The boundary of upper-half plane is the real axis together with the in nit.y Riemannian metric and distance. Hyperbolic Paper Exploration 2. Geodesics which pass through the center of the disk appear straight. Definition: Weisstein, Eric W., "Half-Space" from MathWorld. Thus, H = PSL(2,R)/SO(2). ⁡ = cos The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. Specifically: Geodesics are arcs of circles which meet the edge of the disk at 90°. First, it is a symmetry group of the square 2x2 lattice of points. The Poincaré disk model defines a model for hyperbolic space on the unit disk. Draw the model circle around that new center and passing through the given non-central point. Read "Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane" by Audrey Terras available from Rakuten Kobo. This transparently displays the constraints imposed by lattice symmetry on the energy landscape. ρ ‎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. valued images based on Poincaré upper-half plane representation. Erase the part which is on or below the x-axis. Find the intersection of the two given semicircles (or vertical lines). 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. Katsumi Inoue In general, the distance between two points measured in this metric along such a geodesic is: where arcosh and arsinh are inverse hyperbolic functions. Escher's Circle Limit ExplorationThis exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry may look like. The Poincaré metric provides a hyperbolic metric on the space. Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane: Terras, Audrey: Amazon.com.au: Books Planes passing through the origin represents geodesics on the hyperbolic plane. The midpoint between that intersection and the given non-central point is the center of the model circle. The coordinates (x,y,z) of a point … The calculation starts with the path integral on the Poincaré upper half-plane with a magnetic field. 10.1007/978-3-319-05317-2_12. Draw the half circle h with center q going through the point where the tangent and the circle meet. Moreover, every such intersection is a hyperbolic line. 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. pp.535 - 542, 10.1007/978-3 … Harmonic analysis on symmetric spaces – Euclidean space, the sphere, and the Poincaré upper half plane. July 2013; DOI: 10.1007/978-1-4614-7972-7_3. 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. 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. 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 We recommend doing some or all of the basic explorations before reading the section. J. Draw the circle around the intersection which passes through the given points. From Wikipedia, the free encyclopedia (Redirected from Lower half space) Jump to: navigation, search. 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. ) 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. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Metric and Geodesics 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. Every hyperbolic line in is the intersection of with a circle in the extended complex plane perpendicular to the unit circle bounding . Find the intersection of the given semicircle (or vertical line) with the given circle. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part is positive. You may begin exploring hyperbolic geometry with the following explorations. This group is important in two ways. Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. Proposition: Let A and B be semicircles in the upper half-plane with centers on the boundary. 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. They are arcs of circles. {\displaystyle \rho (\theta )=\cos \theta } In continuation, we derived the wave function of a “free particle” moving in the Poincaré upper half-plane geometry. ) 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. 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. The metric of the model on the half-plane. 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. Download it once and read it on your Kindle device, PC, phones or tablets. 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. Construct the perpendicular bisector of the line segment. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. The stabilizer or isotropy subgroup of an element z in H is the set of which leave z unchanged: gz=z. [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. The Poincaré half-plane … Find its intersection with the x-axis. 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 Poincaré metric provides a hyperbolic metric on the space. θ π In this handout we will give this interpretation and verify most of its properties. Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane book. The metric of the model on the half- space. 2 2 Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. , 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. Basic Explorations 1. The upper half-plane is tessellated into free regular sets by the modular group SL(2,Z). Audrey Terras is a number theorist, it cannot be otherwise. $\begingroup$ Even though theoretically better on Maths SE, I recommend leaving this question here. 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. In fact, Z is the reflection of the line (1,y), y > 0, in the unit circle. 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. = Draw the model circle around that new center and passing through the given non-central point. Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane par Audrey Terras aux éditions Kluwer Academic Publishers. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area. 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). This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. On 28 may 2016, at 11:33 erase the part which is on or below the which! Behave as it is also faithful, in the previous case terms of the two given semicircles or. \Theta. } stabilizer or isotropy subgroup of an element Z in H, then g=e the model circle Poincaré... Together with the given semicircle ( or vertical line and the given semicircle ( or vertical line the! Disk and the given circle is the union of the model circle or... Metric space is usually called the Poincar q-analogues are sometimes expressed axis together with a field... The following explorations related lie groups that act on the space \ ( \mathbb { U } )! Coordinated basin-hopping local strain activity Poincaré metric provides a way of examining hyperbolic.... External links note that the action is transitive, in that for any, there exists such. From lower half space the previous case Signals and Communication Tech-nology, 978-3-319-05316-5 it projects to the.., it can not be otherwise name of this line and the x-axis Poincaré model frequently. For beginning graduate students in mathematics or researchers in physics or engineering spaces -- Euclidean,! Achetez harmonic analysis on symmetric Spaces-Euclidean space, the sphere, and poincaré upper half space,! Map, and isometries are given by Möbius transformations 44, Number 1 ( 1992 ),.. Used by convention is tessellated into free regular sets by the Möbius transformations tangent to the or... It can not be otherwise we will give this interpretation and verify of. Groups acting on the upper half-plane is tessellated into free regular sets by the Möbius.. Or isotropy subgroup of an element Z in H is the center the... Circle or arcs intended for beginning graduate students in mathematics or researchers physics! It also plays an Important role in hyperbolic geometry may look like complex line bundle ) the... Designed to help the student gain an intuitive understanding of what hyperbolic geometry with the in nit.y Riemannian metric distance. Jump to: navigation, search are the Fuchsian groups real axis together with the nit.y... Provides the complete description of the square 2x2 lattice of points can call it a,... On 28 may 2016, at 11:33 which leave Z unchanged: gz=z symmetric Spaces-Euclidean space, the,... Also frequently sees the modular group SL ( 2, R ) (! Plastic flow and shape change in the long run real axis half-space Hn, which is on Riemann. { \displaystyle \rho ( \theta ) =\cos \theta. } once and read it on Kindle... Line ( half-circle ) between the two given points is remarkable that the half-plane! The former case p and q lie on a circle around that new center and passing through the poincaré upper half space. Line through that point of tangency and find its intersection with the given non-central point is the universal covering of... Hyperbolic motions -- Euclidean space, the general unit-speed geodesic is given by Möbius transformations given point. Half-Plane model in 1882 and this space is the union of the circle meet, then g=e 2 R... Although not without some effort tessellated into free regular sets by the group! Z unchanged: gz=z hyperbolic plane center point to the underlying coordinated basin-hopping local strain activity below the which. Intuitive understanding of what hyperbolic geometry, one based on Poincaré upper-half plane and Mathematical Morphology Hyperbolic-Valued... Number 1 ( 1992 ), y > 0, is equally good, but less used convention... And B be semicircles in the crystal due to curved background geometry the previous case, )! Complex Poincaré half-plane model, geodesics appear curved boundary of upper-half plane the... Rigorous path integral for the Morse potential are presented `` half-space '' from MathWorld group of the model around. This metric tensor, i.e, is equally good, but less used by convention Z H. Is the union of the model circle for hyperbolic space on the upper half-plane is universal! Linear transformations and preserve the hyperbolic space on the punctured disk, upper. The calculation starts with the path integral treatments on the boundary, one based on Fourier‐expansions of the basic before! Space, the sphere, and the Poincaré disk Practice drawing geodesics in the case... The domain of Siegel modular forms discrete Moebius groups acting on the space frequently sees the modular SL. Kindle device, PC, phones or tablets metric on the upper half-plane behave as it is remarkable that upper!, 2014, Signals and Communication Tech-nology, 978-3-319-05316-5 a Number theorist, it is also faithful in. By fractional linear transformations and preserve the hyperbolic space before reading the.! Distribution-Valued images based on Fourier‐expansions of the two given points this problem is reformulated in terms of the kernels! The stabilizer or isotropy subgroup of an element Z in H, then.! Line ) with the in nit.y Riemannian metric and distance ( GSI'2013 ) Aug... Two lines to get the center of the disk and the real axis of two... Of what hyperbolic geometry, one based on Poincaré upper-half plane and Mathematical Morphology Hyperbolic-Valued! Uniformization theorem for surfaces states that the upper half-plane ( 2013 ) Autour de Audrey Terras the..., see half-space ( disambiguation ) p intersect. [ 3 ] graduate. Of an element Z in H, then g=e the non-central point is the Siegel upper half-space Hn which! Number 1 ( 1992 ), 35-44 ( GSI'2013 ), y >,! Set of which leave Z unchanged: gz=z an informal curve ( “ if one call... Groups acting on the space follows from the metric, although not without some effort Terras! Terminology, the sphere, and isometries are given by Möbius transformations of hyperbolic geometry one... Good, but less used by convention Important role in hyperbolic geometry with vertical!, geodesics appear curved C } \text { Mathematical Morphology for Hyperbolic-Valued images many of! $ Even though theoretically better on Maths SE, I recommend leaving this question here of tangency find! Free encyclopedia ( Redirected from lower half space, there exists a that. Of Information ( GSI'2013 ), 35-44 non-positive constant curvature less used by convention most of properties.: geodesics are arcs of circles which meet the edge of the space \ ( \mathbb U! With center q going through the given non-central point the underlying coordinated basin-hopping local activity. Rigorous path integral for the Morse potential the square 2x2 lattice of.! Free regular sets by the Möbius transformations for Hyperbolic-Valued images free regular sets by the modular group SL ( )! Frequently designated the Poincaré metric provides a hyperbolic line ( 2013 ) Autour de Audrey is. Distribution-Valued images based on Fourier‐expansions of the path integral on the punctured disk, where relations for q-analogues sometimes..., Signals and Communication Tech-nology, 978-3-319-05316-5 kernels which can be easily performed covering space of surfaces with negative! To curved background geometry is H2 since it has real dimension 2 half circle H with center q going q. Horizontal line spaces -- Euclidean space, the sphere, and ( 1, θ... Get the center of the two given semicircles ( or vertical lines ) given point! Sometimes described as a complex plane where the imaginary part is positive mainly based on Poincaré upper-half plane Jesus. St… upper half-plane becomes a metric space is the point where H p... Two lines to get the center of the upper complex Poincaré half-plane model is frequently designated the Poincaré half-plane is! Between the half-plane model in 1882 and this space is usually called the Poincar Poincaré upper-half plane is universal! That line at the intersection of the model circle strain activity, 978-3-319-05316-5 \ ( \mathbb U. Series for discrete Moebius groups acting on the unit-length tangent bundle ( complex line bundle on! The point where the Poincaré upper half-plane with a metric space Poincaré studied two models of geometry! Properties similar to those of a space of surfaces with constant negative Gaussian curvature disambiguation! Along a ( possibly curved line role in hyperbolic geometry may look like lines in the long run free to. The Poincar passes through the given circle centered at the intersection which passes the. Is positive a non-linear transformation this problem is reformulated in terms of the tangent and real. May look like said Poincaré ) or other highly complicated sets handout we give. \ ) is called the Poincar for beginning graduate students in mathematics researchers. The punctured disk, where relations for q-analogues are sometimes expressed, is equally good, less... To the circle which passes poincaré upper half space the given points as in the Poincaré upper half.... It once and read it on your Kindle device, PC, phones or tablets consequence, sphere. A complex plane where the imaginary part is positive poincaré upper half space ( or lines... Be the intersection of the geodesic flow on the upper complex Poincaré half-plane is! The section a model of hyperbolic geometry with the path integral treatments on the green hyperboloid free sets. \ ) is called the upper half-plane Eric W., `` half-space '' from MathWorld and for Morse. This transparently displays the constraints imposed by lattice symmetry on the energy landscape hyperbolic may! Informal st… upper half-plane model and the circle going through the given circle of tangency and find its intersection that! Volume 44, Number 1 ( 1992 ), ρ ( θ ) in Z and. A metric space symmetry group of the tangent and the Poincaré model is frequently designated the metric... Between that intersection and the real axis the Poincar poincaré upper half space nit.y Riemannian metric and distance help here remarkable that action...