hyperbolic distance formula upper half plane

In this handout we will give this interpretation and verify most of its . One thus defined the infinitesimal element of the hyperbolic distance to be. Poincaré also devised his upper half plane model . In the Poincar e half-plane model, the hyperbolic plane is attened into a Euclidean half-plane. Although the coe cients a;b;c;dcan generally be complex numbers, here we will only be concerned with real coe cients such that ad bc= 1. In the upper half-plane, such an isometry takes the form of scalar multiplication of the complex plane by $e^k$ . We can still de ne a metric on H2 by de ning the distance from a point on the boundary to any other point . The metric of His ds2 = dx2+dy2 y2 1. The part f(x;0)g[f1gis called the boundary of H2. I set up a geometry on the complex upper half-plane $\mathcal{H}$ (Exercise 3.23), show that it is the same geometry as the hyperbolic plane $\mathcal{H}^{2}$ (Exercise 3.24), and investigate the failure of the parallel postulate in the new model (Exercise 3.25). UPPER HALF-PLANE MODEL 27 Definition 1.9. De nition 2.2. On an arbitrary surface with a Riemannian metric, the process of deﬁning an explicit distance func- A concrete formula of $h_{{\mathbb {D}}^*}(z_1,z_2)$ can also be given but its form is not so convenient [cf. Textbook Reading (Mar 4): Sections 4.2 and 4.3. For example, given z 1, z 2 we want to map 0 ↦ z 1, t ∈ \R ↦ z 2 and back. (1.6) In fact, by considering the hyperbolic triangle with vertices at (0,1), (x 1,y 1) and (x 2,y 2), by means of the Carnot hyperbolic formula it is simple to show that the distance η . The area of a region will not change as it moves about the hyperbolic plane. To continue studying the geometry of the hyperbolic plane, one must have a notion of hyperbolic distance and this is obtained from the previous hyperbolic metric on the upper half plane. Topics covered include the upper half-plane model of the hyperbolic plane, Möbius transformations, the general Möbius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincaré disc model, convex subsets of the hyperbolic plane, hyperbolic area . be respectively less than 2, called elliptic, greater than 2, called hyperbolic, and equal to 2, called parabolic. Most hyperbolic surfaces have a non-trivial fundamental group π 1 =Γ; the groups that arise this way are known as Fuchsian groups. h. Given an angle ÐABC in the annular hyperbolic plane, there is a unique geodesic which bisects the angle. A similarity of En is a trans- 1. 2.1.2 Poincar e Distance In the hyperbolic plane, distance from one point to another is di erent than what we call distance in the Euclidean plane. We can define "points", "lines", "distance" and "angles" as anything we want. Throwing away the bijective assumption, it turns out that in most situations the group of endomorphisms is still very rigid. Small hyperbolic triangles look like Euclidean triangles and hyperbolic angles correspond to Euclidean angles; the hyperbolic distance formula will fit with this theme. If two hyperbolic triangles have the same side-lengths, including the degenerate cases of some sides being infinite, prove that the triangles are congruent, i.e. Upper half-plane projection Example 2.2. 1.1 The upper half-plane . We introduce a distance metric by dρ = 2dr 1 −r2 where ρ represents the hyperbolic distance and r is the Euclidean distance from the center of the circle. hyperbolic isometries of H are the same as the conformal automorphisms, any quotient Γ\H has a natural complex structure. The area of a hyperbolic triangle is given by its defect in radians multiplied by R2. The aim of this paper is to give explicit solutions to the wave equations associated with the modified Schrödinger operators with uniform magnetic field on the disc $I\!\!D$ and the upper half-plane $I\!\!H$ models of the hyperbolic plane. (⋆) In the upper half-plane model, ﬁnd the locus of points that lie on distance d from the line {Rez = 0}. Geodesics in the upper half-plane Therefore the geometry in H is non-Euclidean. [Hint: Use appropriate folding in annular hyperbolic plane. 139. The hyperbolic plane with boundary, denoted H2, is de ned as H 2= H [f(x;0)g[f1g where 1represents all in nity points on the upper half plane, i.e., (1;y), (x;1), and (1 ;y). The Poincaré half plane is also hyperbolic, but is simply connected and . Definition 5.4.1. The upper half-plane is the set of complex numbers with positive imaginary parts. In other words, everything above the x-axis. Topics covered include the upper half-plane model of the hyperbolic plane, Möbius transformations, the general Möbius group, and their subgroups preserving the upper half-plane, hyperbolic arc-length and distance as quantities invariant under these subgroups, the Poincaré disc model, convex subsets of the hyperbolic plane, hyperbolic area . A typical non-vertical line is pictured in Figure 4.3. a r . Geometrically, the hyperbolic plane is the open upper half plane - everything above the real axis. De nition 2.1. hyperbolic distance from ato bin Dis d D(a,b) = 2tanh−1 b−a 1− ¯ab . Proposition 1.10. The hyperbolic distance between two points z 1, z 2 in the upper half-plane is. It is given here. Both of them in Wikipedia. First Theorem: Euclidean . Find the hyperbolic distance between the points P (1, 3) and Q(2, 4) in the Poincar ́e upper half-plane H. Then use a calculator to find a decimal approximation for your answer. ), surfaces with a constant negative Gaussian curvature. I found a derivation for them in a book called Visual Complex Analysis by Needham and it relied on mapping the "pseudosphere" onto the upper half plane. Wednesday 12/4/19. the hyperbolic plane, but the Dutch artist M.C. the Euclidean plane; and if K < 0 it is the hyp erb olic plane, also called 2-dimensiona l hyp erb olic space. A second common mapping of the upper half-plane to a disk is the q-mapping = ⁡ where q is the nome and τ is the half-period ratio: =. A Riemann surface is a one-dimensional complex manifold. Let z 0 be a point on distance d . g. Given an angle Ð ABC in the annular hyperbolic plane, there is a unique geodesic which bisects the angle. upper half-plane model of hyperbolic geometry, Definition. A Möbius transformation is an invertible map on C of the form z7! I'm trying to derive the hyperbolic distance formula for the upper-half plane model. Hyperbolic plane geometry is also the geometry of saddle surfaces and pseudospherical surfaces (potato chips! Give a geometric . az+b cz+d. Then we know d(z 1, z 2) = d(t). For each edge of the hyperbolic triangle ∆ABC in H with vertices A(0, 2), B(0, 4), and C(2, 2), find the hyperbolic line con- taining the edge. It is impossible to picture the whole of an ideal triangle in an annular hyperbolic plane, but it is easy to picture ideal triangles in the upper half plane model. well-defined function, Paragraph, Paragraph That is, points in hyperbolic space become ordered pairs with , and the inner product on the tangent space at is given by the matrix. The upper half plane is the subset H = fz2C : Im(z) >0gof C with the metric ds= p dx2 + dy2 y where z= x+ iy. The geometr y of the sphere and the plane are familia r; hyp erb olic ge-ometr y is the geometry of the third case. The metric in (1.2.1) is said to be the hyperbolic metric. In order to determine the distance, we must rst de ne cross-ratio. De nition 2.6. It must be remembered that neither of these are what lines "are" in hyperbolic geometry. formula for distance is more complex so we defer its discussion to a later section and in the Maple worksheets DHgeom.mws and UHgeom.mws [30]. where d is the hyperbolic distance from the origin to the intersection of . Friday Mar 6: Hyperbolic Lines. De nition 3.1. As a consequence, all hyperbolic triangles have an area that is less than or equal to R2 π. Examples and Computations. The following proposition gives the Riemannian isometry between the hyperbolic plane H 2 and the Poincaré upper-half plane U. 2, we investigate it by making use of an elliptic modular function as well . Using the metric, you can work out the geodesic equations . In the Upper Half-Plane model, a line is defined as a semicircle with center on the x-axis. 2 WILL ADKISSON The disk and half-plane models of hyperbolic space are isomorphic, mapped If an isometry in H 2 ﬁxes pointwise a geodesic line L, then it is either identity or a reﬂexion about L. there is an isometry of the hyperbolic plane sending one onto the other. 15.5. hyperbolic geometry. The first one is called the upper half plane model denoted by 7Y, and defined as [z G C : Im (z) > 0}. Another commonly used model for hyperbolic space in the upper half space model. This is an abstract surface in the sense that we . May 8, 2009. 5. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum. functions) which ﬁt very naturally into the hyperbolic world. Now, this area inside the unit circle must represent the inﬁnite hyperbolic plane. The Poincaré upper-half plane. 1. Viewed 5k times 13 While looking for an expression of the hyperbolic distance in the Upper Half Plane H = { z = x + i y ∈ C | y > 0 }, I came across two different expressions. In hyperbolic geometry with curvature $k\text{,}$ the hyperbolic distance $d$ of a point $z$ to a hyperbolic line $L$ is related to the angle of parallelism $\theta$ by the formula Most books and websites define the hyperbolic distance element and the corresponding shortest paths in the upper half plane with no explanation. ( ( z 1, z 2, q 2, q 1)) . Once we deﬁne the hyperbolic plane as a set of points, we will deﬁne what we mean by the lengths of curves in the hyperbolic plane. A (hyperbolic) reﬂexion in H 2 is a conjugate of z !z¯ by M 2 so it ﬁxes pointwise a unique geodesic line. In Sect. In the Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles.And in the Beltrami-Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean . Uniformization implies that any Riemann surface can be realized as a quotient of the Riemann sphere C∪{∞}, the complex plane C, and the upper half-plane H. Uniformization implies that any Riemann surface can be realized as a quotient of the Riemann sphere C∪{∞}, the complex plane C, and the upper half-plane H. Viewing the hyperbolic plane using the upper half-plane model, conjugate $T$ to a hyperbolic isometry that preserves the $y$-axis and translates it in the positive direction. In the upper half plane model, use Part e to transform AB to a vertical line and then search for the semicircle that is perpendicular bisector in the upper half plane model.] A Möbius transformation is an invertible map on C of the form z7! A typical non-vertical line is pictured in Figure 4.3. a r . The upper half-plane model, The upper half-plane is the set of complex numbers with positive imaginary part: . (3.3.12) (3.3.12) d ( z 1, z 2) = ln. The upper half plane is the subset H = fz2C : Im(z) >0gof C with the metric ds= p dx2 + dy2 y where z= x+ iy. So, here is a model for a hyperbolic plane: As a set, it consists of complex numbers x + iy with y > 0. Under the hyperbolic metric in the upper half-plane, the shortest distance between two points is along a vertical line or an arc of a circle perpendicular to the boundary (the real axis). Let γ( ) ( ) ( )t x t iy t= + be path so the hyperbolic distance between two points (a, b) on the upper half plane with metric 22 2 2 dd d x y s y + = is defined by 2 ( ) ( ) 1 22 infimumof d t t xt y t t ′′ + The group of orientation preserving isometries . Hi! Definition of hyperbolic length and its formula. 1.3 The Upper Half-Plane Model Now we turn to hyperbolic geometry. We calculate the length in the upper half-plane model after normalizing the triangle to the following: We use the formula derived for the hyperbolic distance in the upper half-plane using complex variables: \[\cosh d(z_1,z_2)-1 . Although the coe cients a;b;c;dcan generally be complex numbers, here we will only be concerned with real coe cients such that ad bc= 1. In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. 1. A hyperbolic polygonis a closed convex set in the hyperbolic plane, that can be expressed as the intersection of a (locally finite) collection of closed half-planes. Let U ⊂ Cbe the upper half-plane, consisting of points z with Im(z) > 0. Suc h sur face s look the same at ev ery p oin t and in ev ery directio n and so oug ht to ha ve lots of symmet ries . We express the area formula in terms of polar coordinates. De nition 1.1. ⁡. A small segment in the hyperbolic plane is approximated to the first order by a Euclidean segment. This means that our standard distance formula will not work. hyperbolic isometries of H are the same as the conformal automorphisms, any quotient Γ\H has a natural complex structure. Orientation preserving isometries under this metric are M obius transformations. Theorem 7.3.3. when the circle is completely inside the halfplane and touches the boundary a horocycle centered around the ideal point when the circle intersects the boundary orthogonal a hyperbolic line when the circle intersects the boundary non- orthogonal a hypercycle. Beltrami-Poincar e's Half-Plane Model Consider all points P(x;y) for which y > 0, this is the upper half-plane. The Quantum Ergodic Theorem on Hyperbolic Surfaces 8 Acknowledgments 16 References 17 1. Review of the Hyperbolic Plane The hyperbolic plane can be modeled by the upper half plane with the hy-perbolic metric. Using a formula relating the Schrödinger operator with uniform magnetic field on the hyperbolic plane and the Schrödinger operator with Morse . Definition (4-2). formula for distance is more complex so we defer its discussion to a later section and in the Maple worksheets DHgeom.mws and UHgeom.mws [30]. As a set, the hyperbolic plane is just U. Figure 1.2.1. The expression of the hyperbolic distance between two arbitrary points (x 1,y 1) and (x 2,y 2) is instead given by coshη= (x 1 −x 2)2 +y2 1 +y 2 2 2y 1y 2. Hyperbolic straight lines in the upper half-plane model are clines that intersect the real line at right angles. To find the distance function, start with a point's distance from the origin. In the page Poincaré Half Plane Model it is explicitly stated that the distance of z, w ∈ H is: The second model will be referred to as the unit disc model and will be denoted by C. The arc-length differential determines an area differential and the area of a region will also be an invariant of hyperbolic geometry. 2.1 Visualizing the Hyperbolic Plane There are two models of the hyperbolic plane that we will use, each one having its own advantages. Given a curve , its length is thus given by. the bounding plane, or half-lines perpendicular to it. We saw in the previous post that the group of automorphisms of a compact Riemann surface X is always finite and can be bounded in terms of its genus. All such points will be considered as "points\ in this model. 12. The Poincar e upper half plane model for hyperbolic geometry 1 The Poincar e 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 upper half-plane model of hyperbolic space, H, consists of the upper half of the complex plane, not including the real line; that is, the set H= fz= x+ iyjy>0g. A Riemann surface is a one-dimensional complex manifold. (What locally finite means is a bit complicated to explain, it requires some deep analysis. Let γ( ) ( ) ( )t x t iy t= + be path so the hyperbolic distance between two points (a, b) on the upper half plane with metric 22 2 2 dd d x y s y + = is defined by 2 ( ) ( ) 1 22 infimumof d t t xt y t t ′′ + In order to understand the hyperbolic distance $h_X(w_1,w_2)$ when one of $w_1,w_2$ is close to a puncture, we should take a careful look at the hyperbolic geodesic nearby the puncture. Moreover, we can see that Möbius transformation is hyperbolic isometries that form a group action PSL (2,Â) on the upper half plane model. See Figure 14. Given a curve , its length is thus given by. be respectively less than 2, called elliptic, greater than 2, called hyperbolic, and equal to 2, called parabolic. below].. The distance from origin to point z = t ∈ \R is a simple line integration d(t) = ∫ t0 2dt 1 − t2dt = ln(1 + t 1 − t) Using this neat formula and the isometric group, we can compute distance between any pair of points. The following theorem provides another formula relating the angle of parallelism to a point's distance to a line. #2. Compass and straightedge constructions See also: Compass and straightedge constructions The upper half plane model H. The hyperbolic length of curves and hyperbolic distance in H. Hyperbolic lines are semicircles with center on the x-axis and vertical half lines in H. There is a unique hyperbolic line through any two points P and Q in H. The shortest path between two points in H is given by . The most straightforward way of working with hyperbolic space is by thinking of it as the upper half-plane with metric . 5 The hyperbolic plane 5.1 Isometries We just saw that a metric of constant negative curvature is modelled on the upper half space Hwith metric dx 2+dy y2 which is called the hyperbolic plane. Lobatchevsky's formula. Escher was the first person . Figure 7 shows the same congruent tracts as Figure 3, but seen in the upper half-space model. In the notation of the previous sections, τ is the coordinate in the upper half-plane >.The mapping is to the punctured disk, because the value q=0 is not in the image of the map.. The space U U is called the upper half-plane of C. C. Why and how are there always infinitely many hyperbolic lines parallel to a given hyperbolic line L and passing through a given point p not lying on L? Given a curve , its length is thus given by. As part of the attening, many of the lines in the hyperbolic plane appear curved in . 2. hyperbolic geometry. The hyperbolic distance: definition, and main formula. This note may be used in model-based courses on the classical geometries. However, we will describe . The Poincaré metric on the upper half-plane induces a metric on the q-disk At first glance it appears that there must . Modular forms are fundamental objects in modern number theory; they were famously used in Wiles' proof of Fermat's last theorem.They are functions defined on the upper half-plane which are invariant under isometry, which is a (hyperbolic-) distance-preserving map from the upper half-plane to itself.So properties of hyperbolic geometry become important when studying modular forms. The quotient space H²/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The hyperbolic plane. Figure 7: Another view of the hyperbolic world divided into congruent tracts. Solution: Subdividing the pentagon into 3 triangles, we see that S = 3π −5π 2 = π 2. The upper half-plane is the set of complex numbers with positive imaginary parts. 1 The Hyperbolic Plane The distance function, I Def The hyperbolic distance between two points z1;z2 2H is dhyp (z1;z2) := inff'hyp (c)g where the inf is taken over all continuous pw C1 paths c from z1 to z2 in H. When there is no ambiguity we will denote from now on d = dhyp and '= 'hyp to simplify notations. az+b cz+d. The model includes all points (x,y) where y>0. Models. By virtue of Lemma 3.1, the claim is a consequence of the fact that T z is preserving the orientation of curves, An explicit formula for the hyperbolic distance d p 0, U n (z) in terms of the . In this model, hyperbolic space is mapped to the upper half of the plane. One thus defined the infinitesimal element of the hyperbolic distance to be. De nition 1.1. ILO1 calculate the hyperbolic distance between and the geodesic through points in the hyperbolic plane, ILO2 compare diﬀerent models (the upper half-plane model and the Poincar´e disc model) of hyperbolic geometry, ILO3 prove results (Gauss-Bonnet Theorem, angle formulæ for triangles, etc as The other model of the hyperbolic plane is the upper half plane model {z ∈ C : 4.5 Law of sines and cosines 31 Im(z) > 0}. To continue studying the geometry of the hyperbolic plane, one must have a notion of hyperbolic distance and this is obtained from the previous hyperbolic metric on the upper half plane. One thus defined the infinitesimal element of the hyperbolic distance to be. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.. (Euclidean similarities are hyperbolic isometries). The upper half plane is denoted by H. Hyperbolic Geometry and PSL(2,R) There are several models of hyperbolic space, but for the purposes of this paper we will restrict our view to the Lobachevski upper half plane. Classifying endomorphisms of a hyperbolic Riemann surface. Definition (4-2). The geodesics in the upper half space model are lines perpendicular to the x-axis and semi-circles . Final Example and relation to length. The Poincaré upper-half plane is one of the models of hyperbolic plane which is defined by: U ≔ (u, v) ∈ R 2 ∣ v > 0, equipped with the Riemannian metric g ̃ ≔ d u 2 + d v 2 v 2. The x-axis is not a part of this geometry. De nition 1.1. Note that dρ → ∞ as r → 1. H r (a,0) Models of Hyperbolic Geometry Solution: Consider the isometry z → kz for k > 0. To continue studying the geometry of the hyperbolic plane, one must have a notion of hyperbolic distance and this is obtained from the previous hyperbolic metric on the upper half plane. Keywords: The Upper Half-Plane Model, Möbius Transformation, Hyperbolic Distance, Fixed Points, The Group PSL (2,Â) 1. For instance, Möbius transformation is classified according to the invariant points. Find an area of a right-angled hyperbolic pentagon. De nition 2.2. The points in are defined to be the usual Euclidean points, and lines are defined to be either half-lines perpendicular to the real axis, or half-circles with centers on the real axis ( image source ): Given two points P 1 (x 1, y 1) and P 2 (x 2, y 2) in the Poincaré upper half-plane model of hyperbolic plane geometry with x 1 ≠x 2, a Cartesian equation of thebowed geodesic passing through P 1 and P 2 and an integral expression for the hyperbolic distance between P 1 and P 2 are developed. In the upper half plane model, use Part f to transform AB to a vertical line and then search for the semicircle that is perpendicular bisector in the upper half plane model.] c. Geodesics and distances on H2. Hyperbolic Geometry and PSL(2,R) There are several models of hyperbolic space, but for the purposes of this paper we will restrict our view to the Lobachevski upper half plane. It can be used to calculate the length of curves in H the same way the Euclidean metric % dx2 +dy2 is used to calculate the length of curves on the Euclidean plane. Let I = [0,1] be the As in Euclidean geometry, each hyperbolic triangle has an incircle. Formula for the sphere inversion. This set is denoted H2. If P, Q, A, and B are distinct points in R2, then their cross-ratio is [P;Q;A;B] = PBQA PAQB VKint. De nition 2.1. For example, the shortest hyperbolic path between the points and is the top arc of the circle , which passes through both points and is perpendicular to the . Starting with the hyperbolic plane, the upper half plane with the hyperbolic lines being all half-lines perpendicular to the x-axis, together with all semi-circles with center on the x-axis. If you want to read further on this, look at Beardon [2], Chapter $7 .$ Let $$ The main question: What are lines in Hyperbolic Upper-Half Plane ? metric to the hyperbolic plane, one introduces coordinates on the pseudosphere in which the Riemannian metric induced from R3 has the same form as in the upper half-plane model of the hyperbolic plane. The upper half-plane model of hyperbolic geometry has space U U consisting of all complex numbers z z such that Im ( z) >0, z) > 0, and transformation group U U consisting of all Möbius transformations that send U U to itself. In the upper half plane model an ideal triangle is a triangle with all three vertices either on the x-axis or at infinity. The other model of the hyperbolic plane is the upper half plane model {z ∈ C : 4.5 Law of sines and cosines 31 Im(z) > 0}. , z 2 ) = d ( z 1, z 2 ) = d ( z 1 z. < span class= '' result__type '' > coordinates in hyperbolic plane appear in. 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Elliptic modular function as well H²/Γ of the hyperbolic distance from ato bin Dis d d ( z,...

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