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

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1answer
28 views

How to build a hexagon according to Poincaré model?

Given a side, I know how to build a hexagon in the euclidean geometry. How can i build it in the hyperbolic geometry according to the Poincaré model? By translating every step using hyperbolic circle ...
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1answer
35 views

Conics and Loci Question (Hyperbolae and Circles)

A circle has the equation $x^2 + y^2 = r^2$. Tangents are drawn from a point $P(x_1,y_1)$ to the circle and these touch the circle at points $A$ and $B$. If the position of $P$ can vary and the locus ...
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1answer
56 views

Parallelism preservation of hyperbolic rigid motions on the half plane model

I need to proof (Under Hilbert axiomatization) that hyperbolic rigid motions, with respect to the metric $ d:\mathbb{H}^2 \times\mathbb{H}^2\rightarrow\mathbb{R}: d(A,B) =\left| \log \left( ...
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1answer
19 views

Parametrizaction of a Hyperboloid

I do not understand why when you revolve a hyperbola around a circle the respective parameters (cosh (v) and cos (u)) are multiplied by each other to get the parametric form of the hyperboloid. I ...
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1answer
42 views

Hyperbolic area

Define Hyperbolic area of a subset $E$ of the unit disk $D$ to be $\displaystyle 4\int \int_E \frac{dx dy}{(1-|z|^2)^2}$. Show that the hyperbolic area is invariant under conformal self maps of ...
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1answer
72 views

Rulings of One Sheet Hyperboloid

Let $M$ be a hyperboloid of one sheet satisfying $x^2+y^2-z^2=1$. Show that $x(u,v)=(\frac{uv+1}{uv-1},\frac{u-v}{uv-1},\frac{u+v}{uv-1})$ gives a parametrization of $M$ where both sets of parameter ...
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1answer
37 views

Rotate a point around another point in the Poincaré Hyperbolic Disk

Suppose I have a point $P = (x_1,y_1)$ in the Poincaré disk model. How do I rotate it about another point $Q = (x_2,y_2) \neq(0,0)$ by a Euclidean angle $\alpha$? If $Q = (0,0)$ this is simple, just ...
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0answers
41 views

Harmonic maps between Riemann surfaces

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the ...
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2answers
78 views

Geodesics in Poincare Disk

I would like to find the geodesics in the Poincare disk. I know that the metric is $$\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$$ so $$s=\int \frac{\sqrt{1+y'^2}}{1-x^2-y^2}\, dx$$ Then I try to find y(x) using ...
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32 views

uniquess of hyperbolic numbers

I'm trying to prove, the uniqueness of hyperbolic numbers, like the complex numbers are unique, but since hyperbolic numbers aren't a field, I can't use the ideas of this. Are there theorems of ...
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107 views

How to parameterize these pretty hyperbolic (Amsler) surfaces?

I've seen the attached images describing surfaces of negative curvature. I was wondering if there exist such surfaces with constant Gaussian negative curvature. To this end, I attempted to model the ...
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0answers
107 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers ...
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1answer
51 views

This map is an isometry (in the Riemannian sense) of the hyperbolic plane. Why is the following a proof of it?

I'm making my way through a textbook on elementary undergraduate geometry. The author has defined the notion of an isometry between two subsets of $\mathbb{R}^2$ equipped with a Riemannian metric. It ...
4
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1answer
26 views

Examples of Lattices in $\operatorname{Isom}(H^n)$ for all $n \geq 2$?

I just had an exam today where I was asked to give an example of a lattice in $\operatorname{Isom}(H^n)$ for all $n \geq 2$, and with bonus points if I could give cocompact and noncocompact examples. ...
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1answer
58 views

Triangle inequality for hyperbolic metric of logarithm of cross ratio.

Consider the Poincaré Half-Plane model of the Hyperbolic Space $ \mathbb{H}^2 $. I need to proof that the following d function is a metric.$ d:\mathbb{H}^2\rightarrow\mathbb{R}, d(A,B) = \big{|} log ...
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1answer
38 views

Length of the arc of a hypercycle

I am still puzzeling to get a nice equation for the arclength of an hypercycle. (I asked a similar question (less developed) about a year ago that was never answered, now i am a bit further, i ...
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2answers
56 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola [closed]

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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48 views

How do I solve for the volume of a hyperboloid using a double integral in polar coordinates?

Here is the problem text, with my attempts at solving it at the bottom: Suppose you are part of a team designing a water tank in the shape of a hyperboloid. The tank is to have a top radius a of 2 ...
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0answers
79 views

problem with proof 'horoball QI extension theorem'

I'm reading the book of Drutu and Kapovich "Lectures on geometric group theory". In the proof of Mostow rigidity theorem, they say that they can extend an $\rho$-equivariant function ...
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0answers
40 views

Fundamental domain of a Fuchsian group is non-compact if the group contains parabolic element

Suppose a discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ acts on $\mathbb{H}^2$. Why is the fundamental domain non-compact if $\Gamma$ contains a parabolic element? Thanks in advance.
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2answers
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Need to change variables in equations with cosh.

i have these five functions: $x=\tau \cosh(s)$ $q=\tau \sinh(s)$ $y= \sinh(s)$ $p= \cosh(s)$ $u= 1/2*\tau*\cosh(2s)+1/2*\tau$ I need to write $u$ in terms of $x$ and $y$ I know the answer is ...
2
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1answer
32 views

Non-equivalent metrics on $PSL_2(\mathbb{R})$

I am reading a paper on continued fractions and it uses the following result on Lie Groups: Fix an arbitrary left-invariant metric $d$ on $PSL_2(\mathbb{R})$ ... This phrase really throws me ...
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1answer
56 views

How to analytically derive the geometric properties of a hyperbola and programatically use them to graph?

I have the following equation: $$ \left(10^{\left\lfloor\frac{\ln\sqrt k}{\ln10}\right\rfloor-1}+x\right) \left(10^{\left\lfloor\frac{\ln\sqrt k}{\ln10}\right\rfloor-1}+y\right) = k $$ Entering the ...
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1answer
16 views

Finding the second point of intersection from a normal on a hyperbola

The question is phrased as follows: "A rectangular hyperbola, W, has equation xy = 12" a) Show that the gradient of the normal, N, to W at the point P(2,6), is 1/3. b) Hence find an equation for ...
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0answers
26 views

congruency of triangles in hyperbolic and spherical geometry

In Euclidean geometry, we have the following congruencies of triangles: side-side-side, side-angle-side, angle-angle-side = angle-side-angle (because of the angle sum) and side-side-angle (only if the ...
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1answer
58 views

Equalities and inequalities for quadrilaterals in hyperbolic space

In euclidean space any quadrilateral satisfies equalities and inequalities $$a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2$$ $$a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2$$ where $a,b,c,d$ are the side lenghts, ...
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0answers
31 views

Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
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0answers
49 views

Hartshorne Exercise 41.2 - Altitudes of a triangle have a common perpendicular [hyperbolic]

This is my first post here ever, so don't be too rude, if i missed something. My question refers to exercise 41.2 "GEOMETRY:EUCLID AND BEYOND" from Robin Hartshorne. You can easily find the book as ...
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1answer
40 views

Discrete group of isometries of a finitely compact metric space is countable.

This question comes from Ratcliffe's Foundations of Hyperbolic Manifolds. Let $X$ be a finitely compact metric space (i.e. all closed metric balls are compact). Prove that a discrete group ...
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0answers
58 views

Kobayashi distance on the Siegel upper half space

Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article "inavariant metrics on ...
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0answers
42 views

Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian ...
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1answer
56 views

Relationship between regular tessellations in general space

Using the notation for regular tessellations $\{p,q\}$ denoting the tessellation consisting of p-gons , q of which meet at each vertex [e.g. $\{3,6\}$ in $\mathbb{R}^{2}$ is the equilateral triangle ...
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1answer
19 views

Incidence axioms for the upper half plane (complex plane).

The axiom I am checking for this question is I1: "For any two distinct points A,B there exist a unique line L containing both points." Show that I1 is satisfied in the upper half of the complex plane ...
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0answers
37 views

another pythagorean theorem in hyperbolic geometry

on https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry it says However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition ...
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2answers
67 views

What is the radius of the inscribed circle of an ideal triangle

I wanted to calculate the radius of the inscribed circle of an ideal triangle. and when i dat calculate it i came to $\ln( \sqrt {3}) \approx 0.54 $ (being arcos(sec (30^o)) but then at ...
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1answer
36 views

Where are the vertices of the universal cover of the genus 2 torus octagon?

The universal cover of the genus 2 torus is hyperbolic plane and the fundamental domain is a octagon. Here is a picture, which I took from here. Is there a closed form for the points of set of the ...
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1answer
49 views

In hyperbolic geometry, prove that parallel lines are not equidistant

In Euclidean Geometry, parallel lines are equidistant. In hyperbolic geometry, it appears that parallel lines are $not$ equidistant. Is there a proof that supports this, or is it supposed to be ...
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1answer
35 views

Hyperbolic geometry question concerning lengths between parallel lines

Theorem (H16). If: $l$ and $m$ are parallel lines, $j$ is a common perpendicular intersecting $l$ at point A and $m$ at point B, and C and E are points on $l$ so that C is between A and E, Then: ...
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1answer
99 views

Lie groups for beginners: Lie group of hyperbolic geometry

I am trying to understand Lie groups and their relation to (2 dimensional) hyperbolic geometry. as far as I understand it (which is not very far, I am pushing my understanding here) the Lie-group ...
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1answer
54 views

Conformal Map from Intersection of Two Discs and Half-Plane

I have one of those "find the map" problems that is really giving me a lot of trouble. Let $B_1(1)$ be the ball of radius $1$ centered at $1$. We have the following domain: $\mathbb{D} \cap B_1(1) ...
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2answers
50 views

How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
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1answer
26 views

Isoperimetric inequality in the Poincare disc

In the Poincare disc model, a version of the isoperimetric inequality states that $L(\partial(A))>c\mu(A),$ where $\mu$ is the hyperbolic area and $L(\gamma)=\int_0^1 ...
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0answers
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spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...
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1answer
54 views

Volumes of hyperbolic manifolds

In a talk I attended the speaker said that the volume of a closed hyperbolic manifold is a topological invariant. Are known volumes rational? Irrational? Transcendental?
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45 views

Hyperbolic Geometry - Parabolic Matrix?

In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ ...
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0answers
35 views

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of ...
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51 views

Prove $A \cdot B\leq -1$, where $A$ and $B$ are in $\mathbb{H}^2$

Let $A$ and $B$ be in $\mathbb{H}^2$. I need to prove that the lorentzian dot product between $A$ and $B$ is less than or equal to $-1$. I have no idea where to start.
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1answer
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Non euclidean lines (finding endpoints of semicircle)

A non euclidean line in $\mathbb{RP}^1$ in terms of reflections about the unit circle can be written in the form $A+B(\overline{w}+w)+C(\overline{w}w)=0$ Where $w=\frac{1}{\overline{z}}$ The ...
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24 views

sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
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0answers
46 views

Misinterpretations of Hilbert's Theorem?

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ ...