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

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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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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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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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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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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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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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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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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 ...
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43 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 ...
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Length of a hypercycle.

I was a bit puzzeling about what is the length of a hypercycle, horocycle and the line segment between two points. and found out that if $h$ is the length of one of the two horocycles between P and ...
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52 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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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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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 ...
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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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50 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 ...
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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 ...
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Creating a Hyperbola with a Flashlight

I ran into this problem in a textbook and was intrigued by it. Conics are generally formed through different cuts one can make with the shape of a cone. But, there have been recent discussions on ...
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Problem in understanding models of hyperbolic geometry

I recently started reading The Princeton Companion to Mathematics. I am currently stuck in the introduction to hyperbolic geometry and have some doubts about its models. Isn't the hyperbolic space ...
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Gromov's boundary at infinity, drop the hypothesis on hyperbolicity

It's an easy result that if we have two quasi isometric hyperbolic spaces, then their Gromov boundaries at infinity are homeomorphic. I found online these notes where at page 8, prop 2.20 they seem ...
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Collinear Points and Congruent triangles

How might one show that three points are collinear? I am in hyperbolic geometry and am showing that two parallel lines also have a common perpendicular. I have two parallel lines $m$ and $l$ cut by a ...
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33 views

Trisection of a hyperbolic line/segment

I'm wondering how to trisect a line/segment in $\mathbb{H}^2$ (using the Poincaré Disk model). Bisection of a hyperbolic line seems rather straightforward (e.g. as described in the paper Compass and ...
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27 views

Noneuclidean Geometry - how do I find $(x,y,z)$ in $\mathcal{S}$?

I've been asked to find $(x,y,z)$ in $\mathcal{S}$. I'm stuck on the question attached because although it gives the formula of how to find $\pi_\mathcal{s}$ (stereographic projection), I'm not sure ...
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Thread constructions in the Poincaré's disc

I just came across the following image (source) and realised something that should have been obvious to me a while ago: it should be possible to construct the envelopes of curves in the Poincaré disc ...
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constructing an segment with a known length in hyperbolic geometry

I am studying Arlan Ramsay's and Robert Richtmeyer's " Introduction to hyperbolic geometry" On page 255/6 it gives how to construct an segment with an absolute length of $ \ln (\sqrt{2} +1) $ (via ...
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38 views

Coordinate on the boundary of Riemann surface

Let $\Sigma$ be a Riemann surface with boundary. Question: Is there canonical way to parameterise the boundary components up to shift? By shift I mean change of coordinate $\phi$ to $\phi + c$. ...
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signature of a 2-dim linear subspace

Prop: If $U$ is a 2-dim linear subspace of $\mathbb{R}^{n,1}$ (Eculidean space with Lorentz-scalar product) with $U \cap H^n \neq \emptyset$, then the restriction $<\cdot ,\cdot>|_{U}$ has ...
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calculating Hyperbolic distance

To calculate the hyperbolic distance we use the formula $$\left|\frac{w-z}{1-\bar wz}\right|$$ I want to apply this to the following pair of points: \begin{align*} w&=\frac{-1}{\sqrt{3}}\space ...
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Hyperbolic geometry when the curvature is constant and negative but not -1

Help I am getting completely confused Hyperbolic geometry is the geometry of surfaces of a constant negative Gaussian curvature, in most formula's it is almost assumed this constant negative ...
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Canonical map from fundamental group to Fuchsian group?

Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ ...
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75 views

Length of geodesic representative on hyperbolic surfaces

Let $S$ be a closed oriented hyperbolic surface. Let $x,y \in S$ and let $\alpha,\beta$ be two geodesic arcs with endpoints $x$ and $y$. Let $\alpha \beta$ be the closed piecewise geodesic curve ...
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49 views

A casual definition of Hyperbolic Space

I'm writing an article about a lecture that mentioned hyperbolic space. I wondered if anyone had a friendly way of describing it to the general public. (I will rewrite any definitions in my own ...
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Two transformation groups of the hyperbolic plane are isomorphic?

I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane. I've just been told that $SO(2,1)$, the indefinite ...
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How to solve an Hyperbolic triangle when all is given except angle C and side c)

Another Hyperbolic triangle problem (all given except angle $\angle C$, and side $c$) I thought that after asking How to solve an hyperbolic Angle Side Angle triangle? I could solve all hyperbolic ...
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How to solve an hyperbolic Angle Side Angle triangle?

If from an hyperbolic triangle $ \triangle ABC$ the angels $\angle A$, $\angle B$ and side $c$ are given. (ASA triangle, a side and both adjacent angles are given) How can I calculate the remaining ...
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Decision Boundary of A Single Perception with Logistic Function

I am currently studying neural networks and have been trying to reason about this for a while to no avail. I understand that given a perceptron(such as above) with f as a step function, any ...
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98 views

Dehn twist as isometries on hyperbolic surface

[I am editing the question to a most correct and precise one thanks to comments of Lor and studiosus] Let (S,g) be a compact hyperbolic surface. On a simple closed geodesic $\gamma $ I can realized a ...
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Advantage of using Hyperbolic Trigonometric functions?

Is there any added advantage of using Hyperbolic Trigonometric functions? Since you can always use normal trigonometric functions in all cases: $$\left.\begin{array}{ccc} ...
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Hyperbolic Geometry: Question about the Transitivity of Möbius transformations

I was confronted with this exercise in the book Hyperbolic Geometry by Anderson which states: In each case, find $m \in Möb(\mathbb{H})$ such that the property holds, or prove that no such $m$ ...
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Is there a name for this special, “most parallel” ultraparallel line in hyperbolic geometry?

Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P. However, there's one line M which is ...
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Commutator of hyperbolic isometries

Let $f,g \in PSL(2,R)$ be isometries of $\mathbb{H}^2$ of hyperbolic type. Let $h=[f,g]$ be their commutator. Is there an explicit geometric criterion to determine if $h$ is hyperbolic, parabolic or ...
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Relation between length of arc of horocycle and length of chord?

In Hyperbolic geometry: What is the relation between the length of the arc of a horocycle between two points and the length of the chord (segment) between the two points? Also what is the relation ...
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Complex hyperbolic Trigonometry

When faced with the equation $\cos{z}=\sqrt{2}$ I want to solve for z so I break it up into a sum $z=x+iy$ and get: $\cos{z}=\cos{x}\cosh{y}-i \sin{x} \sinh{y}$ equating real and imaginary parts I ...
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Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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190 views

What's the point of the Poincaré disc model?

I'm trying to work out the point of the Poincaré disc model (excuse the pun). As far as I can tell, it's a disc, on which the only permitted lines are a line straight across the middle, and arcs of ...
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Poincare disk and Poincare half plane

My book claims that the Möbius transforms are isometries of the Poincare half plane model. Thus, the metric is preserved under these maps. But I know that the Poincare disk can be derived from ...
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Triangle Identity leads to another Euclidean parallel.

Referring to TriangleIdentity by 伍柒貳 a while ago, considering $\bigtriangleup$ ABC, it is proved that: $$\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C (1*) $$ I want to take angle $A = ...
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Constructing a quadrilateral in hyperbolic plane

Suppose we have a set of angles, $\{\alpha_1, \alpha_2, \alpha_3, \alpha_4 \}$, and we want to construct some quadrilateral at Poincare Disc Model with angle $\alpha_i$ at vertex $i$. The question ...
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Bi-asymptotic geodesics in Visibility manifolds

I'm thinking about some properties of geodesics in visibility spaces. Here I give some definitions: A Riemannian manifold $(M,g)$ with riemannian distance $d$ is said to be a visibility manifold if ...
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constant-curvature Riemannian metric for Bring's surface

There is a well-known and very symmetric space that is called either "Bring's curve" or "Bring's surface", depending upon the context. (Bring was a Swedish mathematician in the 18th century.) Let's ...
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We can get the geodesic only by the first geodesic equation in hyperbolic plane?

The two geodesic equations are: $$\frac d{ds} (Eu' + Fv') = 1/2(E_u {u'}^2 + 2F_u u'v' + G_u {v'}^2) $$ $$\frac d{ds} (Fu' + Gv') = 1/2(Ev{u'}^2 + 2F_v u'v' + G_v{v'}^2) $$ And in the hyperbolic ...