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. 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