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

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2
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5answers
452 views

Distance between points in hyperbolic disk models

I was puzzeling with the distance between points in hyperbolic geometry and found that the same formula is used for calculating the length in the Poincare disk model as for the Beltrami-Klein model ...
1
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0answers
83 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
5
votes
2answers
799 views

The shape of Pringles potato chip

Why the shape of Pringles potato chip is hyperbolic paraboloid? I found several articles that say the shape is hyperbolic paraboloid, but cannot find out why it is so. Does anyone have reasonable ...
1
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1answer
52 views

What does the logarithm of a hyperbolic line look like?

At a fixed point $p$ of hyperbolic $n$-space $H$, there is the exponential map from flat $n$-space to $H$ taking straight lines through the origin of the flat space to hyperbolic lines through $p$ in ...
-1
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2answers
130 views

Please explain the shortest path between two points in non-euclidean geometry. [closed]

Please explain it for those with inferior knowledge of mathematics (using easy to understand words): e.g., kids and adults with no knowledge of mathematics (calculus, algebra , etc.) , or rather from ...
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votes
0answers
50 views

Internal angle formula of a generalized polygon as a function of side length and apothem

I am looking to compute the internal angle of a generalized regular polygon (spherical, euclidean, or hyperbolic) as a function of its apothem and side length. I know the equation for a euclidean ...
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1answer
53 views

Is the usual topology on the upper half plane same as that induced by Riemannian metric?

The upper half plane is a Riemannian manifold, with the Riemannian metric given by $(ds)^2 = (dx^2+dy^2)/y^2$ and thus has a metric topology induced by this metric. Is this topology same as the ...
4
votes
2answers
84 views

Distance from a point to a line in the hyperbolic plane

I have two questions: What is the distance from a point to a line in the hyperbolic plane? Fix a line $L$ in the hyperbolic plane. What does the set of points of distance $d$ from $L$ look like?
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1answer
61 views

Fundamental Domain for Congruence (mod 2) Group

How can I show that the area between the circles $|z|=1$, $|z+\frac{1}{2}|=\frac{1}{2}$, $|z-\frac{1}{2}|=\frac{1}{2}$ in the upper-half plane (here's a picture) is a fundamental domain for the ...
2
votes
1answer
84 views

Definition of Hyperbolic n-space

I am a little bit confused about the definition of hyperbolic $n$-space. How do we see $\mathbb{H}^n$ as a homogeneous space model? We can think, the Poincare upper half plane $\mathfrak{H}$ as ...
2
votes
2answers
30 views

algorithm for reducing to the fundamental domain $\mathbb{H}/SL(2,\mathbb{Z})$

I have $y << 1$ and $x \in [0,1]$ uniformly chosen at random and I want to find its representative in the fundamental domain with $\big|\mathrm{Re} \; \tau \big|< \frac{1}{2}$ and $|\tau| ...
2
votes
2answers
81 views

Verifying Möbius Transformations using Hyperbolic Geometry

Verify that every transformation from $$H = \left\{Tz = e^{i\theta} \frac{z-z_0}{1-z_0 z} \right\}$$ can be written as $Tz = \frac{az-b}{\bar{b}z+\bar{a}}$ with $|a|^2 - |b|^2 = 1$. The book gives ...
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3answers
266 views

working in hyperbolic geometry

I wonder if anyone can provide me with a simple step-by-step proof in hyperbolic geometry of a fact that does not hold in Euclidean geometry. I imagine an answer to be a series of statements, such ...
0
votes
1answer
40 views

Non-Euclidean Translations and Rotations

If $f(z) = z + 1$ and $g(z) = -\frac{1}{z}$ show that $$ g f g^{-1}(z) = \frac{z}{1-z}. $$ I don't know how to solve this question please help.
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votes
2answers
39 views

What is the group $\Gamma$ such that $\mathbb{H}/\Gamma$ is a genus-n torus

We know that the universal cover of genus-n torus is a unit disk ($n\ge2$), which is conformal to upper half plane $\mathbb{H}$, with automorphism group $SL(2,\mathbb{R})$. Thus the genus-n torus can ...
3
votes
2answers
83 views

Shortest words in a group with finite presentation

Suppose we're given a group with presentation G=, where both the generating set and the relations are finite. Given a word $w$ in the elements of $X$, I would like to know whether this word is ...
2
votes
1answer
51 views

Broken geodesics in the hyperbolic plane and bending angles

Let $\gamma$ be an infinite broken geodesic in the hyperbolic plane, that is a curve formed by consecutive geodesic segments. Assume also that each of these segments is longer than a certain positive ...
1
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1answer
162 views

Geodesic hyperbolic metric

For a hyperbolic metric on the upper half plane $H = \{(u,v)\in \mathbb{R}^2 \ | \ v>0\},$ how can I prove that the vertical lines are geodesics and that the intersection of any circle centered on ...
1
vote
0answers
91 views

what is the use of the hyperboloid model for hyperbolic geometry?

I am quite new to hyperbolic geometry so even an answer that this question doesn't make any sense can be very helpful. As far as i understand: There are different models of a plane where hyperbolic ...
1
vote
1answer
67 views

area form of the Poincare half plane

For the upper half plane $\{(u,v)|v>0\}$, its area form is $du\wedge dv/v^2$. How to compute the area between the u axis and the curve $\alpha(t)=(r\cos t, r\sin t)$, $0< t < \pi$? Is this ...
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1answer
104 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
2
votes
2answers
102 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) ...
0
votes
2answers
72 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 ...
0
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3answers
100 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?
3
votes
2answers
102 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 ...
0
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0answers
33 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 ...
1
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1answer
65 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 ?
3
votes
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 ...
5
votes
0answers
74 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 ...
4
votes
0answers
235 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 ...
2
votes
1answer
71 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 ...
0
votes
2answers
64 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 ...
1
vote
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
votes
1answer
83 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.
0
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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 ...
0
votes
1answer
63 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
200 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?
1
vote
1answer
219 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
votes
1answer
204 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 ...
2
votes
1answer
203 views

Distance in hyperbolic geometry

In Euclidean geometry, we have that the distance between two points $p$ and $q$ in $\Re^n$ is $\sqrt{(p_1^2-q_1^2) + (p_2^2-q_2^2) + \ldots + (p_n^2-q_n^2) }$ (if we denote the points by $p = (p_1, ...
2
votes
1answer
121 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 ...
5
votes
1answer
336 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 ...
0
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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
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 ...
3
votes
1answer
129 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
45 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 ...
0
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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 ...
4
votes
1answer
539 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 ...