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

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4
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1answer
541 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 ...
89
votes
4answers
2k views

Hyperbolic critters studying Euclidean geometry

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...
0
votes
1answer
223 views

Hyperbolic geometry

Post Number: 45 Posted on Friday, 22 March, 2013 - 04:48 pm: I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this? Show that there ...
1
vote
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?
13
votes
2answers
275 views

How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?

This is in response to comments and the answer by user studiosus to this question: As for Beltrami's work: Consistency of a geometry from (post) Hilbert viewpoint has nothing to do with ...
21
votes
9answers
3k views

What are the interesting applications of hyperbolic geometry?

I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...
3
votes
1answer
287 views

Riemann surface with punctures corresponds to a hyperbolic surface with cusps

I am reading a paper on Riemann surfaces and the author used the fact that $\{$Riemann surfaces with genus $g$ and $n$ punctures$\}$ is in one-to-one correspondence with $\{$ hyperbolic surfaces ...
5
votes
3answers
847 views

Expression of the Hyperbolic Distance in the Upper Half Plane

While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia. ...
4
votes
1answer
1k views

Möbius Transforms that preserve the unit disk

Say I wish to prove that every möbius transformation of the unit disk onto itself can be written in the form $A(z) = e^{i\theta}\frac{z+a}{1+\bar{a}z}$, where $\theta$ is a real number and $a$ is a ...
3
votes
2answers
550 views

Does anyone know a good hyperbolic geometry software program?

We are currently using this program called NonEuclid but it is a little frustrating to use sometimes and I was wondering if anyone knows another program for hyperbolic geometry.
5
votes
0answers
120 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
5
votes
2answers
348 views

Symbolic coordinates for a hyperbolic grid?

Rephrasing     (one year later)    (original question is below) Apparently the original question wasn't clear, or nobody knows an answer (or both). So I will try to rephrase it. Look at your ...
4
votes
2answers
96 views

An axiomatic treatment of hyperbolic trigonometry?

I would like to see results derived in hyperbolic trigonometry synthetically, i.e. just by working from axioms, for example the ones given by Hilbert (or even Tarski). Most authors seem to discuss ...
4
votes
6answers
226 views

Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?

I can faintly visualize some way of deducing this formula with exponential functions but forgot it. How do you remember it? Suppose you just forget whether it is plus-or-minus there, how do you find ...
4
votes
2answers
409 views

The law of sines in hyperbolic geometry

What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the ...
2
votes
2answers
130 views

Is there a surface in Euclidean space that admits elliptic geometry?

As I understand, on a pseudosphere, a surface of constant negative curvature, we can realize a part of the hyperbolic plane (but not the entire plane due to Hilbert's 1901 theorem) and use this for ...
2
votes
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 ...
2
votes
2answers
155 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 ...
1
vote
1answer
262 views

Hyperbolic area and $SL_2$

Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that: a. The measure $\mu$ is invariant under all $g \in ...
4
votes
2answers
166 views

List of connected Lie subgroups of $\mathrm{SL}(2,\mathbb{C})$.

I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation. One can find 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?
4
votes
1answer
303 views

Models of hyperbolic geometry

Wikipedia states the following: [The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model ...
3
votes
0answers
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 ...
2
votes
1answer
113 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 ...
2
votes
0answers
50 views

limit set of Kleinian grouos with closed manifolds as quotient

I'm trying to convince myself that if $M\cong\mathbb{H}^3/G$ is a closed hyperbolic 3-manifold then the limit set $\Lambda(G)$ equals the whole Riemann sphere $S_\infty^2$. My idea of the proof goes ...
2
votes
3answers
281 views

Gromov boundary — TFAE

I am a newcomer to hyperbolic geometry and was trying to understand some of it in the context of dynamics, for reading certain literature. Let a discrete subgroup $G$ of $SL_2(\mathbb R)$ act on the ...
1
vote
1answer
31 views

Distance from point to line segment in Poincaré disk model

I'm trying to build a geometric datastructure in hyperbolic space. For that purpose, I'm using the Poincaré disk model. The distance between two points can be calculated with the hyperbolic law of ...
1
vote
1answer
120 views

hyperbolic geometry (and circle ) construction problem

Was thinking about hyperbolic geometry, the Poincare Disk Model and Sweikarts constant and combined them all in a construction puzzle that I was unable to solve. My construction puzzle: Given: A ...
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vote
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 ...
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 ...
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1answer
105 views

How does my Beltrami-Klein model look?

Did I sketch the picture right based off of the specific instructions given in the problem?
1
vote
1answer
169 views

A book to study about hyperbolic plane, hyperbolic translations, etc.

In this paper, page $6$, the authors state the following: The translations of the hyperbolic plane are defined as products of two central symmetries; the set of hyperbolic translations forms a ...
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votes
1answer
39 views

which surfaces have (for a large area) a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
0
votes
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 ...
-1
votes
1answer
140 views

Length of a curve on $S^2$

$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$? $2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates? ...