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

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Constructing shapes in hyperbolic space

I'm trying to get started writing a game that uses the order-4 dodecahedral honeycomb in hyperbolic space. I'm representing points as 4-vectors of the form ...
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24 views

What is the hyperbolic plane equivalent to translation in euclidean space

in euclidean plane one can move polygons like rectangles, triangles etc. around by isometries, e.g. translations. For instance if we consider a rectangle with midpoint $0\in\mathbb{R}²$ then the image ...
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Four-line-locus problem [on hold]

I have a question that i need help figuring out. It is "Show that the symptom of a hyperbola solves a four-line-locus problem". This is related to Apollonius work on the conic sections. If someone ...
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1answer
23 views

Question about hyperbolic space-forms

I have a map between hyperbolic space-forms $\varphi:B^n/\Gamma \longrightarrow B^n/H$ (where $\Gamma, H$ are discrete groups of isometries that act freely), and a lift to a map $\tilde{\varphi}:B^n ...
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1answer
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Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
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39 views

Invariance of measure on upper half plane

The upper half plane has the measure $|y|^{-2}dxdy$. Show that it is invariant under the action of $SL(2, \mathbb{R})$. I don't understand what any of this means. First, I don't understand what they ...
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here is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$.

There is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$. My question is, isn't $z \rightarrow kz$ an isometry for ...
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Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that ...
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33 views

Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
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2answers
70 views

Simple proof of existence of hyperbolic triangles

I've studied the hyperbolic plane by analytically building up the hyperboloid model, the Klein—Beltrami disc, the Poincaré disc, and the half-plane model from scratch. Now I'd like to prove that, ...
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1answer
198 views

Distance in hyperbolic geometry

In Euclidean geometry, we have that the distance between two points $p$ and $q$ in $\Re^n$ is $\sqrt{(p_1^2-q_1^2) + (p_2^2-q_2^2) + \ldots + (p_n^2-q_n^2) }$ (if we denote the points by $p = (p_1, ...
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1answer
42 views

Geometries (Euclidean and Projective)

We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it's true in Coordinate Geometry. It makes ...
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1answer
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Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
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1answer
33 views

hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

On http://en.wikipedia.org/wiki/Hypercycle_%28geometry%29 I found the statement. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their ...
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2answers
47 views

How to construct a circle in a the Poincare Disk model

How can I construct an circle with centre C going trough point P in a Poincare disk?. I found an script of how to do it in the "Poincaré Disk Model of Hyperbolic Geometry"toolkit from the geometers ...
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1answer
32 views

Distance-preserving coordinate transformations for the poincaré disc

Following this question, I'm looking for a coordinate transformation which leaves distances unchanged. Does such a transformation exist? The isometries for the poincaré disk looked promising, but only ...
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1answer
51 views

Representation of cell location in hyperbolic plane

I want to represent an order-5 square tiling (image from Wikipedia; more text below image): Obviously for a simple grid I can uniquely refer to a given square by its ...
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1answer
14 views

calculate the curvature of a surface with a Lambert quadrilateral

I was wondering how can I calculate the curvature of a surface? For example: Given a Lambert quadrilateral ABCD (see http://en.wikipedia.org/wiki/Lambert_quadrilateral ) with: $ DA \bot AB $, $ AB ...
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Hyperbolic geometry and orientation reversing isometries.

In Quasi-cluster algebras from non-orientable surfaces by Dupont and Palesi, one can read the following on page 11: I don't understand why the 'following relations' in the image included hold. ...
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1answer
37 views

Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
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Hyperbolic metric of arbitrary curvature.

I've been trying to find this online, in books, etc, but I can never find the expression for the metric on the unit disk $$\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$$ that has constant ...
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1answer
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The distance in Lobachevski (Hyperbolic) space

I need to find the distance from the point provided in the hyperboloid model with a vector $x$ where $\langle x,x\rangle=-1$ to the hyperplane $H_e$ with a normal vector $e$, where $\langle ...
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1answer
27 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 ...
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1answer
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Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
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What is a manifold with cusp?

What I am primarily currently learning about is hyperbolic geometry and methods to find hyperbolic structures on triangulated manifolds. I see phrases such as 'cusp ends' and 'manifold with one cusp' ...
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Submanifold of a Kobayashi hyperbolic manifold

Let $M$ be complex manifold which is Kobayashi hyperbolic. Let $N$ be a submanifold of $M$ obtained as the zeroes of an analytic submersion $f : M \rightarrow R$, $R$ complex manifold. Question : ...
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1answer
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Fundamental domain of a Fuchsian group which is not locally finite

I am trying to understand Example 9.2.5 in Beardon's book The Geometry of Discrete Groups. The goal is to construct a fundamental domain of a Fuchsian group which is not locally finite. Definitions ...
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1answer
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Hyperbolic (and related) structures on open unit disk

I am facing some confusion about different structures on the open unit disk $D:=\{ z \in \mathbb{C}, |z|<1 \}$. By Riemann Mapping Theorem we know there is just one complex structure on $D$, up to ...
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1answer
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In the hyperbolic geometry, is there a range of $\pi$?

In Euclidean space, $\pi$ is the constant value $3.14159\dots$ But I tried to measure the value of $\pi$ and found that $\pi$ is not constant! So I wonder if there is a range of $\pi$. If so, is ...
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1answer
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What are Straight lines in the Gans Disk model of the Euclidean plane?

The answer of Blue ( http://math.stackexchange.com/a/1464/88985 ) to Hyperbolic critters studying Euclidean geometry made me interested in the Gans Disk model of the euclidean plane. Blue writes: ...
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1answer
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Limit sets of representations of once-punctured torus groups and circle packings

Let $\rho\colon\pi_1(T_1)\to PSL(2,\mathbb{C})$ be a faithful representation of the fundamental group of a once-punctured torus. If both the components of the convex core in the quotient manifold are ...
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1answer
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Hyperbolic Triangles and Uniform thinness

My textbook states that all triangles in hyperbolic space are uniformly thin in the following way: If $ABC$ is a triangle and $x$ is a point on one side, then there exists a point $y$ on one of the ...
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1answer
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Show how to map the semicircle $x^2 +y^2 = 1$, $y > 0$, onto $(x−1)^2+y^2 = 4$, $y > 0$, by a combination of $z \to z+l$ and $z \to kz$.

I need some help with this one! One can begin to understand the geometric significance of linear fractional transformations of the half plane by studying the simplest ones, $z \to z+l$ and $z \to kz$ ...
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the fundamental group acts on half upper plan

Let $S$ be a compact oriental surface without boundary of genus $g\ge 2$, then its universal covering is $\mathbb{H}^2$, I am confused with 2 facts following: (1) $\rho:\pi_1(S)\hookrightarrow ...
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1answer
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Are there other models for 2 dimensional hyperbolic geometry?

I was a bit browsing the internet for models for (2-dimensional) hyperbolic geometry. and realised that besides the well known Poincare half plane model Poincare disk model Beltrami-Klein disk ...
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Showing geodesics in $\mathbb{H}^2$ have unique common perpendicular or common endpoint

If we let $\mathbb{H}^2$ be the hyperbolic plane and we let $\gamma_1,\gamma_2$ be geodesics which do not intersect. I have a question which asks me to show that either $\gamma_1$ and $\gamma_2$ have ...
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48 views

Showing that triangles in $\mathbb{Z}$ are thin

If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this ...
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1answer
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Hyperbolic metric spaces

I'm trying to prove a simple proposition wich is in Burago's "A Course in Metric Spaces" (Exercise $8.4.5$, p.$287$). Before exposing my problem, let me give some definitions. A metric space $(X,d)$ ...
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2answers
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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 ...
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1answer
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oriented surface of genus g and m punctures

Let $S_{g,m}$ be an oriented surface of genus g and m punctures, what's the condition to ensure $S_{g,m}$ is hyperbolic? If $g\ge 2$, I know it is hyperbolic, how about g=0 and g=1? Thanks in advance. ...
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1answer
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Definition of complex hyperbolic geometry

I am trying to read about complex hyperbolic geometry.But I couldnot find a basic definition for it. Is it just the special case of hyperbolic geometry where we work with complex numbers in the model. ...
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1answer
251 views

Is there a hyperbolic geometry equivalent to Möbius transformations in spherical geometry?

There is a sense in which all "interesting" properties of functions in spherical geometry are invariant under conjugation by a Möbius transformation. The reason is that the Möbius transformations ...
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1answer
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Measuring distance on the Poincare disk

I've seen several different ways to measure distance on the Poincare disk i.e Riemann metric/manifold (which I don't understand). However the method we're taught is using $\tanh^{-1}$ and complex ...
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Parallel postulate in hyperbolic geometry question [closed]

Let $L =\{y=0\}\cap\mathcal{H}^{2}$, and let $P=(3,2,2)$. Show that the parallel postulate fails in $\mathcal{H}^{2}$ by giving two lines $L',L'' \in \mathcal{H}^{2}$ with $P\in L',L''$ and $L\cap ...
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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 ...
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148 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
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1answer
66 views

hyperbolic geometry proof with parallel lines

We are assuming hyperbolic geometry in this proof. Prove that for every line $l$ and external point P (im assuming point $P$ is not on line $l$), there are an infinite number of distinct lines ...
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Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
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1answer
39 views

Showing reflections are hyperbolic isometries in $\mathbb{D}$.

I am interested in showing that isometries in $\mathbb{D}$ are either conformal self-maps in $\mathbb{D}$ or they are compositions of conformal self-maps with $z\mapsto \bar{z}$. It is given that ...
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Structure of $x^2 + xy + y^2 = z^2$ integer quadratic form

The pythagorean triples $x^2 + y^2 = z^2$ can be solved in integers using rational parameterization of solutions to $x^2 + y^2 = 1$. It goes through $(1,0)$, then consider the line $y = -k (x - 1)$ ...