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

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1answer
102 views

Distance between a point and the origin in Poincaré Disk

How to calculate the distance between the origin and the point $p=(a,0)$, $a>0$, using the metric $g = \frac{4}{(1-x^2-y^2)^2}(dx^2+dy^2)$? I don't how to use correctly these $dx^2$'s
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2answers
90 views

Dirichlet Domain of a Fuchsian Group

Recall that a Fuchsian group is a discrete group $\Gamma \leq PSL(2, \mathbb{R})$ and that a Dirichlet domain for $\Gamma$ is a set $D \subset \mathbb{H}^2$ of the form $$\{z \in \mathbb{H}^2: d(z, p) ...
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2answers
71 views

Prove that a transformation from the hyperbolic group can not be loxodromic.

Prove that a transformation from the hyperbolic group can not be loxodromic. I know a loxodromic λ = kei$^\theta$ with k not equal to 1 and theta not equal to 0. But I'm unsure how to go after that, ...
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0answers
15 views

software to decide whether a 2-generator subgroup of PSL(2,R) is discrete/free

Gilman developed an algorithm with polynomial complexity that, given two elements in PSL(2,R), decides whether the group they generate is free/discrete or not. I was wondering whether anybody ever ...
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3answers
91 views

parametrise equation of a hyperbola

Any point on an ellipse can be wrttien as $(a\cos\theta,b\sin\theta)$, How could we genarilse this to a hyperbola?
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2answers
92 views

Constructing a differential equation for hyperbolic crochet

There is plenty of information about hyperbolic geometry and its melding with crochet, however I have yet to find an exact equation for determining the number of stitches in each row. I will try to ...
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0answers
32 views

How to get the smallest sample size with max probability in Hypergeometric Distribution

A body of students has 30 male students and 20 female students. Suppose a sample of n students are drawn from this population. What is the smallest n that can yield the maximum probability to have 5 ...
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1answer
60 views

$2$-dimensional Hyperbolic space with fundamental group ${\bf Z}$ and constant curvature $-1$

$$ d\rho^2 + \cosh^2\rho\ d\theta^2$$ Only one ? Is there any other example ?
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1answer
72 views

complete metric on a Riemann Surface

I'm reading a book and found a sentence I don't understand: Every Riemann surface $S$ supports an essentially unique complete metric of constant curvature $1$, $0$ or $-1$. Every point of ...
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0answers
73 views

Tilings of the Hyperbolic plane

Given a tiling of the hyperbolic plane projected onto a unit disc such as this which can be considered as a graph. I then define some functions: $f(r) =$ number of graph nodes contained within the a ...
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0answers
199 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
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1answer
63 views

Is there an algebraic method for hyperbolic rotations?

Given a 2d vector, how do you rotate it in space? You could use a rotation matrix, $$\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta ...
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2answers
63 views

Why is it said that every point in hyperbolic space is a saddle point?

I have read that since hyperbolic space has a constant negative curvature (a concept that I think I understand), every point is a saddle point. I am trying to understand what that means. Can we say ...
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1answer
64 views

Problem about alternate angle on poincare disc model.

If two alternate angles are same, two poincare lines are parallel. (i.e. If two poincare lines cut by a transversal have a pair of congruent alternate interior angles, then the two poincare lines are ...
3
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1answer
81 views

non-discrete group isomomorphic to a discrete group

I am trying to find an example of a discrete group of Möbius transformation that is isomorphic (algebraically) to a non-discrete group. Can someone please help finding such groups.
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1answer
48 views

Solve $d_h(A,B)$ on a Poincare Disc

Consider △ABC on a poincare disc. On △ABC, $\angle C = \theta(radian)$, $d_h(B,C)=d_h(A,C)=b$ In this situation, solve $d_h(A,B)$. To me, it is hard because I have no experience. Is there someone ...
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1answer
61 views

Poincare disc model problem. find $d_h(A,B)$

Consider $\triangle ABC$ on a poincare disc. On $\triangle ABC$, $\angle C=90^\circ$, $d_h(B,C)=a$ and $d_h(A,C)=b$ In this situation, find $d_h(A,B)$. I'm taking a course but I cannot follow ...
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1answer
196 views

prove that the sum of the angles in any triangle is less than 180 in hyperbolic geometry (or poincare model).

We could use poincare disc model as a hyperbolic geometry model. I have difficulty understanding poincare disc model. So is there someone to help?
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1answer
202 views

Covert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels x units in Euclidean space, how much ...
2
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1answer
189 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
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1answer
179 views

Distance in hyperbolic geometry

This is probably a bad question, but an answer could help me to understand. In Euclidean geometry, we have that the distance between two points $p$ and $q$ in $\Re^n$ is $\sqrt{(p_1^2-q_1^2) + ...
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1answer
108 views

Which of the (non-)Euclidean planes can we embed into non-Euclidean 3-space?

I read the answers to this very interesting question and saw that we can in fact embed the Euclidean plane into hyperbolic 3-space using what is called a horosphere. However, as Hilbert showed us, the ...
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1answer
269 views

What is the relationship between hyperbolic geometry and Einstein's special relativity?

I am a third year math student writing a term paper on hyperbolic geometry and I would like to understand its relationship with special relativity. I have read that the hyperboloid model of hyperbolic ...
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1answer
61 views

What would be called as the shape of $xy=10$ in 3-dimensional space?

As title says, what would be called as the shape of $xy=10$ in 3-dimensional space? It doesn't seem to be paraboloid nor hyperboloid...
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0answers
67 views

Prove the Lobachevsky-Bolyai formula for the Klein model

I want to prove that e^(-d) = tan(Π(d)/d) in the Beltrami-Klein Model for the angle of parallelism in correspondence to the distance d, where d is the klein distance d(AB) = (1/2)|ln((AB,PQ)). A hint ...
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0answers
40 views

Equidistant Gergonne point in trebly asymptotic triangle, a consequence of Gergonne's Theorem in the Klein-Beltrami Model?

It is a theorem in hyperbolic geometry that inside every trebly asymptotic triangle (ABC) there is a unique Gergonne point G equidistant from all sides. Show that in the Beltrami-Klein model ...
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0answers
26 views

Proof: Every H-Projection is product of H-Reflections

I have some problems with the proof of the statement above (sorry for a bad translation I guess). First of all some definitions: Definition 1 (möbius-transformation): Let $A \in \mathbb{C}^{2 ...
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1answer
128 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
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0answers
43 views

JSJ-decomposition of a non-hyperbolic 3-manifold

Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the ...
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0answers
45 views

boundary of word-hyperbolic group

Let $G$ be a word-hyperbolic group and let $\partial G$ be its (Gromov) boundary. Do there exist criteria that imply that all non-trivial finite order elements of $G$ act fixed-point freely on ...
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5answers
75 views

Easy explanation of non-abelianness of hyperbolic curves

I'm looking for easy proofs (or just an easy proof) of the following statement: Let X be a hyperbolic Riemann surface, i.e., $X$ is a Riemann surface and the universal covering of $X$ is the complex ...
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1answer
490 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
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1answer
113 views

Seifert manifolds and Fuchsian group

A Fuchsian group is a discrete subgroup of $PSL(2, \mathbb{R})$. Let $M$ be a Seifert manifold (maybe with boundary) and $t \in \pi_1(M)$ the class of a regular Seifert fiber. Hempel claims in his ...
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1answer
170 views

Questions about triangles, n-gons, and tessellations of the hyperbolic plane

Why there are infinitely many regular tessellations of the hyperbolic plane? Can there be a triangle made up of three straight lines in the hyperbolic plane? I know it's impossible since the angle ...
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0answers
51 views

Drawing graphics on a pseudosphere

I'm pretty sure this question is going to be very difficult to answer, so I will do my best to explain my problem, and maybe, just maybe someone will have a good answer. I have written a program ...
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1answer
73 views

What is complex conjugation in Hyperbolic Geometry

Suppose we are working in the hyperbolic plane, that is the set $$ \mathbb{H}^2 = \{ u + iv \colon v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ Now, formally, I can rewrite ...
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1answer
106 views

justifying reflection across line in beltrami-klein model

Justify the following construction of the Klein reflection A' of A across m. Let Λ be an end of m and P be the pole of m. Join Λ to A and let this line cut y (which is the circle, my note) ...
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1answer
48 views

Simple closed geodesic around two hyperbolic cusps.

Consider a connected hyperbolic $2$-manifold $M$ with cusps. Consider a simple closed geodesic in $M$, which dissects $M$ into two components. Assume that one of the components contains exactly two ...
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1answer
74 views

projection onto the nullspace of the Laplacian on a conformally compact surface

Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
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1answer
44 views

Verifying the vertices of a Hyperbola

In this exercise it is required to verify that the points O & A(4,0) are the vertices of the Hyperbola H , as you can see in the marked part. Can someone help verify that ?
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1answer
59 views

Computing the volume of a hyperbolic knot

Could anyone show me or refer me to a link where the volume of a hyperbolic knot, say, the figure-8 knot, is computed (well, in fact estimated) explicitly and not only having the procedures outlined?
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1answer
81 views

Quotient under a Fuchsian Group

Let $D$ be the unit disk in the complex plane, and let $H$ be a fuchsian group generated by one fixed point free element, say $a$. What it the quotient $D/H$? Attempt: The quotient is biholomorphic ...
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1answer
72 views

A Fuchsian Group?

Let $p_k := e^{\pi/2 i k}$, $k \in \{0, 1,2,3\}$. Let $b_k$ the geodesic of the hyperbolic disk connecting $p_k$ and $p_{k+1(\text{mod}4)}$. For instance, $p_0$ and $p_1$ are connected by the lower ...
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2answers
137 views

Hyperbolic Geometry - reference request

I need some information about Hyperbolic Geometry. For example, Spherical Geometry is a subsection of Hyperbolic Geometry or no? Can you suggest to me a book or some other reference to help me ...
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77 views

Question regarding the projective models of the anti-de-Sitter spaces and good online references for learning them from the scratch? (Specifics below)

As my title says above, I am trying to find answers to and also good online reference where I can find complete description of projective models of hyperbolic space, de-Sitter space and anti-de-Sitter ...
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1answer
108 views

Isometries of a hyperbolic quadratic form

I am reading an article that says "The group of isometries (of a hyperbolic space) of a hyperbolic quadratic form in two variables is isomorphic to the semi-direct product $\mathbb{R} \rtimes ...
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1answer
123 views

Isometry fixing two points of a geodesic line

Let $H$ be a hyperbolic space, and let $\Gamma \subset H$ be a geodesic line, i.e., the image of an isometry from $\mathbb{R}$ to $H$. If $f$ is an isometry of $H$ that fixes two distinct points of ...
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1answer
190 views

Arc length parameter s

Consider the metric $$ds^2 = \frac{dx^2+dy^2}{y^2}.$$ Assume $R>0, a\in\mathbb{R}$. Consider the curve $$\gamma(\theta)=(a+R\sin\theta,R\cos\theta)$$ for $-\frac{\pi}{2}\leq\theta\leq ...
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2answers
144 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
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1answer
91 views

Möbius transformation that preserve distance for two pair of given points in $\mathbb{H}$.

I need to prove that for a given two pair of points $(z_1,z_2)$ and $(w_1,w_2)$ in $\mathbb{H}$ (Poincaré's upper half plane), where $d_{\mathbb{H}}(z_1,z_2)=d_{\mathbb{H}}(w_1,w_2)$, there is an ...