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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 two-dimensional 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 Cartesian-style coordinate systems2.1 Axial coordinates 2.2 Lobachevsky coordinates 2.3 Horocycle-based coordinate system3 Model-based 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
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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
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Euclidean Plane
Two-dimensional space
Two-dimensional space
or bi-dimensional 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. Two-dimensional space
Two-dimensional space
can be seen as a projection of the physical universe onto a plane
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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
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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")
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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")
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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
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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 n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space
Minkowski space
and m-planes 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
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Poincaré Disk Model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional 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 half-space model, it was proposed by Eugenio Beltrami
Eugenio Beltrami
who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry
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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 n-dimensional 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
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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 two-dimensional 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
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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
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Nikolai Lobachevsky
Nikolai Ivanovich Lobachevsky
Lobachevsky
(Russian: Никола́й Ива́нович Лобаче́вский, IPA: [nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj] ( listen); 1 December [O.S
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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
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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 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
Poincaré disk model
and the Klein disk model
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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
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