional plane &sigma sub p sub in the tangent space at p . It is the Gaussian curvature of that section &mdash the surface ... curvature in fact depends only on the 2 plane sub p sub in the tangent space at p spanned by u and v . It is called the sectional curvature of the 2 plane &sigma sub p sub , and is denoted K &sigma sub p sub . Manifolds with constant sectional curvature Riemannian manifold s with constant sectional curvature are the most simple. These are called space form s. By rescaling the metric there are three possible cases negative curvature &minus 1, hyperbolic geometry zero curvature, Euclidean geometry positive curvature 1, elliptic geometry The model manifolds for the three geometries ... complete , simply connected Riemannian manifolds of given sectional curvature. All other complete constant curvature manifolds are quotients of those by some group of isometry isometries . If for each point in a connected Riemannian manifold of dimension three or greater the sectional curvature is independent of the tangent 2 plane, then the sectional curvature is in fact constant on the whole manifold. Toponogov s theorem Toponogov s theorem affords a characterization of sectional curvature in terms ... has non negative curvature, then for all sufficiently small triangles math d z,m 2 ge tfrac12d z,x ... more details
Image Gaussian curvature.PNG thumb From left to right a surface of negative Gaussian curvature hyperboloid , a surface of zero Gaussian curvature cylinder geometry cylinder , and a surface of positive Gaussian curvature sphere . In differential geometry , the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvature s, sub 1 sub and sub 2 sub , of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances ... curvature is defined as math Kappa kappa 1 kappa 2 , math . where math kappa 1 math and math kappa 2 math are the Principal curvature principal curvatures . Alternative definitions It is also given ... where math nabla i nabla mathbf e i math is the covariant derivative and g is the metric tensor . At a point p on a regular surface in R sup 3 sup , the Gaussian curvature is also given by math K mathbf p det S mathbf p , math where S is the shape operator . A useful formula for the Gaussian curvature ... of f vanishes this can always be attained by a suitable rigid motion . Then the Gaussian curvature ... curvature Image Hyperbolic triangle.svg thumb The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle. The surface integral of the Gaussian curvature over some region of a surface is called the total curvature . The total curvature of a geodesic ... on a surface of positive curvature will exceed math pi math , while the sum of the angles of a triangle on a surface of negative curvature will be less than math pi math . On a surface of zero curvature, such as the Euclidean plane , the angles will sum to precisely math pi math . math ... states that Gaussian curvature of a surface can be determined from the measurements of length on the surface ... of the Gaussian curvature of a surface S in R sup 3 sup certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the inner ... more details
In mathematics , the mean curvature math H math of a surface math S math is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedding ... curvature math K i math given at math p math . Of those curvatures math K i math , at least one is characterized ... curvatures math kappa 1, kappa 2 math are known as the principal curvature s of math S math . The mean curvature at math p in S math is the average of curvatures harv Spivak 1999 loc Volume ... loc Volume 4, Chapter 7 , for a hypersurface math T math the mean curvature is given as math H frac 1 n sum i 1 n kappa i . math More abstractly, the mean curvature is the trace of the second fundamental form divided by n or equivalently, the shape operator . Additionally, the mean curvature math ... embedded hypersurfaces, math vec n math a unit normal vector, and math g ij math the metric tensor . A surface is a minimal surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface math S math , is said to obey a heat equation heat type equation called the mean curvature flow equation. The sphere is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true ... space For a surface defined in 3D space, the mean curvature is related to a unit Surface normal normal of the surface math 2 H nabla cdot hat n math where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal the curvature is positive if the surface ... normal the doubled mean curvature expression is math begin align 2 H & nabla cdot left frac nabla ... S r S left scriptstyle sqrt x 2 y 2 right math . Mean curvature in fluid mechanics An alternate definition ... surface is a surface which has zero mean curvature at all points. Classic examples include the catenoid ... . An extension of the idea of a minimal surface are surfaces of constant mean curvature . See ... more details
Image Minimal surface curvature planes en.svg thumb 300px right Saddle surface with normal planes in directions ... have different curvature s for different normal planes at p . The principal curvatures at p , denoted k sub 1 sub and k sub 2 sub , are the maximum and minimum values of this curvature. Here the curvature of a curve is by definition the multiplicative inverse reciprocal of the radius of the osculating circle . The curvature is taken to be positive if the curve turns in the same direction ... the curvature takes its maximum and minimum values are always perpendicular, a result of Leonhard Euler ... from the spectral theorem because they can be given as the principal axes of a symmetric tensor ... 2 sub of the two principal curvatures is the Gaussian curvature , K , and the average k sub 1 sub k sub 2 sub 2 is the mean curvature , H . If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface . For a minimal surface , the mean curvature is zero at every point. Formal definition Let M be a surface in Euclidean ... eigenvectors principal directions , then the sectional curvature of M at p is given by math K .... The Monkey saddle is one surface with an isolated flat umbilic. Lines of curvature The lines of curvature or curvature lines are curves which are always tangent to a principal direction they are integral curve s for the principal direction fields . There will be two lines of curvature through each ... of curvature form one of three configurations star , lemon and monstar derived from lemon star ref ... of curvature near umbilics widths 150px Image TensorLemon.png Lemon Image TensorMonstar.png Monstar Image TensorStar.png Star gallery In these figures, the red curves are the lines of curvature for one family of principal directions, and the blue curves for the other. When a line of curvature has a local extremum of the same principal curvature then the curve has a ridge differential geometry ... more details
Unreferenced date December 2009 Seealso Space form In mathematics , constant curvature in differential geometry is a concept most commonly applied to surface s. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points. The circle has constant curvature, also, in a natural but different sense. The standard surface geometries of constant curvature are elliptic geometry or spherical geometry which has positive curvature , Euclidean geometry which has zero curvature, and hyperbolic geometry pseudosphere geometry which has negative curvature . Since Riemann surface s can be taken to have constant curvature, there is a large supply of other examples, for negative curvature. For higher dimensional manifold s, constant curvature is usually taken to mean constant sectional curvature , and a complete manifold of this kind is called a space form . As in the case of surfaces, there are three types of geometries elliptic, flat, or hyperbolic according to whether the curvature is positive, zero, or negative. The universal cover of a manifold of constant sectional curvature is one of the model spaces sphere, Euclidean space, hyperbolic space , and the study of space forms is thus generalized crystallography. Spherical manifold Flat manifold Hyperbolic manifold Seealso Curvature of Riemannian manifolds DEFAULTSORT Constant Curvature Category Differential geometry Category Riemannian geometry nl Constante kromming zh ... more details
Wikt refer Radius of curvature optics Radius of curvature applications , in geodesy and materials science The reciprocal of the curvature , in differential geometry Radius , for a sphere lingo The radius of the osculating circle in differential geometry of curves Minimum railway curve radius dab ... more details
In geometry , center of curvature of a curve is found at a point that is at a distance equal to the radius of curvature lying on the normal vector . It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the center of curvature. See also Curvature Differential geometry of curves References Citation last1 Hilbert first1 David author1 link David Hilbert last2 Cohn Vossen first2 Stephan author2 link Stephan Cohn Vossen title Geometry and the Imagination publisher Chelsea location New York edition 2nd isbn 978 0 8284 1097 8 year 1952 geometry stub Category Curves Category Differential geometry am ca Centre de curvatura es Centro de curvatura ... more details
In Riemannian geometry , the geodesic curvature math k g math of a curve lying on a submanifold of the ambient space measures how far the curve is from being a geodesic. For instance it applies to Curvature Curves on surfaces curves on surfaces . The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold the normal curvature math k n math and the one that comes from the curve itself. The curvature math k math of the curve is related to these two by math k sqrt k g 2 k n 2 math . In particular geodesics have no geodesic curvature they are straight , and that is their definition, so that math k k n math , which explains why they appear to be curved in ambient space whenever the submanifold is. Definition Consider a curve lying ... s math , with unit tangent vector math T math . The geodesic curvature is the norm of the projection ... curvature is the norm of the projection of math dT ds math on the normal bundle to the submanifold at the point ... Euclidean space. The normal curvature of math S 2 math is identically 1. Great circles have curvature math k 1 math , which implies zero geodesic curvature, thus they are geodesics. Smaller circles of radius math r math will have curvature math 1 r math and geodesic curvature math k g sqrt 1 r 2 r math . Some results involving geodesic curvature The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold math M math . It does not depend on the way the submanifold math M math sits in math bar M math . On the contrary the normal curvature ... Codazzi equations . The Gauss Bonnet theorem . See also Curvature Darboux frame Gauss Codazzi equations ... Surfaces isbn 0 486 63433 7 . springer id G g044070 title Geodesic curvature first Yu.S. last Slobodyan year 2001 . External links Mathworld urlname GeodesicCurvature title Geodesic curvaturecurvature Category Curvature mathematics Category Differential geometry of surfaces Category Riemannian geometry ... more details
Height of curvature in the tooth can be defined as the line encircling a tooth at its greatest bulge to a selected path of insertion. Locations of height of curvature For the outer surfaces of all teeth, the height of curvature is located in the cervical third of the teeth. In the inner surfaces of anterior teeth , both upper & lower teeth, the height of curvature is also located in the cervical third of the tooth. In the posterior teeth , both in upper and lower jaw, the height of contour is found at the middle third of the inner surface of the tooth. The lower second premolar proposes an exception as its height of curvature in inner surface is located in the Occlusion dentistry occlusal third of the inner surface. Functions The height of contours have great functions to mouth oral cavity They allow the food to be deflected allowing proper degree of massage to the gingiva . They prevent the food of being accumlated at the tooth. Holding the gingiva under definite tension. Mal developed height of curvature In case of under developed curvature, gingival recession may result. In case of over developed curvature, food will accumlate and there will be no massage to the gingiva and chronic inflammation may result. Unreferenced date December 2009 Category Dentistry definitions ... more details
File Winding Number Around Point.svg thumb 300px This curve has total curvature 6 , and index turning number 3, though it only has winding number 2 about p . In mathematics mathematical study of the differential geometry of curves , the total curvature of a immersion mathematics immersed plane curve is the integral of curvature along a curve taken with respect to arclength math int a b k s ,ds. math The total curvature of a closed curve is always an integer multiple of 2&pi , called the index of the curve, or turning number it is the winding number of the unit tangent about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces. Comparison to surfaces This relationship between a local invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher dimensional Riemannian geometry such as the Gauss Bonnet theorem . Invariance According to the Whitney Graustein theorem , the total curvature is invariant under a regular homotopy of a curve it is the degree of the Gauss map . However, it is not invariant under homotopy passing through a kink cusp changes the turning number by 1. By contrast, winding ... no 180 angles have well defined total curvature, interpreting the curvature as point masses at the angles. The total curvature of a curve &gamma in a higher dimensional Euclidean space equipped with its ..., and computing the total curvature of the resulting curve. That is, the total curvature of a curve ... sub n &minus 1 sub is last Frenet curvature the torsion of curves torsion of the curve and sgn is the signum function . The minimum total curvature of any three dimensional curve representing a given ... title On the Total Curvature of Knots first John W. last Milnor authorlink John Milnor journal ... total curvature year 2007 id arxiv math 0606007 . Category Curves Category Curvature mathematics Category ... more details
Infobox Disease Name PAGENAME Image Spinal column curvature.png Caption Different regions curvatures of the vertebral column DiseasesDB ICD10 ICD10 M 40 m 40 ICD10 M 41 m 40 , ICD10 Q 76 3 q 65 ICD10 Q 76 4 q 65 ICD9 ICD9 737 , ICD9 756.1 ICDO OMIM MedlinePlus eMedicineSubj eMedicineTopic MeshID D013121 Although spinal curvature or curvature of spine can refer to the normal concave and convex curvature of the spine, in clinical contexts, the phrase usually refers to deviations from the expected curvature, even when that difference is a reduction in curvature. Types include kyphosis , lordosis , and scoliosis . The thoracic and sacral pelvic curves develop in the fetus. Around 6 months after birth the cervical curve appears which helps hold the head up. Around one year of age the lumbar curve develops which helps with balance and walking. The cervical and lumbar curves are considered secondary curves whereas the thoracic and sacral curves are primary. medicine stub Dorsopathies Congenital malformations and deformations of musculoskeletal system Category Deforming dorsopathies Category Congenital disorders of musculoskeletal system la Curvatura columnae vertebralis ... more details
orphan date April 2010 Membrane curvature is the geometrical measure or characterization of the curvature ... concentration is reached. Basic Geometry of Curvature A biological membrane is commonly described ..., R1 and R2, are called the principal radii of curvature, and their inverse values are referred ... Brandeis University, Waltham,1970 ref . math c1 1 R1 math math c2 1 R2 math File Curvature radii.JPG ... shapes, such as cylinder, plane, sphere and saddle. Analysis of the principal curvature is important ... Saddle Even though often membrane curvature is thought to be a completely spontaneous process, thermodynamically speaking there must be factors actuating as the driving force for curvature to exist. Currently, there are some postulated mechanisms for accepted theories on curvature nonetheless, undoubtedly .... Driving forces for membrane Curvature Lipid Spontaneous Curvature Perhaps the most simple and intuitive driving force in membrane curvature is the natural spontaneous curvature exhibited by some ... spontaneously negative or positive curvature. Lipids such as DOPC dioleyl phosphatidyl choline ... curvature ref Martens, S., McMahon, H. T. Nature Reviews . 9, 543 556 2008 ref . On the other ..., in other words they exhibit positive spontaneous curvature ref Kamal, M et al. Measurement of the membrane curvature preference of phospholipids reveals only weak coupling between lipid shape and leaflet curvature. PNAS 2009 vol. 106 52 pp. 22245 50 ref . The table below lists experimentally determined ... L is the length of the cylinder, J sub B sub is the difference between the spontaneous curvature, J ... and plasma membrane . Journal of Cell Biology 148, 45 58 2000 . ref . So, the spontaneous curvature ... to produce such curvature. The lipids cholesterol, dioleoylphosphatidylethanolamine DOPE and diacylglycerol ... the potential to generate a large membrane curvature. However, even for these lipids, the required ... can Induce Curvature As mentioned previously, some biologically occurring lipids do exhibit spontaneous ... more details
In mathematics , the curvature of a measure defined on the Euclidean plane R sup 2 sup is a quantification of how much the measure s distribution of mass is curved . It is related to notions of curvature in geometry . In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov accordingly, it may be referred to as the Melnikov curvature or Menger Melnikov curvature . Melnikov and Verdera 1995 established a powerful connection between the curvature of measures and the Cauchy integral formula Cauchy kernel . Definition Let be a Borel measure on the Euclidean plane R sup 2 sup . Given three distinct points x , y and z in R sup 2 sup , let R x , y , z be the radius of the Euclidean circle that joins all three of them, or if they are Line geometry collinear . The Menger curvature c x , y , z is defined to be math c x, y, z frac 1 ... 0 if any of the points x , y and z coincide. The Menger Melnikov curvature c sup 2 sup ... to the curvature of at a given point x math c 2 mu x iint mathbb R 2 c x, y, z 2 , mathrm d mu y mathrm ... The trivial measure has zero curvature. A Dirac measure sub a sub supported at any point a has zero curvature. If is any measure whose support measure theory support is contained within a Euclidean line L , then has zero curvature. For example, one dimensional Lebesgue measure on any line or line segment has zero curvature. The Lebesgue measure defined on all of R sup 2 sup has infinite curvature ... has curvature 1 r . Relationship to the Cauchy kernel In this section, R sup 2 sup is thought ... boundedness of the Cauchy kernel to the curvature of measures. They proved that if there is some ... > 0. Here c sub sub denotes a truncated version of the Menger Melnikov curvature in which ... Analytic capacity a discrete approach and the curvature of measure journal Sbornik Mathematics Mat ... 3 Category Curvature mathematics Category Geometry Category Measure theory ... more details
In mathematics , the Menger curvature of a triple of points in n dimension al Euclidean space R sup n sup is the Multiplicative inverse reciprocal of the radius of the circle that passes through the three points. It is named after the Austria n United States American mathematician Karl Menger . Definition Let x , y and z be three points in R sup n sup for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let R sup n sup be the Plane mathematics Euclidean plane spanned by x , y and z and let C be the unique circle Euclidean circle in that passes through x , y and z the Circumscribed circle circumcircle of x , y and z . Let R be the radius of C . Then the Menger curvature c x , y , z of x , y and z ... spanned by x , y and z . Another way of computing Menger curvature is the identity math c x,y,z frac ... spanned by x , y , z . Menger curvature may also be defined on a general metric space . If X is a metric ... mathbb R 2 math . Define the Menger curvature of these points to be math c X x,y,z c f x ,f y ... x,y,z is independent of the choice of f . Integral Curvature Rectifiability Menger curvature can be used ... measure restricted to the set math E math . ref cite journal last Leger first J. title Menger curvature ... of Mathematics ref . The basic intuition behind the result is that Menger curvature measures how straight ... der Mosel title Regularizing and self avoidance e ects of integral Menger curvature journal Institut ... journal last Yong Lin and Pertti Mattila title Menger curvature and math C 1 math regularity of fractals ... Capacity, Rectifiability, Menger Curvature and the Cauchy Integral publisher Springer year 2000 ... See also Curvature of a measure Menger Melnikov curvature of a measure External links cite web url ... Curvature last Leymarie first F. accessdate 2007 11 19 year 2003 month September archiveurl http ... 2111&ndash 2119 doi 10.1090 S0002 9939 00 05264 3 issue 7 Category Curvature mathematics Category Geometry ... more details
This article is about the curvature of affine plane curves, not to be confused with the curvature of an affine connection . Special affine curvature , also known as the equi affine curvature or affine curvature , is a particular type of curvature that is defined on a plane curve that remains unchanged ... area . The curves of constant equi affine curvature k are precisely all non singular conic section ... with k 0 are hyperbola s. The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle , the unique circle making second order contact with the curve at the point. In the same way, the special affine curvature of a curve at a point P is the special affine curvature of its hyperosculating conic , which is the unique conic making fourth order contact ... curvature refers to a differential invariant of the affine group general affine group , which may readily obtained from the special affine curvature k by k sup 3 2 sup d k d s , where s is the special affine arc length. Where the general affine group is not used, the special affine curvature k is sometimes confusingly also called the affine curvature harv Shirokov 2001b . Formal definition Special affine arclength To define the special affine curvature, it is necessary first to define the special ... with respect to its special affine arclength. Special affine curvature Suppose that s is a curve parameterized with its special affine arclength. Then the special affine curvature or equi affine curvature is given by math k s det begin bmatrix beta s & beta s end bmatrix . math Here ... the special affine curvature is math begin align k t & frac x y x y x y x y 5 3 frac 1 2 left ... with respect to x harvnb Blaschke 1923 loc 5 harvnb Shirokov 2001a . Affine curvature Suppose as above ... with the special affine arclength described above . The second is referred to as the affine curvature ... affine arclength with constant affine curvature k . Let math C beta s begin bmatrix beta s & beta ... more details
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