# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 $-k$ ...