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

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constructing an equilateral triangle in the Beltrami klein model

I am puzzeling with the following: Using the beltrami klein disk of hyperbolic geometry (see https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model ) (PS not the poincare disk model) and given ...
3
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2answers
70 views

Travel across the Poincaré disk model of the hyperbolic plane

A is a point on the Poincaré disk model of the hyperbolic plane. B is a second point, d hyperbolic distance away from A. The hyperbolic ray AB passes through A at angle θ. How might one find the ...
2
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2answers
206 views

Proving pseudo-hyperbolic distance is distance

The pseudo-hyperbolic distance on the unit disk is defined as: $$\rho(z,w)=\left|\dfrac{z-w}{1-\bar wz}\right|.$$ I'd like to prove it's a distance. The real problem is, as always, the triangle ...
4
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2answers
107 views

Translating a Euclidean proof to hyperbolic language..

User HyperLuminal asked for help to prove the following statement: Connecting the feet of the altitudes of a given triangle, we obtain another triangle for with the altitudes of the original ...
2
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1answer
51 views

Question about two definitions of Teichmuller space for a surface of genus $g$

There are many equivalent definitions of Teichmuller space for a surface of genus $g\ge 2$. One of them concerns the complex structure: the Teichmuller space $\mathcal{T}(g)$ is the set of the ...
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1answer
146 views

Instruct geometer moths so you can learn about their true geometry.

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...
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0answers
250 views

Circumference of hyperbolic circle is $2\pi \sinh r$

I'm looking for a proof that in the Poincare disk model the circumference of a circle of radius $r$ is $2\pi \sinh r$. I have seen this result in many places but I haven't been able to find a proof. ...
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1answer
61 views

Distance in the $y$-axis of the hyperbolic plane

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems: In the upper half-plane model, ...
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2answers
105 views

Constructions of perpendicular in hyperbolic plane

Consider the disc model of hyperbolic plane $\mathbb{D}^2$ and a line $g$ through the origin $(0,0)\in \mathbb{D}\subset\mathbb{C}$, i.e. a diameter of the circle $\partial \mathbb{D}=S^1$. Let ...
4
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1answer
380 views

Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry

I was trying to compare the metric tensor at the wikipedia pages of the Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model and the metric tensor of the Poincare disk model at ...
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1answer
35 views

Show that $\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4 $

I have been reading about "mean value theorems in number theory" such as $$\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4 $$ How to prove such a result? One source says it is ...
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1answer
75 views

What is the name of this (circumscribed) triangle?

I am meeting the following triangle more and more in my investigations of ideal triangles in the Beltrami Klein model of hyperbolic geometry. That made me wonder: is there a name for it? (And does it ...
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2answers
93 views

An interesting point of a triangle. (Help needed to prove a statement.)

Consider a triangle whose sides are segments of $\color{red}{\text{line}}$, $\color{blue}{\text{line}}$, $\color{green}{\text{line}}$ falling in the circum-circle $c$. Let ...
2
votes
2answers
118 views

Finding angles of hyperbolic triangles

I am trying to learn about how to find the angles of hyperbolic triangles. Now below is a problem: It has all the steps but I am not understanding the concept (the ones that are underlined in green ...
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2answers
147 views

Hyperbolic Ideal Triangle

I have everything pretty much figured out everything but I need help proving the unique point formed by the three perpendiculars in the picture
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1answer
40 views

Jacobian for Partial Iwasawa Coordinates

I am working through Terras' Harmonic Analysis, V2, and am stuck on I believe a notational point. We are asked to show that for ...
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1answer
83 views

circumscribe a regular polygon around a circle in hyperbolic geometry [duplicate]

In the hyperbolic plane, let a circle of radius r be given. If we want to circumscribe a regular polygon with n sides around this circle (i.e., if we want the sides of the polygon to be tangents of ...
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2answers
182 views

For what $n$ does a hyperbolic regular $n$-gon exist around a circle?

Does there exist a relationship in terms of $r$ and $n$ to represent how large $n$ must be if $r$ of the circle is given in the hyperbolic plane? (The edges of the regular $n$-gon are tangent to the ...
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1answer
27 views

Hyperbolic quadrilaterals : Opposite sides of the quadrilateral cannot intersect

Suppose that a hyperbolic quadrilateral $ABCD$ satisfies $h(A, B) = h(C, D), h(B, C) = h(A, D)$. Mark each of the following claims about the quadrilateral as true or false: Opposite angles of the ...
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1answer
60 views

Euclidean circle in complex plane

I am reading Anderson's Hyperbolic Geometry and am having trouble with one of the Exercises in Chapter 1: Consider the unit circle $\mathbb{S}^1=\{z \in \mathbb{C} \text{ s.t. }|z|=1\}$. Let $A$ be a ...
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1answer
161 views

Circles inscribed in regular polygons in hyperbolic geometry

Does the radius of a circle matter when determining the number sides of a regular polygon in hyperbolic geometry? The sides must be tangent to the circle. Can't I just use an equilateral triangle ...
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1answer
28 views

Hyperbolloid Model Translations

Although the hyperboloid model of hyperbolic geometry has natural analogues of reflections and rotations, I am having trouble finding any linear transformation which is distance preserving and ...
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1answer
37 views

Given a non-ideal hyperbolic triangle and the Euclidean comparison triangle with equal side lengths, are the interiors of the two bi-Lipschitz?

Fix three finite real numbers $p,q,r > 0$. Up to isometry, there is a unique 2-simplex $\Delta$ in the Euclidean plane bounded by a geodesic triangle with these three reals $p,q,r$ as side-lengths. ...
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1answer
41 views

Prove this equality about hyperbolic right triangles

If K is the area of a hyperbolic right triangle ABC in which the right angle is at C, prove that $$ \sin K=\frac{ \sinh a \sinh b}{1+\cosh a\cosh b}$$ My attempt at the solution: I basically need ...
3
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1answer
86 views

Finding the hyperbolic length in a hyperbolic right triangle

Question: In a hyperbolic right angled triangle, the two legs have hyperbolic lengths of $3$ and $4$. What is the hyperbolic length of the hypotenuse? Is this larger or smaller than $5$? I'm having ...
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2answers
86 views

Hyperbolic Geometry and Circles

How does the angle of parallelism relate to the arc of a circle and a point outside? In Hyperbolic Geometry, I'm trying to figure out what happens to the "visibility" of a circle when a point ...
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0answers
41 views

Scale and the models of the hyperbolic plane

I was reading somewhere (sorry I always forget where) that the scale of the Poincare Half plane is y (the vertical) So at the boundary line the scale is $ 0 $ or $ ( 1 : \infty ) $. at the ...
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3answers
208 views

Why do lines in the poincare model meet the infinite edge at right angles?

I know the lines are generated by projecting geodesics on a hyperboloid to a plane and the boundary of the disk comes from the asymptotic cone around the hyperboloid, but I just don't see why the ...
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1answer
29 views

What is vectors straddle a plane mean?

There is a condition in a paper, saying that two vectors straddle a plane. How can we transfer this condition to a equation? Because I have another 5 equations and need this one to solve 6 unknowns. ...
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1answer
22 views

Cusp-end in the universal covering

Let $M$ be a n-dimensional hyperbolic manifold with finite volume. Then as a consequence of the Margulis-Lemma we have a decomposition in different types of ends. So let $C$ be a cusp-end. Then there ...
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1answer
20 views

Finding the euclidean centers of the geodesics AB, AC, and BC

I am trying to learn about finding the angles in hyperbolic geometry and I am trying to understand this example given in Stahl's Introduction to topology and geometry. You can notice that there is a ...
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1answer
36 views

Can an isometry of the hyperbolic plane that maps a circle to a disjoint circle have a fixed point?

Can an isometry of the hyperbolic plane that maps a circle (centred on the real line) to a disjoint circle (also centred on the real line) have a fixed point? By disjoint, I mean that the two circles ...
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1answer
47 views

Construct a circle cutting two other circles at right angles

I have the following problem: On a line $l$ on this line are the centers of two circles $C_1$ and $C_2$ . Circles $C_1$ and $C_2$ do not intersect and are not tangent to eachother. (but one could be ...
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0answers
140 views

Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
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1answer
54 views

Spherical and Hyperbolic lines in the Extended Complex Plane.

We work in the Extended Complex Plane: $ \mathbb{C} \cup (\infty)$. Basically, say we have two points, $z_1$ and $z_2$. It can be shown that, on stereographic projection of the Riemann Sphere onto ...
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0answers
33 views

Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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1answer
67 views

Find the hyperbolic distance in the upper hyperbolic plane

Let $A=(0,112), B=(0,126), C=(98,112)$ be points in the hyperbolic upper half plane H. Find the hyperbolic distances $d_h(A,B), d_h(A,C), d_h(B,C)$. Every answer should be in the form of a ...
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1answer
61 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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1answer
244 views

Find the 3 angles of the hyperbolic triangle

A(0,5) B(0,2) C(4,2) In Euclidean geometry the three points given are the vertices of a right-angled triangle. Find the three angles of the hyperbolic triangle with vertices A,B,C. Find the ...
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1answer
33 views

How to formulate the hyperbolic parallel postulate for more than dimensions?

To formulate the hyperbolic parallel postulate for the hyperbolic 2 dimensional (plane) is easy: Given any line ''L'' and point ''P'' not on ''L'', there are at least two distinct lines passing ...
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2answers
93 views

Why do we use cosh to define the angle between two vectors in hyperbolic geometry?

I can kind of see why this works when we use the regular dot product, but I don't understand why this is still true when we use the dot product adapted for hyperbolic geometry?
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1answer
61 views

Is there any textbook about computing the automorphism group of the triangle group?

For example computing the automorphism group of the 2 genus surface made by triangles (12,2,3) in the hyperbolic plane. In addition,if you know the trick of the computing the automorphism groups like ...
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0answers
64 views

curves in Poincare half space (3 dimensional hyperbolic geometry)

Okay maybe I am going a bit ahead of my self The Poincare half plane still has many mysteries for me But still I was puzzeling about the 3 dimensional variant of it. So lets assume an hyperbolic 3 ...
0
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1answer
47 views

Proving limit on angle of a hyperbolic right triangle

I'm trying to prove that for a right triangle $\Delta ABC$ with right angle $B$, the angle $BAC \le \sin^{-1}(sech AB)$ I'm not really able to find a way to bring this proof together. I've tried ...
2
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1answer
98 views

The locus of points forming a right angle, in nonzero curvature

Given a line segment $AB$ in the Euclidean plane, the locus of points which form a right angle with $A$ and $B$ is known to be a circle, with $AB$ as a diameter. Is this also true for a geodesic ...
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0answers
66 views

Find the hyperbolic length of the geodesic segment

I'm reading my textbook and I'm trying to make sense of this example. So the place with the red star shows the actual process of calculating the hyperbolic length. My question is how they get the ...
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0answers
78 views

Calculating hyperbolic length

So, I am looking at a question and I'm having a hard time solving it. So I know the formula but my question is first, what is $\alpha$ and what is $\beta$? So I calculated some values: I found ...
3
votes
3answers
190 views

Reference: In every free homotopy class is a unique minimizing closed geodesic

Does anyone know a reference for the following result: Let $M$ be a compact hyperbolic manifold/manifold with strict negative curvature . Then in every non-trivial free homotopy class of $M$ there ...
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1answer
61 views

Fundamental group of a compact hyperbolic manifold

Let $M$ be a compact hyperbolic manifold, and $\tilde M = H^n$ the universal covering. Now let $\Gamma$ be the group of Decktransformations. So we have $\tilde M / \Gamma = M$. My question: Is it ...
2
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0answers
42 views

Compute hyperbolic length of the arc of the circle

Compute the hyperbolic length of the arc of the circle $ x^2 + y^2 = 25$ that lies between (3, 4) and (4, 3). From my notes I know the formula is $$ \ln \frac{{\csc \beta - \cot \beta }}{{\csc ...