Differential geometry

Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus to study problems in geometry. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and nineteenth century. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian In Mathematics, the differential geometry of surfaces deals with smooth Surfaces with various additional structures most often a Riemannian metric Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. It is closely related with differential topology and with the geometric aspects of the theory of differential equations. In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Grigori Perelman's proof of the Poincare conjecture using the techniques of Ricci flow demonstrated the power of the differential-geometric approach to questions in topology and highlighted the important role played by the analytic methods. Grigori Yakovlevich Perelman (Григорий Яковлевич Перельман born 13 June 1966 in Leningrad, USSR (now St In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among In Differential geometry, the Ricci flow is an intrinsic Geometric flow —a process which deforms the metric of a Riemannian manifold —in this case in Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

1 Branches of differential geometry
2 Bundles and connections
3 Intrinsic versus extrinsic
4 Applications of differential geometry
5 See also
6 Notes
7 References
8 External links

Branches of differential geometry

Riemannian geometry

Main article: Riemannian geometry

Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric, a notion of a distance expressed by means of a positive definite symmetric bilinear form defined on the tangent space at each point. Elliptic geometry is also sometimes called Riemannian geometry. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point "infinitesimally", i. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. e. in the first order of approximation. Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all admit natural analogues in Riemannian geometry. Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically The notion of a directional derivative of a function from the multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable For the Mechanical engineering and Architecture usage see Isometric projection. This notion can also be defined locally, i. e. for small neighborhoods of points. Any two regular curves are locally isometric. However, Theorema Egregium of Gauss showed that already for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same. Gauss's Theorema Egregium (Latin "Remarkable Theorem" is a foundational result in Differential geometry proved by Carl Friedrich Gauss that concerns the Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat. In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express An important class of Riemannian manifolds is formed by the Riemannian symmetric spaces, whose curvature is constant. They are the closest to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry

Pseudo-Riemannian geometry

Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed A special case of this is a Lorentzian manifold which is the mathematical basis of Einstein's general relativity theory of gravity. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i. In Mathematics, particularly Differential geometry, a Finsler manifold is a Differentiable manifold M with a Banach norm defined over In Mathematics, particularly Differential geometry, a Finsler manifold is a Differentiable manifold M with a Banach norm defined over e. a Banach norm defined on each tangent space. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis A Finsler metric is a much more general structure than a Riemannian metric. A Finsler structure on a manifold M is a function F : TM → [0,∞) such that:

F(x, my) = mF(x,y) for all x, y in TM,
F is infinitely differentiable in TM − {0},
The vertical Hessian of F² is positive definite.

Symplectic geometry

Main article: Symplectic geometry

Symplectic geometry is the study of symplectic manifolds. Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that In Linear algebra, a skew-symmetric (or antisymmetric) matrix is a Square matrix A whose Transpose is also its negative In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where e. , a nondegenerate 2-form ω, called the symplectic form. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: dω = 0.

A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically Non-degenerate skew-symmetric bilinear forms can only exist on even dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Lagrange on analytical mechanics and later in Jacobi's and Hamilton's formulation of classical mechanics. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System Analytical mechanics is a term used for a refined highly mathematical form of Classical mechanics, constructed from the Eighteenth century onwards as a formulation Carl Gustav Jacob Jacobi ( December 10, 1804 - February 18, 1851) was a Prussian Mathematician, widely considered to be Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.

By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry Darboux's theorem is a Theorem in the mathematical field of Differential geometry and more specifically Differential forms, partially generalizing The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré-Birkhoff theorem, conjectured by Henri Poincaré and proved by George Birkhoff in 1912. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician George David Birkhoff ( 21 March 1884, Overisel Michigan - 12 November 1944, Cambridge Massachusetts) was an American It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then the map has at least two fixed points. ^[1]

Contact geometry

Main article: Contact geometry

Contact geometry deals with certain manifolds of odd dimension. In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2n+1)-dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the Near each point p, a hyperplane distribution is determined by a nowhere vanishing 1-form $α$ , which is unique up to multiplication by a nowhere vanishing function:

$H_p = \ker\alpha_p\subset T_{p}M.$

A local 1-form on M is a contact form if the restriction of its exterior derivative to H is a non-degenerate 2-form and thus induces a symplectic structure on H_p at each point. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms If the distribution H can be defined by a global 1-form $α$ then this form is contact if and only if the top-dimensional form

$\alpha\wedge (d\alpha)^n$

is a volume form on M, i. In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold. e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.

Complex and Kähler geometry

Complex differential geometry is the study of complex manifolds. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, An almost complex manifold is a real manifold $M$ , endowed with a tensor of type (1,1), i. In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually e. a vector bundle endomorphism (called an almost complex structure)

$J:TM\rightarrow TM$ , such that $J 2 = - 1$ . In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space.

It follows from this definition that an almost complex manifold is even dimensional.

An almost complex manifold is called complex if $N J = 0$ , where $N J$ is a tensor of type (2,1) related to $J$ , called the Nijenhuis tensor (or sometimes the torsion). In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how An almost Hermitian structure is given by an almost complex structure J, along with a riemannian metric g, satisfying the compatibility condition $g (J X, J Y) = g (X, Y)$ . In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M An almost hermitian structure defines naturally a differential 2-form $ω J, g (X, Y): = g (J X, Y)$ . In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is The following two conditions are equivalent:

$N J = 0 and d ω = 0,$
$\nabla J=0,$

where $\nabla$ is the Levi-Civita connection of $g$ . In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i In this case, $(J, g)$ is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In Mathematics, a Kähler manifold is a Manifold with unitary structure (a ''U''(''n''-structure) satisfying an Integrability condition In particular, a Kähler manifold is both a complex and a symplectic manifold. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

CR geometry

CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds. In Mathematics, a CR manifold is a Differentiable manifold together with a geometric structure modeled on that of a real Hypersurface in a Complex In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n,

Bundles and connections

The apparatus of vector bundles, principal bundles, and connections on them plays an extraordinarily important role in the modern differential geometry. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G In Geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and A smooth manifold always carries a natural vector bundle, the tangent bundle. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires in addition some way to relate the tangent spaces at different points, i. e. a notion of parallel transport. In Geometry, parallel transport is a way of transporting geometrical data along smooth curves in a Manifold. An important example is provided by affine connections. In the mathematical field of Differential geometry, an affine connection is a geometrical object on a Smooth manifold which connects nearby Tangent For a surface in R³, tangent planes at different points can be identified using the flat nature of the ambient Euclidean space. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Riemannian geometry, the Levi-Civita connection serves a similar purpose. Elliptic geometry is also sometimes called Riemannian geometry. In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i More generally, differential geometers consider spaces with a vector bundle and a connection as a replacement for the notion of a Riemannian manifold. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In this approach, the bundle is external to the manifold and has to be specified as a part of the structure, while the connection provides a further enhancement. In physics, the manifold may be the spacetime and bundles and connections correspond to various physical fields. 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

Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object. The simplest results are those in the differential geometry of curves. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way.

The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of curvature and other structures such as connections, so there is a price to pay. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry In Geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and

These two points of view can be reconciled, i. e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem. The Nash embedding theorems (or imbedding theorems) named after John Forbes Nash, state that every n-dimensional Riemannian manifold can be isometrically )

Applications of differential geometry

Below are some examples of how differential geometry is applied to other fields of science and mathematics.

In physics, differential geometry is the language in which Einstein's general theory of relativity is expressed. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 According to the theory, the universe is a smooth manifold equipped with pseudo-Riemannian metric, which described the curvature of space-time. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry 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 Understanding this curvature is essential for the positioning of satellites into orbit around the earth. This article is about artificial satellites For natural satellites also known as moons see Natural satellite. Differential geometry is also indispensable in the study of gravitational lensing and black holes. A gravitational lens is formed when the light from a very distant bright source (such as a Quasar) is "bent" around a massive object (such as a cluster of A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e
In economics, differential geometry has applications to the field of econometrics^[2]. Economics is the social science that studies the production distribution, and consumption of goods and services. Econometrics is concerned with the tasks of developing and applying Quantitative or Statistical methods to the study and elucidation of economic principles
Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry. Geometric model(ling is the construction or use of geometric models Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data
In engineering, differential geometry can be applied to solve problems in digital signal processing ^[3]. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Digital signal processing ( DSP) is concerned with the representation of the signals by a sequence of numbers or symbols and the processing of these signals
In physics, the use of differential forms is useful in the study of electromagnetism. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of
In physics, differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Symplectic manifolds in particular can be used to study Hamiltonian systems. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Classical mechanics, a Hamiltonian system is a Physical system in which Forces are Velocity invariant

Notes

^ It is easy to show that the area preserving condition (or the twisting condition) cannot be removed. In Mathematics, the term integral geometry is used in two ways which although related imply different views of the content of the subject This is a list of Differential geometry topics See also Glossary of differential and metric geometry and List of Lie group topics. This is a Glossary of terms specific to Differential geometry and Differential topology. Algebra Theory of equations Hisab Algebra Theory of equations Hisab An understanding of Calculus and Differential equations is necessary for the understanding of Nonrelativistic physics. Discrete differential geometry is the study of discrete counterparts of notions in Differential geometry. Note that if one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.
^ Paul Marriott and Mark Salmon, "Applications of Differential Geometry to Econometrics".
^ Jonathan H. Manton, "On the role of differential geometry in signal processing" [1].

References

Theodore Frankel (2004). The geometry of physics: an introduction, second edition. ISBN 0-521-53927-7.
Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry (5 Volumes), 3rd Edition.
do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7. A classical geometric approach to differential geometry without the tensor machinery.
do Carmo, Manfredo Perdigao (1994). Riemannian Geometry.
McCleary, John (1994). Geometry from a Differentiable Viewpoint.
Bloch, Ethan D. (1996). A First Course in Geometric Topology and Differential Geometry.
Gray, Alfred (1998). Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. .
Burke, William L. (1985). Applied Differential Geometry.
ter Haar Romeny, Bart M. . Front-End Vision and Multi-Scale Image Analysis. ISBN 1-4020-1507-0.

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