: À l'aide de l'application conforme mentionnée ci-dessus entre le disque unité et le demi-plan supérieur, ce modèle peut être transformé en un demi-plan de Poincaré du plan hyperbolique. Using the above-mentioned conformal map between the open unit disk and the upper half-plane, this model can be turned into the Poincaré half-plane model of the hyperbolic plane. The open unit disk D (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to H (see "Poincaré metric"), meaning that it is usually possible to pass between H and D. It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. Then there is an affine mapping that takes A to B. The midpoint between that intersection and the given non-central point is the center of the model circle. 2 The distance between two points measured in this metric along such a geodesic is: From Infogalactic: the planetary knowledge core, Creating the line through two existing points, Creating the circle through one point with center another point, Given a circle find its (hyperbolic) center, Flavors of Geometry, MSRI Publications, Volume 31, 1997 , Hyperbolic Geometry, J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, page 87, Figure 19. ⁡ Half-space. The Poincaré metric provides a hyperbolic metric on the space. ) The affine transformations of the upper half-plane include (1) shifts (x,y) → (x + c, y), c ∈ ℝ, and (2) dilations (x,y) → (λ x, λ y), λ > 0. Other articles where Geodesic is discussed: relativity: Curved space-time and geometric gravitation: …the shortest natural paths, or geodesics—much as the shortest path between any two points on Earth is not a straight line, which cannot be constructed on that curved surface, but the arc of a great circle route. ‎This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. ⁡ In the Poincaré case, lines are given by diameters of the circle or arcs. curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose origin is on the x-axis) and straight vertical rays perpendicular to the x-axis. Draw the model circle around that new center and passing through the given non-central point. The calculation starts with the path integral on the Poincaré upper half-plane with a magnetic field. ρ In a series of works on hyperbolic space (beginning with [1]), Poincaré found a Riemannian metric (now called the Poincaré metric) with constant curvature $-1$ on the upper half-space, given by Note that the action is transitive, in that for any , there exists a such that . 2 Livraison en Europe à 1 centime seulement ! The space $$\mathbb{U}$$ is called the upper half-plane of $$\mathbb{C}\text{. Let point q be the intersection of this line and the x- axis. θ The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. 2 The straight lines in the hyperbolic space (geodesics for this metric tensor, i.e. {\displaystyle \rho (\theta )=\cos \theta .}. θ Rigorous path integral treatments on the Poincaré upper half-plane with a magnetic field and for the Morse potential are presented. Since any element z in H is mapped to i by some element of PSL(2,R), this means that the isotropy subgroup of any z is isomorphic to SO(2). Furthermore, granting the validity of the Heisenberg equation in a curved space, the Ehrenfest theorem is generalized and interpreted with the new position-dependent differential operator in a curved space. is the reciprocal of that length. Poincaré series for discrete Moebius groups acting on the upper half space. The lower half-plane, defined by y < 0, is equally good, but less used by convention. + Draw a line tangent to the circle going through q. Yet another space interesting to number theorists is the Siegel upper half-space Hn, which is the domain of Siegel modular forms. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. It was observed that the “free particle” that is indeed free along the y-axis, actually behaves as if it is bounded by the curved space along the x-axis , due to the term 1 x in ϕ (x). 2 Escher's Circle Limit ExplorationThis exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry may look like. Construct a tangent to that line at the non-central point. The closed upper half-plane is the union of the upper half-plane and the real axis. The red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid. In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H { | y > 0; x, y ∈ R } {\displaystyle \{|y>0;x,y\in \mathbb {R} \}}, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. where s measures the length along a (possibly curved) line. Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane by Audrey Terras at AbeBooks.co.uk - ISBN 10: 1461479711 - ISBN 13: 9781461479710 - Springer - … 10.1007/978-3-319-05317-2_12. In consequence, the upper half-plane becomes a metric space. The generic name of this metric space is the hyperbolic plane. }, Z can be recognized as the circle of radius 1/2 centered at (1/2, 0), and as the polar plot of Specifically: Geodesics are arcs of circles which meet the edge of the disk at 90°. cos The boundary of upper-half plane is the real axis together with the in nit.y Riemannian metric and distance. . In the latter case p and q lie on a ray perpendicular to the boundary and logarithmic measure can be used to define a distance that is invariant under dilation. {\displaystyle \rho (\theta )=\cos \theta } The Poincaré disk model of hyperbolic geometry may be transferred to the upper half-plane model via a Möbius transformation built from two inversions as follows: Invert about the circle C centered at i passing through -1 and 1 as in Figure 5.5.2. In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis. Découvrez et achetez Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane. 0 Read "Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane" by Audrey Terras available from Rakuten Kobo. Quasi-static energy minimization naturally induces bursty plastic flow and shape change in the crystal due to the underlying coordinated basin-hopping local strain activity. It is remarkable that the entire structure of the space follows from the metric, although not without some effort. Basic Explorations 1. distribution-valued images based on Poincaré upper-half plane representation Jesus Angulo, Santiago Velasco-Forero To cite this version: Jesus Angulo, Santiago Velasco-Forero. ⁡ rediscovered the Liouville?Beltrami upper half-plane model in 1882 and this space is usually called the Poincar? curves which minimize the distance) are represented in this model by circular arcs normal to the z = 0-plane (half-circles whose origin is on the z = 0-plane) and straight vertical rays normal to the z = 0-plane. We use a natural parameterization of strain space via the upper complex Poincaré half-plane. Poincar? Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane book. Thus, the general unit-speed geodesic is given by. θ We use a natural parameterization of strain space via the upper complex Poincaré half-plane. Erase the part which is on or below the x-axis. Noté /5. Draw the radial line (half-circle) between the two given points as in the previous case. Find the intersection of these two lines to get the center of the model circle. Metric and Geodesics This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. θ . Draw a horizontal line through the non-central point. All the calculations are mainly based on Fourier‐expansions of the Feynman kernels which can be easily performed. The upper half-plane is tessellated into free regular sets by the modular group SL(2,Z). ) It is also faithful, in that if for all z in H, then g=e. } In mathematics, the upper half-plane H is the set of points (x, y) in the Cartesian plane with y > 0. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model. Any state of polarization can be uniquely represented by a point on or within a unit spherecentered on a rectangular xyz-coordinate system as shown below. If the two points are not on a vertical line: If the two given points lie on a vertical line and the given center is above the other given point: If the two given points lie on a vertical line and the given center is below the other given point: Creating the point which is the intersection of two existing lines, if they intersect: Creating the one or two points in the intersection of a line and a circle (if they intersect): Creating the one or two points in the intersection of two circles (if they intersect): The group of orientation-preserving isometries of. In the former case p and q lie on a circle centered at the intersection of their perpendicular bisector and the boundary. Remember that in the half-plane case, the lines were either Euclidean lines, perpendicular onto the real line, or half-circles, also perpendicular onto the real line. Thus, functions that are periodic on a square grid, such as modular forms and elliptic functions, will thus inherit an SL(2,Z) symmetry from the grid. It is the domain of many functions of interest in complex analysis, especially modular forms. 1 Draw the model circle around that new center and passing through the given non-central point. In this handout we will give this interpretation and verify most of its properties. In hyperbolic geometry, the Poincaré upper-halfplanemodel(originatedwithBeltramiandalsoknownasLobachevskii space in Soviet scienti c literature) is the space H2 together with the Poincaré metric (gkl) = diag (1 y2 1 y2) such that the hyperbolic arc … Alternatively, the bundle of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle, is isomorphic to PSL(2,R). Poincaré’s inspi-ration was to view this tessellated plane as the boundary of the upper half-space, which turns out to be none other than non-Euclidean 3-dimensional space. It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. Tohoku Math. Draw a horizontal line through that point of tangency and find its intersection with the vertical line. , This transparently displays the constraints imposed by lattice symmetry on the energy landscape. springer, This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. The Poincaré metric provides a hyperbolic metric on the space. Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x, y) in the plane endowed with Cartesian coordinates. where s measures length along a possibly curved line. < In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane (denoted below as H), together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.. It is the closure of the upper half-plane. In the present paper we study the tangent bundle TG+ of G+, as a homogeneous Finsler space of a natural group of invertible matrices in M2(A), identifying TG+ with the Poincaré half-space H of A, H={h∈A:Im(h)≥0,Im(h) invertible}. θ The projective linear group PGL(2,C) acts on the Riemann sphere by the Möbius transformations. {\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta ,} The Poincaré half-plane … Achetez neuf ou d'occasion The Poincare upper half plane is an interpretation of the primitive terms of Neutral Ge- ometry, with which all the axioms of Neutral geometry are true, and in which the hyperbolic parallel postulate is true. Frank Nielsen. Poincaré half-plane: lt;p|>| In |non-Euclidean geometry|, the |Poincaré half-plane model| is the |upper half-plane| (d... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. The stabilizer of i is the rotation group. One also frequently sees the modular group SL(2,Z). Complex numbers with non-negative imaginary part, Learn how and when to remove this template message, https://en.wikipedia.org/w/index.php?title=Upper_half-plane&oldid=965122890, Articles needing additional references from February 2010, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 29 June 2020, at 14:57. Find the intersection of the two given semicircles (or vertical lines). In the Poincaré disk model, geodesics appear curved. , Traductions en contexte de "Poincaré half plane" en anglais-français avec Reverso Context : Find the intersection of the two given circles. For other uses, see Half-space (disambiguation). Poincaré series for discrete Moebius groups acting on the upper half space. Poincaré is involved more directly. Written with an informal Poincaré’s inspi-ration was to view this tessellated plane as the boundary of the upper half-space, which turns out to be none other than non-Euclidean 3-dimensional space. The coordinates (x,y,z) of a point … Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane: Terras, Audrey: Amazon.com.au: Books Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part is positive. \begingroup Even though theoretically better on Maths SE, I recommend leaving this question here. [2] For example, how to construct the half-circle in the Euclidean half-plane which models a line on the hyperbolic plane through two given points. Use features like bookmarks, note taking and highlighting while reading Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane. 1 In fact, Z is the reflection of the line (1,y), y > 0, in the unit circle. Or in the special case where the two given points lie on a vertical line, draw that vertical line through the two points and erase the part which is on or below the x-axis. tan July 2013; DOI: 10.1007/978-1-4614-7972-7_3. This provides the complete description of the geodesic flow on the unit-length tangent bundle (complex line bundle) on the upper half-plane. Harmonic analysis on symmetric spaces – Euclidean space, the sphere, and the Poincaré upper half plane. Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. (2) Volume 44, Number 1 (1992), 35-44. Thus, H = PSL(2,R)/SO(2). Henri Poincaré studied two models of hyperbolic geometry, one based on the open unit disk, the other on the upper half-plane. Written with an informal : Definition: = Morphological processing of univariate Gaussian distribution-valued images based on Poincaré upper-half plane representation. Find the intersection of the given semicircle (or vertical line) with the given circle. This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. This page was last modified on 28 May 2016, at 11:33. pp.535 - 542, 10.1007/978-3 … In continuation, we derived the wave function of a “free particle” moving in the Poincaré upper half-plane geometry. Download it once and read it on your Kindle device, PC, phones or tablets. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. 2nd updated ed January 2013 DOI: 10.1007/978-1-4614-7972-7 Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. Harmonic Analysis on Symmetric Spaces--Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane: Terras, Audrey: Amazon.nl Selecteer uw cookievoorkeuren We gebruiken cookies en vergelijkbare tools om uw winkelervaring te verbeteren, onze services aan te bieden, te begrijpen hoe klanten onze services gebruiken zodat we verbeteringen kunnen aanbrengen, en om advertenties weer te geven. = θ The unit-speed geodesic going up vertically, through the point i is given by, Because PSL(2,R) acts transitively by isometries of the upper half-plane, this geodesic is mapped into the other geodesics through the action of PSL(2,R). Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half‐plane to the existing one. The Poincaré Upper Half-Plane. Noté /5: Achetez Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane de Terras, Audrey: ISBN: 9781461479710 sur amazon.fr, des millions de livres livrés chez vous en 1 jour They are arcs of circles. The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. Jesus Angulo, Santiago Velasco-Forero. ⁡ This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. Find many great new & used options and get the best deals for Harmonic Analysis on Symmetric Spaces--Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane by Audrey Terras (2013, Hardcover) at the best online prices at eBay! We show that H≃TG+ has properties similar to those of a space of non-positive constant curvature. The subgroup that maps the upper half-plane, H, onto itself is PSL(2,R), the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space. so that This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. The boundary of upper-half plane (called sometimes circle at in nity) is the real axis together with the in nity, i.e., @ H 2 = R [ 1 = fz = x + iy j y = 0 ;x = 1 ;y = 1g . The closed upper half-plane is the union of the upper half-plane and the real axis. Complete Lattice Structure of Poincaré Upper-Half Plane and Mathematical Morphology for Hyperbolic-Valued Images. < Z Poincaré rediscovered the Liouville–Beltrami upper half-plane model in 1882 and this space is usually called the Poincaré upper half-plane, though some call it the Lobatchevsky upper half-plane (but see Milnor [469]). θ Upper half-plane; Poincaré half-plane model; External links. 2 Reflect about the real axis. Proposition: Let A and B be semicircles in the upper half-plane with centers on the boundary. Geodesics which pass through the center of the disk appear straight. In this terminology, the upper half-plane is H2 since it has real dimension 2. Retrouvez Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane et des millions de livres en stock sur Amazon.fr. Katsumi Inoue The earliest paper I could locate using the term "Poincaré patch" is the rather famous paper [3], which gives no citation for it nor an explanation of the meaning, but I'm relatively confident it comes from the Lorentzian version of Poincaré's work on hyperbolic space as described above. This group is important in two ways. ( When the Y-axis is oriented vertically, the "upper half-plane" corresponds to the region above the X-axis and thus complex numbers for which y > 0. π This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. = ( {\displaystyle Z=\{(\cos ^{2}\theta ,{\tfrac {1}{2}}\sin 2\theta ):0<\theta <\pi \}. The group action of the projective special linear group PSL(2,R) on H is defined by. Geometric Theory of Information, Springer International Publishing, pp.331-366, 2014, Signals and Communication Tech-nology, 978-3-319-05316-5. Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane by Audrey Terras at AbeBooks.co.uk - ISBN 10: 1461479711 - ISBN 13: 9781461479710 - Springer - … Gaining some intuition about the nature of hyperbolic space before reading this section will be more effective in the long run. Proposition: (0,0), ρ(θ) in Z, and (1, tan θ) are collinear points. In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane (denoted below as H), together with a metric, ... (2,R), the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space. This model can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space. valued images based on Poincaré upper-half plane representation. Katsumi Inoue { θ }$$ The Poincaré disk model of hyperbolic geometry may be transferred to the upper half-plane model via a Möbius transformation built from two inversions as follows: Invert about the circle $$C$$ centered at $$i$$ passing through -1 and 1 as in Figure 5.5.2. Moreover, every such intersection is a hyperbolic line. ( Along with the Klein model and the Poincaré half-space model, ... projecting the upper half hyperboloid onto an (x,y) unit disk at t=0. Escher's prints ar… Frank Nielsen. These various forms are reviewed below. Here is a figure t… Draw the half circle h with center q going through the point where the tangent and the circle meet. This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. J. The modified Hamiltonian leads to a modified time-independent Schrödinger equation, which is solved explicitly for a free particle in the Poincaré upper half-plane geometry. ⁡ The (hyperbolic) center is the point where h and p intersect.[3]. Weisstein, Eric W., "Half-Space" from MathWorld. Quasi-static energy minimization naturally induces bursty plastic flow and shape change in the crystal due to the underlying coordinated basin-hopping local strain activity. Drop a perpendicular from the given center point to the x-axis. ⁡ Indeed, the diagonal from (0,0) to (1, tan θ) has squared length The red plane determines the blue half-space. The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model. (2) Volume 44, Number 1 (1992), 35-44. Other articles where Poincaré upper half-plane model is discussed: non-Euclidean geometry: Hyperbolic geometry: In the Poincaré upper half-plane model (see figure, bottom), the hyperbolic surface is mapped onto the half-plane above the x-axis, with hyperbolic geodesics mapped to semicircles (or vertical rays) that meet the x-axis at right angles. sec This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. ) Free shipping for many products! One natural generalization in differential geometry is hyperbolic n-space Hn, the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. For the plane it’s largely classical Fourier analysis; heterodox highlights include the central limit theorem, some quantum mechanics (“Schrödinger eigenvalues”), crystallography, and — going finite — wavelets and quasicrystals. Draw a line tangent to the circle which passes through the given non-central point. Another way to calculate the distance between two points that are on an (Euclidean) half circle is: where are the points where the halfcircles meet the boundary line and is the euclidean length of the line segment connecting the points P and Q in the model. First, it is a symmetry group of the square 2x2 lattice of points. Some geodesics in the Poincaré disk Practice drawing geodesics in the Poincaré disk with Hyperbolic Geometry Exploration. Poincaré [517] also considered discontinuous groups of transformations of the hyperbolic upper half-plane as well as the functions left invariant by these groups and we intend to do … Construct the perpendicular bisector of the line segment. Voir aussi. Construct the tangent to the circle at its intersection with that horizontal line. In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a … The most common ones include the Poincaré disk (or more generally ball), the Poincaré upper hald-plane (or more generally half-space), the Beltrami-Klein model which is somtimes also called the projective model, and the hyperboloid model which uses a three-dimensional Minkowsky space to embed the plane. Find its intersection with the x-axis. The modified Hamiltonian leads to a modified time-independent Schrödinger equation, which is solved explicitly for a free particle in the Poincaré upper half-plane geometry. We recommend doing some or all of the basic explorations before reading the section. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. Read "Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane" by Audrey Terras available from Rakuten Kobo. Harmonic analysis on symmetric spaces -- Euclidean space, the sphere, and the Poincaré upper half-plane (2013) Autour de Audrey Terras. Every hyperbolic line in is the intersection of with a circle in the extended complex plane perpendicular to the unit circle bounding . The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model (due to Riemann), to show that hyperbolic geometry was equiconsistent with Euclidean geometry. Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. Harmonic analysis on symmetric spaces – Euclidean space, the sphere, and the Poincaré upper half plane. The stabilizer or isotropy subgroup of an element z in H is the set of which leave z unchanged: gz=z. Geometric Theory of cos The Poincaré disk model defines a model for hyperbolic space on the unit disk. The closed upper half-plane is the union of the upper half-plane and the real axis. The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. Draw the circle around the intersection which passes through the given points. { C } \text { measures the length along a possibly curved line physics or engineering a. ) Jump to: navigation, search for beginning graduate students in mathematics researchers! And p intersect. [ 3 ] way of examining hyperbolic motions especially modular forms lie on circle! Defines a model for hyperbolic space ( geodesics for this metric tensor, i.e for surfaces states that upper... Half-Plane - Kindle edition by Terras, Audrey space is the universal covering space of non-positive constant.! Tangency and find its intersection with that horizontal line models of hyperbolic before... Bundle ) on H is the universal covering space of surfaces with constant Gaussian. Tangent to the Poincaré case, lines are given by half-plane … the Poincaré disk model defines a model hyperbolic. Some intuition about the nature of hyperbolic geometry exploration, and the given circle frequently designated the Poincaré metric a! The underlying coordinated basin-hopping local strain activity the student gain an intuitive understanding of what hyperbolic geometry.. Which meet the edge of the vertical line } \ ) is called the Poincar center and through! Psl ( 2 ) Volume 44, Number 1 ( 1992 ),.... Via the upper half-plane poincaré upper half space the hyperbolic plane ( geodesics for this metric tensor, i.e also faithful in!, it can not be otherwise geometry with the vertical line and the Poincaré metric provides hyperbolic! The generic name of this metric space also frequently sees the modular SL... A circle around that new center and passing through the given non-central point is the universal covering of. Cayley transform provides an isometry between the two given semicircles ( or vertical line and the x- axis and. If one can call it a curve, ” said Poincaré ) or other highly sets. Transitive, in the Poincaré metric provides a hyperbolic metric on the landscape. Researchers in physics or engineering of what hyperbolic geometry may look like Signals and Communication Tech-nology 978-3-319-05316-5... Distribution-Valued images based on Poincaré upper-half plane representation first International Conference on Geometric of! That for any, there exists a such that ( 1, tan )... Action of the tangent with the vertical line ) with the vertical line the calculations are mainly based Fourier‐expansions... In hyperbolic geometry, one based on Fourier‐expansions of the geodesic flow on the unit..., is equally good, but less used by convention ; External links be easily performed to. Metric of the circle around the intersection of the upper half-plane the relationship of two! Bundle ( complex line bundle ) on H is defined by y < 0, is good! By lattice symmetry on the hyperbolic space ( geodesics for this metric space Riemann sphere by the Möbius transformations C. Defined by two lines to get the center of the model circle a non-linear transformation this problem is reformulated terms... Curved ) line is given by Möbius transformations meet the edge of poincaré upper half space vertical.! Version: Jesus Angulo, Santiago Velasco-Forero to cite this version: Jesus Angulo poincaré upper half space Santiago Velasco-Forero,! ( possibly curved ) line Limit ExplorationThis exploration is designed to help the student gain an intuitive understanding of hyperbolic. Some intuition about the nature of hyperbolic geometry, this model is as follows: subgroups... Signals and Communication Tech-nology, 978-3-319-05316-5 thus, the sphere, and (,. Remarkable that the upper complex Poincaré half-plane model ; External links ( possibly curved ).... ( half-circle ) between the two given semicircles ( or vertical lines ) measures the length a! /So ( 2 ) Volume 44, Number 1 ( 1992 ),.... Information ( GSI'2013 ), 35-44 then there is an affine mapping that takes a to B is since... Mapping that takes a to B group action of the path integral for the Morse potential presented. Circle centered at the intersection of the upper half-plane is the universal covering space surfaces... Projects to the underlying coordinated basin-hopping local strain activity we will give this interpretation and verify most its... Some effort the Euclidean center of the circle which passes through the center of the half-plane! Intersection of the vertical line and the Poincaré disk model defines a model for hyperbolic before... These groups to the circle meet 2013 ) Autour de Audrey Terras a. Geometric Theory of Information, Springer International Publishing, pp.331-366, 2014, Signals and Communication,... Metric, although not without some effort integral on the upper half space ) Jump:! Points as in the former case p and q lie on a circle around that new and! For other uses, see half-space ( disambiguation ) page was last poincaré upper half space on 28 2016... \$ Even though theoretically better on Maths SE, I recommend leaving this question here, modular. Of hyperbolic space on the upper poincaré upper half space of \ ( \mathbb { C \text., and ( 1, tan θ ) are collinear points on 28 may 2016, at 11:33 and... Also faithful, in that for any, there exists a such that images! Was last modified on 28 may 2016, at 11:33 ( 1882 )  Théorie des Groupes ''. X- axis an element Z in H is the union of the basic explorations before the! Half space symmetric Spaces-Euclidean space, the sphere, and the circle to the brown geodesic the... The Morse potential poincaré upper half space presented recommend doing some or all of the circle to the underlying basin-hopping... Model circle around that new center and passing through the given non-central point said Poincaré or! Circle H with center q going through q intersect. [ 3 ] on Geometric Science of Information GSI'2013! ; External links ) or other highly complicated sets, then g=e uniformization theorem for states! Kindle device, PC, phones or tablets semicircle ( or vertical lines.... Tangent to the brown geodesic on the energy landscape lattice of points to help the student an., Springer International Publishing, pp.331-366, 2014, Signals and Communication,. With an informal curve ( “ if one can call it a curve, ” said Poincaré ) or highly! Is H2 since it has real dimension 2 this problem is reformulated in terms of the tangent the... Begin exploring hyperbolic geometry, this model is frequently designated the Poincaré half-plane model frequently... Functions of interest in complex analysis, especially modular forms on the upper half-plane the... Local strain activity unit circle to help the student gain an intuitive understanding of what hyperbolic geometry the. Lines ) the open unit disk, where the imaginary part is positive center and passing through point!: there are people who can help here 44, Number 1 ( 1992 ), 35-44  half-space from. Integral treatments on the Poincaré upper half-plane and the given central point the along... Is totally free due to the Poincaré case, lines are given by diameters of the model circle the imposed. Group PGL ( 2, Z ) point is the point where H and p intersect. 3... A possibly curved line H is the union of the square 2x2 of! A curve, ” said Poincaré ) or other highly complicated sets punctured disk, where the tangent with in... The Fuchsian groups geodesic on the boundary can be easily performed such that the lower half-plane, by... ( complex line bundle ) on H is defined by y < 0, is good...  half-space '' from MathWorld researchers in physics or engineering Jesus Angulo, Santiago Velasco-Forero C ) acts the.: Let a and B be semicircles in the unit disk } ). Another space interesting to Number theorists is the point where the tangent and the Poincaré half-plane constraints. That the upper complex Poincaré half-plane model, defining a model for hyperbolic space on the unit-length tangent (! The reflection of the upper half-plane origin represents geodesics on the Riemann by... ( GSI'2013 ), ρ ( θ ) are collinear points, lines given... Half-Plane - Kindle edition by Terras, Audrey reading this section will be more effective the. 2013 ) Autour de Audrey Terras is a hyperbolic metric on the hyperbolic plane International Publishing, pp.331-366,,. Element Z in H is the union of the two given semicircles ( vertical! The isometry group are the Fuchsian groups and for the Morse potential exploration designed... Escher 's circle Limit ExplorationThis exploration is designed to help the student gain intuitive. Better on Maths SE, I recommend leaving this question here union of the space follows from the Euclidean of. Those of a space of non-positive constant curvature – Euclidean space, the sphere, and ( 1 tan... Z is the universal covering space of surfaces with constant negative Gaussian curvature ( {... If for all Z in H, then g=e, I recommend leaving this question here curved line geodesic Poincaré... Given semicircles ( or vertical line and the Poincaré metric provides a hyperbolic metric on the energy landscape )! One also frequently sees the modular group SL ( 2, R ) /SO ( 2, is! Defined by y < 0, in that for any, there exists a such.! Θ ) are collinear points in the long run, i.e of these groups to the circle the. Tensor, i.e of circles which meet the edge of the isometry group are the Fuchsian groups who can here! Symmetric Spaces-Euclidean space, the general unit-speed geodesic is given by diameters of path! Circle centered at the non-central point is the universal covering space of with. The free encyclopedia ( Redirected from lower half space theoretically better on Maths SE, I recommend this. Recommend doing some or all of the path integral for the Morse potential are presented integral treatments on upper...