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Line Coordinates
In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point. Lines in the plane There are several possible ways to specify the position of a line in the plane. A simple way is by the pair where the equation of the line is ''y'' = ''mx'' + ''b''. Here ''m'' is the slope and ''b'' is the ''y''-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates where the equation of the line is ''lx'' + ''my'' + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of ''l'' and ''m'' are the negative reciprocals of the ''x'' and ''y''-intercept respectively. The exclusion of lines passing through the origin can be resolved by using a system of three coordinates to speci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Fractional Transformation
In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional transformation is a ''transformation (function), transformation'' that is represented by a ''fraction'' whose numerator and denominator are ''linear polynomial, linear''. In the most basic setting, , and are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field (mathematics), field. The invertibility condition is then . Over a field, a linear fractional transformation is the restriction (mathematics), restriction to the field of a projective transformation or homography of the projective line. When are integer, integers (or, more generally, belong to an integral domain), is supposed to be a rational number (or to belong to the field of fractions of the integral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultraparallel Theorem
In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel lines has a unique common perpendicular (a hyperbolic line which is perpendicular to both lines). Hilbert's construction Let and be two ultraparallel lines. From any two distinct points and on s draw and perpendicular to with and on . If it happens that AB = CB', then the desired common perpendicular joins the midpoints of AC and BB' (by the symmetry of the Saccheri quadrilateral ACB'B). If not, we may suppose AB < CB' without loss of generality. Let E be a point on the line s on the opposite side of A from C. Take A' on CB' so that A'B' = AB. Through A' draw a line s' (A'E') on the side closer to E, so that the angle B'A'E' is the same as angle BAE. Then s' meets s in an ordinary point D'. Construct a point D on ray AE so that AD = A'D'. Then D' ≠ D. They are the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of events one temporal unit into the future in Minkowski space, the basis of special relativity. Eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split-complex Number
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+yj . The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number with its conjugate is N(z) := zz^* = x^2 - y^2, an isotropic quadratic form. The collection of all split-complex numbers z=x+yj for forms an algebra over the field of real numbers. Two split-complex numbers and have a product that satisfies N(wz)=N(w)N(z). This composition of over the algebra product makes a composition algebra. A similar algebra based on and component-wise operations of addition and multiplication, where is the quadratic form on also forms a quadratic space. The ring isomorphism \begin D &\to \mathbb^2 \\ x + yj &\mapsto (x - y, x + y) \end is an isometry of quadratic spaces. Split-complex numbers have many other na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Dual numbers can be added component-wise, and multiplied by the formula : (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, which follows from the property and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. History Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as , where is the angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal completed the next year, after a regulatory review. Thus, Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ... Well-known products include the '' Methods in Enzymology'' series and encyclopedias such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaak Yaglom
Isaak Moiseevich Yaglom (; 6 March 1921 – 17 April 1988) was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom. Yaglom received a Ph.D. from Moscow State University in 1945 as student of Veniamin Kagan. As the author of several books, translated into English, that have become academic standards of reference, he has an international stature. His attention to the necessities of learning (pedagogy) make his books pleasing experiences for students. The seven authors of his Russian obituary recount "…the breadth of his interests was truly extraordinary: he was seriously interested in history and philosophy, passionately loved and had a good knowledge of literature and art, often came forward with reports and lectures on the most diverse topics (for example, on Alexander Blok, Anna Akhmatova, and the Dutch painter M. C. Escher), actively took part in the work of the cinema club in Yaroslavl and the music club at the House of Composer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Quadric
In mathematics, the lines of a 3-dimensional projective space, ''S'', can be viewed as points of a 5-dimensional projective space, ''T''. In that 5-space, the points that represent each line in ''S'' lie on a quadric, ''Q'' known as the Klein quadric. If the underlying vector space of ''S'' is the 4-dimensional vector space ''V'', then ''T'' has as the underlying vector space the 6-dimensional exterior square Λ2''V'' of ''V''. The line coordinates obtained this way are known as Plücker coordinates. These Plücker coordinates satisfy the quadratic relation : p_ p_+p_p_+p_ p_ = 0 defining ''Q'', where : p_ = u_i v_j - u_j v_i are the coordinates of the line spanned by the two vectors ''u'' and ''v''. The 3-space, ''S'', can be reconstructed again from the quadric, ''Q'': the planes contained in ''Q'' fall into two equivalence classes, where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
