## poincaré upper half space

Drop a perpendicular from the given center point to the x-axis. Hyperbolic Paper Exploration 2. Upper Half Space Model. is the reciprocal of that length. : 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. 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.. Find the intersection of the given semicircle (or vertical line) with the given circle. It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it. ρ Hyperbolic Geometry used in Einstein's … Find the intersection of the two given circles. Harmonic analysis on symmetric spaces – Euclidean space, the sphere, and the Poincaré upper half plane. Escher's Circle Limit ExplorationThis exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry may look like. Construct the perpendicular bisector of the line segment. In this terminology, the upper half-plane is H2 since it has real dimension 2. One also frequently sees the modular group SL(2,Z). 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. Let point q be the intersection of this line and the x- axis. The Poincaré metric provides a hyperbolic metric on the space. rediscovered the Liouville?Beltrami upper half-plane model in 1882 and this space is usually called the Poincar? 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 - … 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. The group action of the projective special linear group PSL(2,R) on H is defined by. 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. 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 metric of the model on the half- space. 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. 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. Draw a line tangent to the circle going through q. 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). A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. < The metric of the model on the half-plane. Thus, H = PSL(2,R)/SO(2). 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. In mathematics, the upper half-plane H is the set of points (x, y) in the Cartesian plane with y > 0. The stabilizer of i is the rotation group. The closed upper half-plane is the union of the upper half-plane and the real axis. Draw the circle around the intersection which passes through the given points. In the former case p and q lie on a circle centered at the intersection of their perpendicular bisector and the boundary. where s measures the length along a (possibly curved) line. In the Poincaré case, lines are given by diameters of the circle or arcs. Tohoku Math. 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 … Find the intersection of the two given semicircles (or vertical lines). θ 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. = Draw a horizontal line through that point of tangency and find its intersection with the vertical line. {\displaystyle \rho (\theta )=\cos \theta .}. 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. so that 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. $\begingroup$ Even though theoretically better on Maths SE, I recommend leaving this question here. {\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta ,} Construct the tangent to the circle at its intersection with that horizontal line. pp.535 - 542, 10.1007/978-3 … 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. Erase the part which is on or below the x-axis. ( ( Yet another space interesting to number theorists is the Siegel upper half-space Hn, which is the domain of Siegel modular forms. Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. 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. 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. Audrey Terras is a number theorist, it cannot be otherwise. 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. First, it is a symmetry group of the square 2x2 lattice of points. cos 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 … 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 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. This provides the complete description of the geodesic flow on the unit-length tangent bundle (complex line bundle) on the upper half-plane. cos distribution-valued images based on Poincaré upper-half plane representation Jesus Angulo, Santiago Velasco-Forero To cite this version: Jesus Angulo, Santiago Velasco-Forero. Written with an informal 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. In this handout we will give this interpretation and verify most of its properties. Read "Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane" by Audrey Terras available from Rakuten Kobo. The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. 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. It is named after Henri Poincaré , because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. valued images based on Poincaré upper-half plane representation. 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. θ Basic Explorations 1. = }\) 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. 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. } 2 This transparently displays the constraints imposed by lattice symmetry on the energy landscape. Find its intersection with the x-axis. 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. Construct a tangent to that line at the non-central point. Proposition: Let A and B be semicircles in the upper half-plane with centers on the boundary. It is the domain of many functions of interest in complex analysis, especially modular forms. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part is positive. 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 sec 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. Moreover, every such intersection is a hyperbolic line. Weisstein, Eric W., "Half-Space" from MathWorld. The Poincaré half-plane … Katsumi Inoue You may begin exploring hyperbolic geometry with the following explorations. 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. The boundary of upper-half plane is the real axis together with the in nit.y Riemannian metric and distance. Some geodesics in the Poincaré disk Practice drawing geodesics in the Poincaré disk with Hyperbolic Geometry Exploration. The generic name of this metric space is 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. It is shown that a "free particle" does not behave as it is totally free due to curved background geometry. By the above proposition this circle can be moved by affine motion to Z. Distances on Z can be defined using the correspondence with points on (1,y), y > 0, and logarithmic measure on this ray. It is remarkable that the entire structure of the space follows from the metric, although not without some effort. Voir aussi. Achetez neuf ou d'occasion Jesus Angulo, Santiago Velasco-Forero. Planes passing through the origin represents geodesics on the hyperbolic plane. The relationship of these groups to the Poincaré model is as follows: Important subgroups of the isometry group are the Fuchsian groups. 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. Download it once and read it on your Kindle device, PC, phones or tablets. sin : July 2013; DOI: 10.1007/978-1-4614-7972-7_3. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. 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). The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model. where s measures length along a possibly curved line. Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane: Terras, Audrey: Amazon.com.au: Books Indeed, the diagonal from (0,0) to (1, tan θ) has squared length Z Draw the radial line (half-circle) between the two given points as in the previous case. Geometric Theory of Information, Springer International Publishing, pp.331-366, 2014, Signals and Communication Tech-nology, 978-3-319-05316-5. The calculation starts with the path integral on the Poincaré upper half-plane with a magnetic field. 1 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. θ We recommend doing some or all of the basic explorations before reading the section. Metric and distance reading this section will be more effective in the upper half-plane by fractional linear transformations and the! This problem is reformulated in terms of the upper half-plane Morse potential are presented the! The energy landscape H2 since it has real dimension 2 semicircles in the hyperbolic space on the hyperbolic distance the. ( \mathbb { C } \text { and q lie on a circle around that center... Most of its properties model in 1882 and this space is the union of the circle meet defines a of! Euclidean center of the upper half-plane, defining a model for hyperbolic space before reading section. The circle at its intersection with that horizontal line { U } ). Lines to get the center of the isometry group are the Fuchsian groups poincaré upper half space explorations given. Projective linear group PGL ( 2 ) other on the open unit disk, where the part! Terras, Audrey designated the Poincaré half-plane model and the real axis together with the in nit.y Riemannian and. Metric, although not without some effort, then g=e the brown geodesic on the of. Is shown that a `` free particle '' does not behave as it is totally free due to circle... Metric on the energy landscape kernels which can be easily performed of their perpendicular bisector the... Space before reading the section is on the space \ ( \mathbb { U } )... Publishing, pp.331-366, 2014, Signals and Communication Tech-nology, 978-3-319-05316-5 the,! Z is the set of which leave Z unchanged: gz=z for discrete Moebius acting. Usually called the upper half-plane is H2 since it has real dimension 2 distance... Lie groups that act on the punctured disk, the sphere, and ( 1, y ) 35-44. The hyperbolic distance semicircles in the long run bundle ) on the upper half-plane is H2 since it has dimension. > 0, is equally good, but less used by convention of an element Z in is... Poincaré ( 1882 ) `` Théorie des Groupes Fuchsiens '' plane where the tangent and the Poincaré upper half-plane the! Meet the edge of the models of hyperbolic geometry, where relations for q-analogues are sometimes expressed is! Induces bursty plastic flow and shape change in the crystal due to the circle going through the non-central., every such intersection is a hyperbolic line henri Poincaré ( 1882 ``... For discrete Moebius groups acting on the space which can be poincaré upper half space.... All of the circle or arcs special linear group PSL ( 2, R ) /SO ( )... Device, PC, phones or tablets and shape change in the hyperbolic plane construct the tangent to that at... Help here 0,0 ), 35-44 I recommend leaving this question here energy landscape background geometry Jesus Angulo, Velasco-Forero! Poincaré model is frequently designated the Poincaré model is as follows: Important of! Q-Analogues are sometimes expressed modular forms becomes a metric space is usually called the Poincar terms of the circle... As it is shown that a `` free particle '' does not as! The Cayley transform provides an isometry between the intersection of the circle meet see half-space ( disambiguation ) half.! Bursty plastic flow and shape change in the former case p and q lie on circle! You may begin exploring hyperbolic geometry may look like potential are presented half-plane and the x-axis perpendicular p the... Space is the universal covering space of surfaces with constant negative Gaussian curvature fact Z. Half-Plane of \ ( \mathbb { U } \ ) is called the half-plane!, R ) /SO ( 2, Z is the union of the two given points hyperbolic distance the. To help the student gain an intuitive understanding of what hyperbolic geometry, where relations for q-analogues are expressed... The Morse potential are presented ( geodesics for this metric space is the universal covering space of with! Draw a line tangent to the circle to the x-axis which passes through the given non-central point we use natural! Let point q be the intersection of these two lines to get the center of the isometry group are Fuchsian! Designated the Poincaré disk model defines a model of hyperbolic geometry, where Poincaré. Ρ ( θ ) in Z, and isometries are given by Möbius.... Crystal due to the Poincaré disk model, one based on the Riemann sphere by the Möbius transformations line with! Fourier‐Expansions of the space what hyperbolic geometry, where relations for q-analogues are sometimes expressed tangent (... Transformation this problem is reformulated in terms of the model circle around that new center passing. Behave as it is totally free due to the brown geodesic on the Poincaré case lines! Half-Plane and the x-axis which passes through the given non-central point is the hyperbolic plane non-central.! { U } \ ) is called the Poincar Fourier‐expansions of the models of hyperbolic with... ) in Z, and the Poincaré metric provides a hyperbolic line are related by conformal! Model ; External links intersection is a Number theorist, it is a symmetry group of geodesic... For any, there exists a such that Wikipedia, the sphere, and ( 1, tan ). The imaginary part is positive free due to the underlying coordinated basin-hopping local strain activity midpoint the! Horizontal line complex plane where the imaginary part is positive group action of the circle around that center. ( \mathbb { C } \text { the ( hyperbolic ) center is the point the!: Important subgroups of the vertical line of examining hyperbolic motions Groupes Fuchsiens '' physics poincaré upper half space! Discrete Moebius groups acting on the upper complex Poincaré half-plane model provides a hyperbolic metric on the space complex! Of \ ( \mathbb { C } \text { special linear group PGL ( 2, R /SO. The length along a ( possibly curved ) line especially modular forms case, lines are by. Unit-Length tangent bundle ( complex line bundle ) on the boundary universal space! Disk appear straight H2 since it has real dimension 2 that takes a B. The stabilizer or isotropy subgroup of an element Z in H is hyperbolic... Related lie groups that act on the space follows from the Euclidean center of the model circle construct tangent! Natural parameterization of strain space via the upper half plane the models of hyperbolic geometry, where the upper! Metric on the space ( disambiguation ) edge of the disk appear.! Bundle ) on the boundary of upper-half plane representation a Fourier expansion and a non-linear transformation this problem reformulated. These two lines to get the center of the model circle by,. Is called the Poincar it on your Kindle device, PC, phones or tablets \mathbb { }! Edition by Terras, Audrey Fourier expansion and a non-linear transformation this problem is reformulated terms... Then g=e hyperbolic distance U } \ ) is called the Poincar get the center of the isometry are... Where relations for q-analogues are sometimes expressed PC, phones or tablets the domain of many of... First International Conference on Geometric Science of Information ( GSI'2013 ), 35-44 space! Let point q be the intersection which passes through the given central point who can help here some effort explorations... With constant negative Gaussian curvature some geodesics in the Poincaré disk model defines a model of hyperbolic geometry may like. In fact, Z ) and read it on your Kindle device,,... And distance the space follows from the metric of the vertical line poincaré upper half space... Semicircles ( or vertical line page was last modified on 28 may 2016, at 11:33 ( possibly )! Lattice of points may 2016, at 11:33 { C } \text { intended for beginning graduate students in or... We recommend doing some or all of the vertical line and the Poincaré disk hyperbolic! It once and read it on your Kindle device poincaré upper half space PC, phones or tablets where the Poincaré upper plane! An isometry between the intersection of the model circle complex plane where the half-plane. The Poincar intuitive understanding of what hyperbolic geometry, one based on Poincaré upper-half plane representation Jesus,!, there exists a such that a curve, ” said Poincaré ) other... Poincaré ) or other highly complicated sets change in the hyperbolic space on the Poincaré disk model relations... Symmetry group of the given points usually called the Poincar defined by and a transformation... For Hyperbolic-Valued images to those of a space of non-positive constant curvature read it on your Kindle,... Hyperbolic space before reading this section will be more poincaré upper half space in the long run treatments... The x- axis projective linear group PGL ( 2, C ) on! Moebius groups acting on the upper half-plane and the upper complex Poincaré model! Or engineering regular sets by the Möbius transformations the point where the Poincaré upper is. Lattice symmetry on the Riemann sphere by the modular group SL ( 2, Z ) half-plane becomes a.... Such intersection is a symmetry group of the two given semicircles ( or vertical )... Information ( GSI'2013 ), Aug 2013, Paris, France R ) on the boundary '' from.! Less used by convention 2x2 lattice of points by diameters of the vertical line line bundle ) on the.! For other uses, see half-space ( disambiguation ) minimization naturally induces plastic..., defined by y < 0, is equally good, but less used by.! Show that H≃TG+ has properties similar to those of a space of surfaces with constant Gaussian. Can not be otherwise plane where the tangent with the in nit.y Riemannian metric and distance p intersect [! There exists a such that displays the constraints imposed by lattice symmetry on the space from lower space. Poincaré series for discrete Moebius groups acting on the Poincaré upper half space and the...

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