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Coordinate Systems For The Hyperbolic Plane In the hyperbolic plane , as in the Euclidean plane , each point can be uniquely identified by two real numbers . Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the twodimensional hyperbolic plane. In the descriptions below the constant Gaussian curvature Gaussian curvature of the plane is −1. Sinh , cosh and tanh are hyperbolic functions [...More...]  "Coordinate Systems For The Hyperbolic Plane" on: Wikipedia Yahoo 

Horocycle In hyperbolic geometry , a HOROCYCLE (Greek : ὅριον + κύκλος — border + circle, sometimes called an ORICYCLE, ORICIRCLE, or LIMIT CIRCLE) is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction . It is the twodimensional example of a horosphere (or orisphere). The centre of a horocycle is the ideal point where all normal geodesics asymptotically converge. Two horocycles who have the same centre are concentric . While it looks that two concentric horocycles cannot have the same length or curvature, in fact any two horocycles are congruent . A horocycle can also be described as the limit of the circles that share a tangent in a given point, as their radii go towards infinity . In Euclidean geometry Euclidean geometry , such a "circle of infinite radius" would be a straight line, but in hyperbolic geometry it is a horocycle (a curve) [...More...]  "Horocycle" on: Wikipedia Yahoo 

Beltrami–Klein Model In geometry, the BELTRAMI–KLEIN MODEL, also called the PROJECTIVE MODEL, KLEIN DISK MODEL, and the CAYLEY–KLEIN MODEL, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or ndimensional unit ball ) and lines are represented by the chords , straight line segments with ideal endpoints on the boundary sphere. The BELTRAMI–KLEIN MODEL is named after the Italian geometer Eugenio Beltrami and the German Felix Klein Felix Klein while "Cayley" in CAYLEY–KLEIN MODEL refers to the English geometer Arthur Cayley Arthur Cayley . The Beltrami–Klein model Beltrami–Klein model is analogous to the gnomonic projection of spherical geometry , in that geodesics (great circles in spherical geometry) are mapped to straight lines [...More...]  "Beltrami–Klein Model" on: Wikipedia Yahoo 

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. An order7 triangular tiling has equilateral triangles with 2π/7 radian internal angles [...More...]  "Hyperbolic Triangle" on: Wikipedia Yahoo 

Nikolai Lobachevsky NIKOLAI IVANOVICH LOBACHEVSKY (Russian : Никола́й Ива́нович Лобаче́вский; IPA: ( listen ); 1 December 1792 – 24 February 1856) was a Russian mathematician and geometer , known primarily for his work on hyperbolic geometry , otherwise known as Lobachevskian geometry . William Kingdon Clifford William Kingdon Clifford called Lobachevsky Lobachevsky the " Copernicus Copernicus of Geometry" due to the revolutionary character of his work [...More...]  "Nikolai Lobachevsky" on: Wikipedia Yahoo 

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 leftlimiting 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 halfplane 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 [...More...]  "Ideal Point" on: Wikipedia Yahoo 

Origin (mathematics) In mathematics , the ORIGIN of a Euclidean space Euclidean space is a special point , usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry . CONTENTS * 1 Cartesian coordinates * 2 Other coordinate systems * 3 See also * 4 References CARTESIAN COORDINATESIn a Cartesian coordinate system Cartesian coordinate system , the origin is the point where the axes of the system intersect. The origin divides each of these axes into two halves, a positive and a negative semiaxis [...More...]  "Origin (mathematics)" on: Wikipedia Yahoo 

Poincaré Disk Model In geometry, the POINCARé DISK MODEL also called the CONFORMAL DISK MODEL, is a model of 2dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk , and the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk. Along with the Klein model and the Poincaré halfspace model , it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré Henri Poincaré , because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. The POINCARé BALL MODEL is the similar model for 3 or ndimensional hyperbolic geometry in which the points of the geometry are in the ndimensional unit ball [...More...]  "Poincaré Disk Model" on: Wikipedia Yahoo 

Hyperboloid Model In geometry , the HYPERBOLOID MODEL, also known as the MINKOWSKI MODEL or the LORENTZ MODEL (after Hermann Minkowski Hermann Minkowski and Hendrik Lorentz ), is a model of ndimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a twosheeted hyperboloid in (n+1)dimensional Minkowski space Minkowski space and mplanes are represented by the intersections of the (m+1)planes in Minkowski space Minkowski space with S+. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the ndimensional hyperbolic space is closely related to the Beltrami–Klein model Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group [...More...]  "Hyperboloid Model" on: Wikipedia Yahoo 

International Standard Book Number The INTERNATIONAL STANDARD BOOK NUMBER (ISBN) is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation (except reprintings) of a book. For example, an ebook , a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, and 10 digits long if assigned before 2007. The method of assigning an ISBN is nationbased and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9digit STANDARD BOOK NUMBERING (SBN) created in 1966. The 10digit ISBN format was developed by the International Organization for Standardization (ISO) and was published in 1970 as international standard ISO 2108 (the SBN code can be converted to a ten digit ISBN by prefixing it with a zero) [...More...]  "International Standard Book Number" on: Wikipedia Yahoo 

Special SPECIAL or SPECIALS may refer to: CONTENTS * 1 Music * 2 Film and television * 3 Other uses * 4 See also MUSIC * Special (album) , a 1992 album by Vesta Williams * "Special" (Garbage song) , 1998 * "Special" (Mew song) , 2005 * "Special" (Stephen Lynch song) , 2000 * The Specials The Specials , a British band * "Special", a song by Violent Femmes on The Blind Leading the Naked * "Special", a song on [...More...]  "Special" on: Wikipedia Yahoo 

Triangle Center In geometry , a TRIANGLE CENTER (or TRIANGLE CENTRE) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles , i.e. a point that is in the middle of the figure by some measure. For example the centroid , circumcenter , incenter and orthocenter were familiar to the ancient Greeks , and can be obtained by simple constructions. Each of these classical centers has the property that it is invariant (more precisely equivariant ) under similarity transformations . In other words, for any triangle and any similarity transformation (such as a rotation , reflection , dilation , or translation ), the center of the transformed triangle is the same point as the transformed center of the original triangle. This invariance is the defining property of a triangle center. It rules out other wellknown points such as the Brocard points which are not invariant under reflection and so fail to qualify as triangle centers [...More...]  "Triangle Center" on: Wikipedia Yahoo 

Gyrovector Space A GYROVECTOR SPACE is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry Euclidean geometry . Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups . Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities (also called boosts  "boosts" are aspects of relative velocities , and should not be conflated with "translations "). This is achieved by introducing "gyro operators"; two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity [...More...]  "Gyrovector Space" on: Wikipedia Yahoo 

Hyperbola In mathematics , a HYPERBOLA (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane , defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows . The hyperbola is one of the three kinds of conic section , formed by the intersection of a plane and a double cone . (The other conic sections are the parabola and the ellipse . A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola [...More...]  "Hyperbola" on: Wikipedia Yahoo 

Gyrovector A GYROVECTOR SPACE is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry Euclidean geometry . Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups . Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities (also called boosts  "boosts" are aspects of relative velocities , and should not be conflated with "translations "). This is achieved by introducing "gyro operators"; two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity [...More...]  "Gyrovector" on: Wikipedia Yahoo 

Hypercycle (geometry) In hyperbolic geometry , a HYPERCYCLE, HYPERCIRCLE or EQUIDISTANT CURVE is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the axis, center, or base line of the hypercycle. The lines perpendicular to the axis, which is also perpendicular to the hypercycle are called the normals of the hypercycle. The segments of the normal between the axis, and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity [...More...]  "Hypercycle (geometry)" on: Wikipedia Yahoo 