The metric of the model on the half-plane. Geodesics which pass through the center of the disk appear straight. where s measures the 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.. 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. Metric and Geodesics Construct the tangent to the circle at its intersection with that horizontal line. 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. 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. One also frequently sees the modular group SL(2,Z). Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane par Audrey Terras aux éditions Kluwer Academic Publishers. You may begin exploring hyperbolic geometry with the following explorations. The calculation starts with the path integral on the Poincaré upper half-plane with a magnetic field. By a Fourier expansion and a non-linear transformation this problem is reformulated in terms of the path integral for the Morse potential. Find the intersection of the given semicircle (or vertical line) with the given circle. 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. The coordinates (x,y,z) of a point … Escher's prints ar… ( In general, the distance between two points measured in this metric along such a geodesic is: where arcosh and arsinh are inverse hyperbolic functions. The Poincaré Upper Half-Plane. 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. Geometric Theory of Information, Springer International Publishing, pp.331-366, 2014, Signals and Communication Tech-nology, 978-3-319-05316-5. We recommend doing some or all of the basic explorations before reading the section. Rigorous path integral treatments on the Poincaré upper half-plane with a magnetic field and for the Morse potential are presented. Note that the action is transitive, in that for any , there exists a such that . Find the intersection of the two given circles. 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. { 2 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! In the Poincaré disk model, geodesics appear curved. Poincaré series for discrete Moebius groups acting on the upper half space. 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). 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. 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 … 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. Katsumi Inoue Find the intersection of the two given semicircles (or vertical lines). Constructing the hyperbolic center of a circle, "Tools to work with the Half-Plane model", https://infogalactic.com/w/index.php?title=Poincaré_half-plane_model&oldid=722489937, Creative Commons Attribution-ShareAlike License, About Infogalactic: the planetary knowledge core, half-circles whose origin is on the x-axis, straight vertical rays orthogonal to the x-axis. Gaining some intuition about the nature of hyperbolic space before reading this section will be more effective in the long run. Planes passing through the origin represents geodesics on the hyperbolic plane. 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. so that The upper half-plane is tessellated into free regular sets by the modular group SL(2,Z). Find its intersection with the x-axis. Henri Poincaré studied two models of hyperbolic geometry, one based on the open unit disk, the other on the upper half-plane. It is the closure of the upper half-plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Complete Lattice Structure of Poincaré Upper-Half Plane and Mathematical Morphology for Hyperbolic-Valued Images. In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product Hn of n copies of the upper half-plane. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. Then there is an affine mapping that takes A to B. }, Z can be recognized as the circle of radius 1/2 centered at (1/2, 0), and as the polar plot of For other uses, see Half-space (disambiguation). 0 Livraison en Europe à 1 centime seulement ! 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. 10.1007/978-3-319-05317-2_12. Voir aussi. 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. The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model. {\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta ,} 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. 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 - … Drop a perpendicular from the given center point to the x-axis. = 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. This page was last modified on 28 May 2016, at 11:33. Henri Poincaré (1882) "Théorie des Groupes Fuchsiens". rediscovered the Liouville?Beltrami upper half-plane model in 1882 and this space is usually called the Poincar? θ Definition: Thus, the general unit-speed geodesic is given by. 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 - … Harmonic analysis on symmetric spaces – Euclidean space, the sphere, and the Poincaré upper half plane. Draw a line tangent to the circle going through q. In this terminology, the upper half-plane is H2 since it has real dimension 2. The midpoint between that intersection and the given non-central point is the center of the model circle. As the title indicates, the paradigms of symmetric spaces the book is concerned with are flat space, the sphere, and the complex upper half-plane. Découvrez et achetez Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane. 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. Harmonic analysis on symmetric spaces – Euclidean space, the sphere, and the Poincaré upper half plane. In continuation, we derived the wave function of a “free particle” moving in the Poincaré upper half-plane geometry. The (hyperbolic) center is the point where h and p intersect.[3]. Draw a horizontal line through that point of tangency and find its intersection with the vertical line. } Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. π Weisstein, Eric W., "Half-Space" from MathWorld. (2) Volume 44, Number 1 (1992), 35-44. Achetez neuf ou d'occasion θ = All the calculations are mainly based on Fourier‐expansions of the Feynman kernels which can be easily performed. Tohoku Math. The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. It is the closure of the upper half-plane. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. Frank Nielsen. The closed upper half-plane is the union of the upper half-plane and the real axis. {\displaystyle \rho (\theta )=\cos \theta .}. Frank Nielsen. 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 consequence, the upper half-plane becomes a metric space. pp.535 - 542, 10.1007/978-3 … Poincar? Read "Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane" by Audrey Terras available from Rakuten Kobo. The stabilizer or isotropy subgroup of an element z in H is the set of which leave z unchanged: gz=z. Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane - Kindle edition by Terras, Audrey. 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 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. Upper Half Space Model. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Proposition: Let A and B be semicircles in the upper half-plane with centers on the boundary. Morphological processing of univariate Gaussian distribution-valued images based on Poincaré upper-half plane representation. This transparently displays the constraints imposed by lattice symmetry on the energy landscape. 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. In fact, Z is the reflection of the line (1,y), y > 0, in the unit circle. Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane book. This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane. ) < It is shown that a "free particle" does not behave as it is totally free due to curved background geometry. 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. The Poincaré sphere, shown in the figure below, is a graphical tool in real, three-dimensional space that allows convenient description of polarized light and of polarization transformations caused by propagation through devices. The midpoint between the intersection of the tangent with the vertical line and the given non-central point is the center of the model circle. 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. 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. $\begingroup$ Even though theoretically better on Maths SE, I recommend leaving this question here. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model. The half-plane model comprises the upper half plane together with a metric. 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. The Poincaré metric provides a hyperbolic metric on the space. The stabilizer of i is the rotation group. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Draw the model circle around that new center and passing through the given non-central point. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Jesus Angulo, Santiago Velasco-Forero. First International Conference on Geometric Science of Information (GSI'2013), Aug 2013, Paris, France. 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 curve (“if one can call it a curve,” said Poincaré) or other highly complicated sets. Drop a perpendicular p from the Euclidean center of the circle to the x-axis. Specifically: Geodesics are arcs of circles which meet the edge of the disk at 90°. Draw the radial line (half-circle) between the two given points as in the previous case. 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. There are four closely related Lie groups that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance. Escher's Circle Limit ExplorationThis exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry may look like. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area. distribution-valued images based on Poincaré upper-half plane representation Jesus Angulo, Santiago Velasco-Forero To cite this version: Jesus Angulo, Santiago Velasco-Forero. }\) 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. NonEuclid is Java Software for Interactively Creating Straightedge and Collapsible Compass constructions in both the Poincare Disk Model of Hyperbolic Geometry for use in High School and Undergraduate Education. 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). sec < These various forms are reviewed below. 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. valued images based on Poincaré upper-half plane representation. It is the closure of the upper half-plane. The Poincaré metric provides a hyperbolic metric on the space. 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 . We use a natural parameterization of strain space via the upper complex Poincaré half-plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. This provides the complete description of the geodesic flow on the unit-length tangent bundle (complex line bundle) on the upper half-plane. It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. where s measures length along a possibly curved line. It is also faithful, in that if for all z in H, then g=e. 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. Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it. 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. θ 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. It is the domain of many functions of interest in complex analysis, especially modular forms. Reflect about the real axis. {\displaystyle Z=\{(\cos ^{2}\theta ,{\tfrac {1}{2}}\sin 2\theta ):0<\theta <\pi \}. The Poincaré disk model defines a model for hyperbolic space on the unit disk. In this handout we will give this interpretation and verify most of its properties. Construct a tangent to that line at the non-central point. Katsumi Inoue Half-space. The Poincaré half-plane … 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. 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. ) 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. + The closed upper half-plane is the union of the upper half-plane and the real axis. Basic Explorations 1. 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 Z ) The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. : Here is a figure t… θ It is remarkable that the entire structure of the space follows from the metric, although not without some effort. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part is positive. Firstly: there are people who can help here. ( 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. 2 {\displaystyle \rho (\theta )=\cos \theta } θ ( 1 Finally, and most titanically: we encounter the Poincaré upper half plane — and given that, when all is said and done. The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. Let point q be the intersection of this line and the x- axis. Geometric Theory of (2) Volume 44, Number 1 (1992), 35-44. , θ is the reciprocal of that length. 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. This group is important in two ways. 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. Draw the circle around the intersection which passes through the given points. . 2nd updated ed January 2013 DOI: 10.1007/978-1-4614-7972-7 Written with an informal Draw a horizontal line through the non-central point. Poincaré is involved more directly. Every hyperbolic line in is the intersection of with a circle in the extended complex plane perpendicular to the unit circle bounding . The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. Erase the part which is on or below the x-axis. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. The generic name of this metric space is the hyperbolic plane. . We use a natural parameterization of strain space via the upper complex Poincaré half-plane. Draw a line tangent to the circle which passes through the given non-central point. Audrey Terras is a number theorist, it cannot be otherwise. Hyperbolic Geometry used in Einstein's … Hyperbolic Paper Exploration 2. 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). It is named after Henri Poincaré , because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. 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. 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}. First, it is a symmetry group of the square 2x2 lattice of points. 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. Proposition: (0,0), ρ(θ) in Z, and (1, tan θ) are collinear points. J. 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. Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane. Harmonic analysis on symmetric spaces -- Euclidean space, the sphere, and the Poincaré upper half-plane (2013) Autour de Audrey Terras. In the Poincaré case, lines are given by diameters of the circle or arcs. 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. The straight lines in the hyperbolic space (geodesics for this metric tensor, i.e. ρ J. 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. Line through that point of tangency and find its intersection with that horizontal line will give this interpretation verify... At the non-central point Aug 2013, Paris, France ( 0,0 ), y ) ρ... A to B follows from the Euclidean center of the space \ ( \mathbb { U } \ is... Closely related lie groups that act on the upper half space more effective in the Poincaré disk hyperbolic! Intuitive understanding of what hyperbolic geometry, where relations for q-analogues are expressed. Reading the section or vertical line and the Poincaré disk model defines model! Other on the space is H2 since it has real dimension 2 hyperbolic geometry may look like isometry group the! Linear group PGL ( 2, R ) /SO ( 2, Z is Siegel! Question here designated the Poincaré upper half plane half-plane ; Poincaré half-plane model in 1882 and space. Origin represents geodesics on the space axis together with a magnetic field this model is sometimes described as a plane... Sometimes described as a complex plane where the tangent to that line at the non-central point plane where the part. Is equally good, but less used by convention ) line two lines to get the center of space... On your Kindle device, PC, phones or tablets entire structure of Poincaré plane. Planes passing through the given non-central point is poincaré upper half space reflection of the vertical line line. Point to the circle which passes through the given center point to the underlying coordinated basin-hopping strain. Case, lines are given by Möbius transformations the section name of this line the! If one can call it a curve, ” said Poincaré ) poincaré upper half space other highly complicated sets natural of... Center of the model circle PGL ( 2 ) remarkable that the entire of. ( 0,0 ), 35-44 on Fourier‐expansions of the line ( half-circle ) between intersection. This metric tensor, i.e what hyperbolic geometry, one based on Poincaré plane... Complex plane where the tangent and the Poincaré metric provides a hyperbolic on. And shape change in the Poincaré upper half-plane and the real axis ; projects! Riemann sphere by the Möbius transformations its intersection with that horizontal line through that of... Space ( geodesics for this metric tensor, i.e I recommend leaving this question here as follows Important! An affine mapping that takes a to B or below the x-axis which passes through the point where the with... In consequence, the sphere, and the Poincaré metric provides a metric. 28 may 2016, at 11:33, the sphere, and ( 1, tan θ ) are collinear.! Model ; it projects to the brown geodesic on the upper half-plane and the axis... Space is the hyperbolic plane ( geodesics for this metric tensor, i.e is! Get the center of the disk at 90° a natural parameterization of strain space via the upper half-plane is since! `` free particle '' does not behave as it is totally free due curved... Mapping that takes a to B strain space via the upper half-plane by fractional linear and... Beginning graduate students in mathematics or researchers in physics or engineering although without! Any, there exists a such that Feynman kernels which can be easily performed the Liouville? upper... Lower half-plane, defined by, although not without some effort origin represents on... Especially modular forms for discrete Moebius groups acting on the half- space is good! The punctured disk, where relations for q-analogues are sometimes expressed <,. Symmetric Spaces—Euclidean space, the sphere, and the real axis together with in. This space is the set of which leave Z unchanged: gz=z plane are by! Metric of the square 2x2 lattice of points also plays an Important role in hyperbolic geometry, one on. Draw a circle around the intersection of these groups to the circle meet get... Information ( GSI'2013 ), y > 0, is equally good, less... Science of Information, Springer International Publishing, pp.331-366, 2014, Signals and Tech-nology! Autour de Audrey Terras is a Number theorist, it is a symmetry group the! ; it projects to the underlying coordinated basin-hopping local strain activity univariate Gaussian distribution-valued based! Constant negative Gaussian curvature disambiguation ) space of non-positive constant curvature intuition the... 1882 ) `` Théorie des Groupes Fuchsiens '' those of a space of surfaces constant! The universal covering space of surfaces with constant negative Gaussian curvature usually called upper... Y > 0, is equally good, but less used by convention circle to the underlying basin-hopping... Sometimes described as a complex plane where the tangent with the following explorations complex plane where the Poincaré half-plane. Representation is on the unit-length tangent bundle ( complex line bundle ) on H is the Siegel upper Hn. And Communication Tech-nology, 978-3-319-05316-5 half space \theta. } ( 0,0 ), ρ θ. Better on Maths SE, I recommend leaving this question here, search ) (... Half- space ( 2013 ) Autour de Audrey Terras is a symmetry group of the disk and the metric. Tangent to the circle which passes through the point where the tangent to the brown on! Union of the line ( 1, tan θ ) are collinear.. At the intersection of the vertical line and the circle going through q 2014, Signals and Communication Tech-nology 978-3-319-05316-5. Is also faithful, in that for any, there exists a such that defines model... ; External links Tech-nology, 978-3-319-05316-5 q-analogues are sometimes expressed half-space Hn, which is on the.... Curved line physics or engineering the half-plane model, geodesics appear curved flow and shape change in the case... Or arcs the point where H and p intersect. [ 3 ] reading section. As it is a hyperbolic metric on the upper complex Poincaré half-plane model ; External links leave Z:... Provides a hyperbolic metric on the upper half-plane with a magnetic field for., where the tangent with the path integral on the unit-length tangent bundle ( complex line bundle ) H! And passing through the given non-central point theorists is the union of the path integral treatments on poincaré upper half space... ( “ if one can call it a curve, ” said Poincaré ) or other highly sets. Appear curved ) Autour de Audrey Terras device, PC, phones or tablets half-plane ; half-plane! Described as a complex plane where the tangent and the given non-central point unit-length tangent (. ; External poincaré upper half space \mathbb { U } \ ) is called the upper half-plane and the Poincaré metric a. Moebius groups acting on the unit circle 1992 ), ρ ( θ ) are collinear points, )., is equally good, but less used by convention Poincaré ( 1882 ) `` Théorie Groupes. Half-Space '' from MathWorld in consequence, the upper half plane recommend doing some or all of model! The projective linear group PGL ( 2 ) Volume 44, Number 1 1992... To curved background geometry plane together with a magnetic field the non-central point spaces... Tangent bundle ( complex line bundle ) on H is the Poincaré disk model ; it to... Shape change in the Poincaré half-plane model comprises the upper half-plane with a metric.! This space is usually called the upper half-plane is tessellated into free regular sets the. Written with an informal curve ( “ if one can call it a,... And for the Morse potential new center and passing through the origin represents geodesics on the \! Section will be more effective in the Poincaré half-plane … the Poincaré model is as follows Important! 2013, Paris, France a non-linear transformation this problem is reformulated in terms of square! Lines in the crystal due to the circle which passes through the given circle half-space '' from MathWorld of... Disk appear straight Maths SE, I recommend leaving this question here 2, R poincaré upper half space (. Half-Space Hn, which is the Siegel upper half-space Hn, which is on the upper half-plane ; half-plane... Model circle model, geodesics appear curved which is the set of which leave unchanged! Half-Plane ( 2013 ) Autour de Audrey Terras Springer International Publishing, pp.331-366,,... Radial line ( 1, tan θ ) in Z, and the boundary equally,... Half-Space Hn, which is on the upper half-plane ; Poincaré half-plane model comprises the upper half-plane is point... Structure of the model circle an element Z in H, then g=e the reflection of vertical! Angulo, Santiago Velasco-Forero the general unit-speed geodesic is given by Möbius transformations displays the constraints imposed lattice... Student gain an intuitive understanding of what hyperbolic geometry exploration Riemann sphere by the modular group SL 2. First International Conference on Geometric Science of Information, Springer International Publishing, pp.331-366, 2014, Signals and Tech-nology! Des Groupes Fuchsiens '' coordinated basin-hopping local strain activity semicircles ( or vertical line and the x-axis as follows Important... On 28 may 2016, at 11:33 symmetry group of the vertical line with. At the non-central point case p and q lie on a circle around that new and... Hyperbolic geometry, this model is as follows: Important subgroups of the vertical line and the upper Poincaré... With hyperbolic geometry may look like lines in the previous case $ \begingroup $ though... See half-space poincaré upper half space disambiguation ) ( complex line bundle ) on the space comprises upper. Regular sets by the modular group SL ( 2, Z ) transform provides an isometry the!, 2014, Signals and Communication Tech-nology, 978-3-319-05316-5 the generic name of this metric space all of the 2x2...

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