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

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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 ...
5
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2answers
645 views

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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2answers
66 views

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 ...
3
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1answer
47 views

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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24 views

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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1answer
32 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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1answer
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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46 views

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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1answer
40 views

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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1answer
35 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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0answers
31 views

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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1answer
33 views

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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2answers
83 views

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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1answer
22 views

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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1answer
61 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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1answer
42 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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1answer
42 views

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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2answers
72 views

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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1answer
54 views

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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16 views

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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1answer
72 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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2answers
52 views

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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1answer
53 views

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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1answer
57 views

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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1answer
46 views

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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1answer
46 views

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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3answers
53 views

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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1answer
39 views

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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1answer
150 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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2answers
94 views

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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1answer
44 views

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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1answer
69 views

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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1answer
40 views

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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1answer
65 views

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 ...
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30 views

Comparing two fundamental domains for $\Gamma(2)$

(0). My question concerns the relation between two different fundamental domains for the group $$ \Gamma(2)= \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix}\in SL_2(\mathbb Z) \; ...
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1answer
22 views

Homothetic transformation in the Poincaré upper half plane

i am interested in finding homothetic transformation in the Poincaré upper half plane. I heard that unlike $\mathbb{R}^n$ we don't have an homothetic transform for every $\lambda \in \mathbb{R}^+$. ...
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Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
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2answers
54 views

hyperbolic tangent vs tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (in a right triangle). Is there a similar definition for the hyperbolic tangent? The reason ...
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69 views

Mapping a line in the hyperboloid model of $\mathbb{H}^2$ to a circle in the Poincaré model

In the hyperboloid model of $\mathbb{H}^2$ a point $P(x,y,z)$ is the intersection of the vector $(x,y,z)$ with the upper sheet of the hyperboloid $x^2 + y^2 - z^2 = -1$, and a line $L(a,b,c)$ is the ...
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1answer
61 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
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1answer
74 views

Models of the hyperbolic plane

Hilbert's theorem tells us that there is no immersion in $\mathbb{R}^3$ with negative Gauß curvature that is complete. Despite, there are some models of surfaces with negative Gauß-curvature like the ...
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1answer
43 views

Proving triangle inequality for hyperbolic distance using contours

I need to prove the triangle inequality for hyperbolic distances. Could someone give me some pointers? I've tried something, but I'm not sure... Is this valid? Could someone look at $\color{red}{(1)}$ ...
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2answers
82 views

Universal Cover of a Surface with Boundary. What does Cantor set on Boundary Correspond to?

I am trying to understand in more detail the answer to: Universal Cover of a Surface (with Boundary) It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a ...
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1answer
220 views

The notion of a right-angled hexagon in hyperbolic geometry

I was hoping someone would help me understand better what a "right-angled hexagon" is in hyperbolic geometry. I know these are glued together somehow to produce hyperbolic pairs-of-pants. The only ...
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38 views

Completion of hyperbolic 3-manifold (Thurston's lecture note)

I need help with a part of Thurston's lecture note. p.55~56 in the 4th lecture note states that when obtaining hyperbolic manifold by gluing polyhedra having some ideal vertices, the set of ...
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1answer
233 views

Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases

There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic): $$ \frac{l(a)}{\sin\alpha}= \frac{l(b)}{\sin\beta}= ...
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1answer
65 views

Determine an hyperbolic midpoint

I am asked to determine the hyperbolic midpoint of the points $0,\frac{1}{2} \in \mathcal{P}$ Q: how do I determine the hyperbolic midpoint and what is actually meant by the midpoint?
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1answer
181 views

Curvature of De Sitter's space: where does the sign comes?

Consider $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, where: $${\rm d}s^2 = {\rm d}x^2 + {\rm d}y^2 - {\rm d}z^2.$$ We have both the hyperbolic space: $$\Bbb H^2(-1) = \{(x,y,z) \in \Bbb L^3 \mid ...
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Pasch's axiom in the hyperbolic plane

Q: I am asked to show that pasch's axiom holds true in the hyperbolic plane. Pasch's axiom Idea: I was thinking of about a circle $\Gamma$ whose centre is the origin of the complex plane. ...