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Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''projective space'') and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "Point at infinity, points at infinity") to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translation (geometry), translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Invariant Theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Line (geometry)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray (optics), ray of light. Lines are space (mathematics), spaces of dimension one, which may be Embedding, embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two Point (geometry), points (its ''endpoints''). Euclid's Elements, Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as Non-Euclidean geometry, non-Euclidean, Project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Point (geometry)
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist. In classical Euclidean geometry, a point is a primitive notion, defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called axioms, that they must satisfy; for example, ''"there is exactly one straight line that passes through two distinct points"''. As physical diagrams, geometric figures are made with tools such as a compass, scriber, or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve. A po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Configuration (geometry)
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points. Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book ''Geometrie der Lage'', in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book ''Anschauliche Geometrie'', reprinted in English as . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be ''realizable'' in that geometry), or as a type of abstract incidence geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Metric (mathematics)
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Projective Differential Geometry
In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group. This is a mixture of the approaches from Riemannian geometry of studying invariances, and of the Erlangen program of characterizing geometries according to their group symmetries. The area was much studied by mathematicians from around 1890 for a generation (by J. G. Darboux, George Henri Halphen, Ernest Julius Wilczynski, E. Bompiani, G. Fubini, Eduard Čech, amongst others), without a comprehensive theory of differential invariants emerging. Élie Cartan formulated the idea of a general projective connection, as part of his method of moving frames; abstractly speaking, this is the level of generality at which the Erlangen program can be reconciled with differential geometry, while it also develops the ol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be Irreducible component, irreducible, which means that it is not the Union (set theory), union of two smaller Set (mathematics), sets that are Closed set, closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Finite Geometry
A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective space, projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius plane, Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometry, inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Synthetic Geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates, and at present called axioms. After the 17th-century introduction by René Descartes of the coordinate method, which was called analytic geometry, the term "synthetic geometry" was coined to refer to the older methods that were, before Descartes, the only known ones. According to Felix Klein Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates. The first systematic approach for synthetic geometry is Euclid's ''Elements''. However, it appeared at the end of the 19th century that Euclid's postulates were not sufficient for characterizing geometry. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Classical Groups
In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph '' The Classical Groups''. The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |