|
Lobachevskian
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.) 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. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, known as the Lobachevsky integral formula. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. Biography Nikolai Lobachevsky was born either in or near the city of Nizhny Novgorod in the Russian Empire (now in Nizhny Novgorod Oblast, Russia) in 1792 to parents of Russian and Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya.Victor J. Katz. ''A history of mathematics: Introduction''. Addison-Wesley. 2009. p. 842. Stephen Hawking. ''God Created the Integers: The Mathematical Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Euclidean Geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point ''A'', which is not on , there is exactly one line through ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloid in (''n''+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and ''m''-planes are represented by the intersections of (''m''+1)-planes passing through the origin in Minkowski space with ''S''+ or by wedge products of ''m'' vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the ''n''-sphere is embedded in (''n''+1)-dimensional Euclidean space. Other models of hyperbolic space can be thought of as map projections of ''S''+: the Beltrami–Klein model is the projection of ''S''+ through the origin onto a plane perpendic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Of Parallelism
In hyperbolic geometry, the angle of parallelism \Pi(a) , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point. Let ''a'' be the length of this perpendicular segment, and \Pi(a) be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel, : \lim_ \Pi(a) = \tfrac\pi\quad\text\quad\lim_ \Pi(a) = 0. There are five equivalent expressions that relate '' \Pi(a)'' and ''a'': : \sin\Pi(a) = \operatorname a = \frac =\frac \ , : \cos\Pi(a) = \tanh a = \frac \ , : \tan\Pi(a) = \operatorname a = \frac = \frac \ , : \tan \left( \tfrac\Pi(a) \right) = e^, : \Pi(a) = \tfrac\pi - \operatorname(a), where sinh, cosh, tanh, sech and cs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Triangle
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three points called ''angles'' or ''vertices''. Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces. Definition A hyperbolic triangle consists of three non-collinear points and the three segments between them. Properties Hyperbolic triangles have some properties that are analogous to those of triangles in Euclidean geometry: *Each hyperbolic triangle has an inscribed circle but not every hyperbolic triangle has a circumscribed circle (see below). Its vertices can lie on a horocycle or hypercycle. Hyperbolic triangles have some properties that are analogous to those of triangles in spherical or elliptic geometry: *Two triangles with the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limiting Parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R). Thus it is useful to make a new definition concerning parallels in neutral geometry. If there are closest parallels to a given line they are known as the limiting parallel, asymptotic parallel or horoparallel (horo from el, ὅριον — border). For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal. If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle. Definition A ray Aa is a limiting parallel to a ray Bb if they are coterminal or if they lie on distinct lines not equal to the line AB, they do not meet, and every ray in the interior of the angle BAa meets the ray Bb. Properties Distinct lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Models Of The Hyperbolic Plane
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models can be divided into physical models (e.g. a model plane) and abstract models (e.g. mathematical expressions describing behavioural patterns). Abstract or conceptual models are central to philosophy of science, as almost every scientific theory effectively embeds some kind of model of the physical or human sphere. In commerce, "model" can refer to a specific design of a product as displayed in a catalogue or show room (e.g. Ford Model T), and by extension to the sold product itself. Types of models include: Physical model A physical model (most commonly referred to simply as a model but in this context distinguished from a conceptual model) is a smaller or larger physical copy of an object. The object being modelled may be small (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supplementary Angles
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection. ''Angle'' is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. History and etymology The word ''angle'' comes from the Latin word ''angulus'', meaning "corner"; cognate words are the Greek ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as ''single elliptic geometry'' whereas spherical geometry is sometimes referred to as ''double elliptic geometry''. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. Definitions In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' converge to ''l'' at ''ideal points''. Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not intersect at an ideal point and such points, although well-defined, do not belong to the hyperbolic space itself. The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model. While the real line forms the Cayley absolute of the Poincaré half-plane model . Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point. Properties * The hyperbolic distance between an ideal point and an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century. Euclid's ''Elements'' has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not sufficient as a basis of Euclidean geometry, other systems, such as Hilbert's axioms without the parallel axiom, are used. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. Properties It might be imagined that absolute geometry is a rather weak system, but that is not the case. Indeed, in Euclid's ''Elements'', the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri–Legendre theorem, which states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
