Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.

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Strong contraction of hyperbolic space

I'm trying to study Hyperbolic geometry, but I can not understand the following statement. Let $X$ be a $δ$-hyperbolic space. Then, there exists $M > 0$ such that for any geodesic $γ$, and ...
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Axioms vs Models in projective and hyperbolic geometry

I'm studying Projective Geometry. The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. ...
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Is the group generated by two loxodromic isometries with a fixed point in common cocompact?

If you have two distinct loxodromic isometries of the hyperbolic plane $\gamma_1, \gamma_2$ such that they have a fixed point in common. For simplicity let's take the half plane model and let the ...
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Side and angle relations in a hyperbolic quadrilateral.

Let $PQRS$ be hyperbolic quadrilateral, i.e. a quadrilateral in $\mathbb{H}$ whose sides are hyperbolic geodesic. Let length$(PQ)=l_1$, and length$(PS)=l_2.$ Also $\angle SPQ=\theta_1$, $\angle ...
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Is this a Trapezium?

I once read that in hyperbolic geometry, two hyperbolas can be parallel. In a trapezium, you have four sides and a pair of parallel lines, therefore is it possible to have a trapezium with two ...
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37 views

Discrete faithful representation in $PSL(2,\mathbb R)$ and horocycles in hyperbolic space

Let $S$ be a closed oriented surface of genus $g>1$. Is the following true ? Let $\alpha,\beta\in \pi_1(S)\backslash \{1\}$ and $\rho:\pi_1(S)\rightarrow PSL(2,\mathbb R)$ be a discrete ...
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54 views

Where does the hyperbolic metric come from?

In hyperbolic geometry, the metric is often defined as $$ds=\frac{\sqrt{dx^2+dy^2}}{y}$$ Where did this metric come from? I have thought long and hard about this question, but have no satisfactory ...
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Prove by showing that CW is asymptotic parallel to DZ in the direction of W and Z

If XABY is a biangle and WCDZ is a figure formed so that AB=CD. angle BAX = angle DCW, and angle ABY = angle CDZ then WCDZ is a biangle. If Two lines AX and BY are asymptotic parallel in the ...
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How to show the Poincaré disk is hyperbolic for some $\delta$

I am trying to prove that the Poincaré disk, $\mathbb{D}$, is $\delta$-hyperbolic with respect to the slim triangle definition for hyperbolicity. I have been stuck for a while on where to begin, ...
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Semiregular tilings of the hyperbolic plane

Consider the irregular quadrilateral tiling of the Euclidean plane depicted by the log-log coordinate grid: I'm wondering if in the Hyperbolic plane exist some analog of this kind of tiling where ...
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Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
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Prove that the gradient of the tangent to $xy=d^2$ is $-\frac{c^2}{4d^2}$

I have this question: Given that $y=mx+c$ is a tangent to $xy=d^2$ prove that $m=-\frac{c^2}{4d^2}$. I'm not sure what direction to take - I tried differentiating the hyperbola equation, but ...
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32 views

measuring curvature

Suppose you are transported to an 2 dimensional hyperbolic world, ( a plane (2 dinensional) manifold with a constant negative curvature ) the only geometrical tools you have are a ruler, a pencil, ...
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52 views

(compact, non-empty boundary )Surface Geodesics on Hyperbolic Geometry

I have a basic knowledge of hyperbolic geometry . I am trying to understand the meaning of "a compact surface S with non-empty boundary(which is neither a disk nor an annulus )with a complete ...
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Fundamental group of a closed hyperbolic surface is Gromov hyperbolic

Does anyone have a reference for the proof of the result in the title? Thanks!
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Fundamental solution of Poisson equation in the Hyperbolic Plane

If we consider the Poisson's equation $$ -\Delta u=f(x), \ \ \mbox{in} \ \ \mathbb{R}^n, $$ we can construct the fundamental solution $$ u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)f(y)dy, $$ where $\Gamma$ is ...
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Fundamental domains of infinite-index subgroups of $SL(2,\mathbb{Z})$

While discussing modular forms associated to different subgroups $\Gamma$ of $SL(2,\mathbb{Z})$, there appeared to be a heuristic relationship between the index $[SL(2,\mathbb{Z}) \colon \Gamma]$ and ...
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measuring the curvature of a pringle

Assuming for the sake of argument that a pringle shaped potato chip has a constant negative curvature, (which according to http://math.stackexchange.com/a/617610/88985 it is not ) see also picture ...
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Trying to understand Lobatschewsky's parallax formula.

Lobatschewsky gave a method to calculate the curvature of space (see Bonola “Non-Euclidean Geometry” § 45) But I don't understand his method. Can somebody explain? I understand that the method now ...
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145 views

Shape made by Beltrami

Beltrami made (out of thin paper or stiff cloth) a model of a surface of constant negative Gauss curvature. The original might have resembled a large saddle shaped Pringles chip, and frills might have ...
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Definition of hyperbolic trig functions

I was doing some homework for my complex analysis class and ran into a personal question. I haven't worked a lot with hyperbolic trig functions (e.g. $\sinh (x)$, $\cosh(x)$, etc.) so this question ...
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Exponential map in the Poincaré upper half plane

I have a question regarding the Poincaré upper half plane. Is there a simple way to express the exponential map? I have been looking unsuccessfully on internet for an expression... Thanks for any ...
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How does one formally verify this picture from hyperbolic geometry?

Suppose we consider some hyperbolic circle with center $iz$ using the upper-half plane model of hyperbolic geometry, and in the interior we have a point $x+iy$. How does one prove that $iy$ is also ...
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Spherical Geometry is to the Sphere just as Hyperbolic Geometry is to the…?

I need to write up a quickie description of Hyperbolic Geometry for non-mathematicians. I am trying to say "Hyperbolic Geoemtry is the Geometry of the surface of a ____" I remember that there is, in ...
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HyperCube questions

I have three hypercube questions. 1) How many nodes does a d-dimensional HyperRing have (as a function of d) ? 2) How many edges ? 3)What is the degree of each node in a HyperRing with n nodes ? I ...
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which surfaces have (for a large area) a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
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“projective maassbestimmung” in Automorphic Functions by Fricke + Klein

I was reading a copy of Fricke and Klein's Theory of Automorphic Forms, and I came across the phrase projective maassbestimmung in the first chapter. Google translate returns: maßbestimmung as ...
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constant $K$ and $k_g$ ovals growth

Referring to my recent post: Ovals of constant $ k_g$ on constant $K$ surfaces, using geodesic polar coordinates with radial geodesic lines built along v=constant around a fixed point on a constant ...
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31 views

Area of pseudospherical segment

Surface area of segment of a sphere radius $a$ at the equator, between two parallels, is given by $ 2 \pi a (z_2-z_1) $,where $z_2, z_1$ are heights of spherical segment at radii of parallel circles ...
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71 views

“Hyperboloid like surface” as hyperbolic plane / pseudosphere

A pseudosphere is an surface wth a constant negative curvature. In most publications, it is almost given that the tracioid (rotated tractrix) is the surface that has a constant negative curvature, ...
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Tractricoid as a pseudosphere (surface with constant negative curvature)

How to motivate/calculate/prove see that the tractricoid, i.e. a tractrix rotated about its asymptote, has a constant negative curvature? What are the hyperbolic lines on a tractricoid and how to see ...
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Good data structure for hyperbolic tiling

Say you're doing something computational where each data point is a tile in a (not necessarily Euclidean) 2-dimensional tiling, for instance, a Life-like cellular automata. You might want a data ...
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simplification of the area of a hyperbolic circle (BONOLA, S 53)

I'm trying to understand the S-53 of "Non-Euclidean Geometry" (BONOLA, R.) in which the formula for the area of a circle of radius r: $$2\pi k^2(\cosh\frac rk -1)$$ is somehow reduced by only applying ...
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Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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What's the connection between “hyperbolic” inner product spaces and the hyperbolic plane?

In Jacobson's Basic Algebra I, in Kaplansky's Linear algebra and geometry and in Artin's Geometric algebra, a hyperbolic plane is defined to be a two-dimensional, nondegenerate inner product space ...
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Why are Euclidean and hyperbolic lengths proportional to first order?

In his book “Three-Dimensional Geometry and Topology”, Thurston constructs a Riemannian metric for the Poincare disk model and begins as follows. He says that, given any (hyperbolic) line segment $s$ ...
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What is the hyperbolic plane equivalent to translation in euclidean space

in euclidean plane one can move polygons like rectangles, triangles etc. around by isometries, e.g. translations. For instance if we consider a rectangle with midpoint $0\in\mathbb{R}²$ then the image ...
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Constructing shapes in hyperbolic space

I'm trying to get started writing a game that uses the order-4 dodecahedral honeycomb in hyperbolic space. I'm representing points as 4-vectors of the form ...
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Question about hyperbolic space-forms

I have a map between hyperbolic space-forms $\varphi:B^n/\Gamma \longrightarrow B^n/H$ (where $\Gamma, H$ are discrete groups of isometries that act freely), and a lift to a map $\tilde{\varphi}:B^n ...
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here is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$.

There is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$. My question is, isn't $z \rightarrow kz$ an isometry for ...
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Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that ...
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Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
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Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
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Simple proof of existence of hyperbolic triangles

I've studied the hyperbolic plane by analytically building up the hyperboloid model, the Klein—Beltrami disc, the Poincaré disc, and the half-plane model from scratch. Now I'd like to prove that, ...
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Geometries (Euclidean and Projective)

We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it's true in Coordinate Geometry. It makes ...
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hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

On http://en.wikipedia.org/wiki/Hypercycle_%28geometry%29 I found the statement. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their ...
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How to construct a circle in a the Poincare Disk model

How can I construct an circle with centre C going trough point P in a Poincare disk?. I found an script of how to do it in the "Poincaré Disk Model of Hyperbolic Geometry"toolkit from the geometers ...
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Distance-preserving coordinate transformations for the poincaré disc

Following this question, I'm looking for a coordinate transformation which leaves distances unchanged. Does such a transformation exist? The isometries for the poincaré disk looked promising, but only ...
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Representation of cell location in hyperbolic plane

I want to represent an order-5 square tiling (image from Wikipedia; more text below image): Obviously for a simple grid I can uniquely refer to a given square by its ...
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calculate the curvature of a surface with a Lambert quadrilateral

I was wondering how can I calculate the curvature of a surface? For example: Given a Lambert quadrilateral ABCD (see http://en.wikipedia.org/wiki/Lambert_quadrilateral ) with: $ DA \bot AB $, $ AB ...