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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.Contents1 Polar coordinate system 2 Cartesianstyle coordinate systems2.1 Axial coordinates 2.2 Lobachevsky coordinates 2.3 Horocyclebased coordinate system3 Modelbased coordinate systems3.1 Beltrami coordinates 3.2 Poincaré coordinates 3.3 Weierstrass coordinates4 Others4.1 Gyrovector Gyrovector coordinates 4.2 Hyperbolic barycentric coordinates5 ReferencesPolar coordinate system[edit]Points in the polar coordinate system with pole O and polar axis L [...More...]  "Coordinate Systems For The Hyperbolic Plane" 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.[1] The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.Contents1 Properties similar to those of Euclidean lines 2 Properties similar to those of Euclidean circles 3 Other properties 4 Len [...More...]  "Hypercycle (geometry)" on: Wikipedia Yahoo 

Euclidean Plane Twodimensional space Twodimensional space or bidimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). In Mathematics, it is commonly represented by the symbol ℝ2. For a generalization of the concept, see dimension. Twodimensional space Twodimensional space can be seen as a projection of the physical universe onto a plane [...More...]  "Euclidean Plane" 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 [...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.[1] 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 Lorentz transformations to represent compositions of velocities (also called boosts  "boosts" are aspects of relative velocities, and should not be conflated with "translations") [...More...]  "Gyrovector Space" 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.[1] 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 Lorentz transformations to represent compositions of velocities (also called boosts  "boosts" are aspects of relative velocities, and should not be conflated with "translations") [...More...]  "Gyrovector" 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 [...More...]  "Hyperbola" 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 with S+. The hyperbolic distance function admits a simple expression in this model [...More...]  "Hyperboloid Model" 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 Klein model and the Poincaré halfspace model, it was proposed by Eugenio Beltrami Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry [...More...]  "Poincaré Disk Model" 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 Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German [...More...]  "Beltrami–Klein Model" 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 [...More...]  "Horocycle" 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 [...More...]  "Hyperbolic Triangle" on: Wikipedia Yahoo 

Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky Lobachevsky (Russian: Никола́й Ива́нович Лобаче́вский, IPA: [nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj] ( listen); 1 December [O.S [...More...]  "Nikolai Lobachevsky" 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 [...More...]  "Origin (mathematics)" on: Wikipedia Yahoo 

Ideal Point In hyperbolic geometry, an ideal point, omega point[1] 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 Poincaré disk model and the Klein disk model [...More...]  "Ideal Point" on: Wikipedia Yahoo 

Lambert Quadrilateral In geometry, a Lambert quadrilateral,[1] named after Johann Heinrich Lambert, is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle. A Lambert quadrilateral Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral [...More...]  "Lambert Quadrilateral" on: Wikipedia Yahoo 