Questions tagged [hyperbolic-geometry]
Questions on hyperbolic geometry, the geometry on manifolds with negative curvature. For questions on hyperbolas in planar geometry, use the tag conic-sections.
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Hyperbola Equation from Foci and Eccentricity
Determine equation in standard form of a hyperbola with foci at $(3,7)$ and $(3,-1)$ with eccentricity $e=2$.
Working:
The equation must be in form $\frac{(y-q)^2}{a^2}-\frac{(x-s)^2}{b^2}=1$, where $...
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Katok's Fuchsian Groups. Lemma 2.2.5 doubt
This is Lemma 2.2.5 from Katok's Fuchsian Groups:
Let $\Gamma$ be a subgroup of $\text{PSL}(2,\mathbb{R})$ acting properly discontinuously on $\mathcal{H}$, and $p \in \mathcal{H}$ be fixed by some ...
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Difficulty in understanding a proof from Katok's Fuchsian Groups.
I am currently reading Fuchsian Groups by Svetlana Katok. The following proposition appears on page 30:
There is a misprint. It is supposed to be $z_0$ not $w_0$.
The next paragraph shows that $\text{...
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What's the exact expression of Green's function on poincare disk?
Consider the Poincare disk $\mathbb{H}^2$, that is the unit disk endowed with the following metric
$$
g_{ij}=\frac{2}{1-|x|^2}\delta_{ij}.
$$
I believe there exists a global Green's function on $\...
- 463
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Distance between points on hyperboloid
I know that the hyperbolic distance between two points in the hyperboloid is
$$d(x, y) = \operatorname{arcosh}(-x \cdot y)$$
using the Minkowski product. The question is how to derive this? I am aware ...
- 11
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Parallel transport on hyperboloid
I have the formula for parallel transport along a geodesic in the hyperboloid model of hyperbolic space but how is it derived?
$$
P_{\mathbf{x} \rightarrow \mathbf{y}}(\mathbf{v})=\mathbf{v}-\frac{\...
- 11
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Geodesics in hyperboloid model of hyperbolic space
What is the proof of
$$\gamma_{\mathbf{x} \rightarrow \mathbf{u}}(t) = \cosh(t)\mathbf{x} + \sinh(t)\mathbf{u}$$
being the formula for a geodesic on the hyperboloid model of hyperbolic space? I haven'...
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Hyperboloid model exponential map [closed]
I am working with the hyperboloid model of hyperbolic space and I am trying to understand its exponential map. However I do not see how this map is derived so I would appreciate either an explanation ...
- 11
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Confusion in definition of properly discontinuous action
In page 27 of Katok's Fuchsian Groups, the following definitions are given.
A family $\{ M_{\alpha} \: \vert \: \alpha \in A \}$ of subsets of $X$ indexed by elements of a set $A$ is called locally ...
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Proving $\cosh\alpha = \frac{\cos a}{1-\cos a}$ for a regular hyperbolic triangle with all side-lengths $\alpha$ and all angles $a$
Consider a regular triangle in the hyperbolic plane $\mathbb{H}^2$ (i.e. a triangle with all sides length $\alpha$ and all angles $a$).
Prove that $\cosh\alpha = \frac{\cos a}{1-\cos a}$.
I think I ...
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Minkowski inner product [closed]
How to show that
“Tangents of the hyperbolic plane ($x^2+y^2-z^2=-1$) satisfy $\langle a,a\rangle_m\ge0$”?
(※ $a$:Tangent. $\langle,\rangle_m$: Minkowski inner product)
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Computationally representing a Fuchsian group
I'd like to learn the math behind this code golf answer, in which the symmetry group of the order-4 pentagonal tiling is represented by integer matrices.
My understanding so far is: In $PSL(2,\Bbb R)$ ...
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Is complex conjugation a Möbius transformation?
This question is probably a little confused. I'm teaching myself some geometry of the hyperbolic plane, and this statement from the Wikipedia page on the Poincaré half-plane model is tripping me up:
...
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Hyperbolic Path Length
$\textbf{Problem}:$ Find the hyperbolic length of the path $t \mapsto t+i$ for $t \in [0,x]$.
Of course I immediateley considered the length formula:
\begin{equation*}
L(\gamma) = \int_0^x ||\gamma'(t)...
- 31
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Why is the uniformizing hyperbolic metric complete?
Introduction: Given a Riemann surface whose universal cover is $\mathbb H$, we have the covering map $\pi:\mathbb H \rightarrow X$. There is a complete hyperbolic metric on $\mathbb H$ where $\...
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Is it true that Fundamental Domain of any congruence subgroup of $\text{SL}_2(\mathbb{Z})$ is a hyperbolic polygon? [closed]
Let $\Gamma$ be a congruence subgroup of $\text{SL}_2(\mathbb{Z})$. Then $\text{D}_{\Gamma}$ is said to be a fundamental domain of $\Gamma$ if for every $\text{z} \in \mathbb{H}$, there exist $\gamma \...
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What is the right measure for the upper half-plane?
I think my question depends on the context of the problem so let me give you some background. I want to measure subsets of $\{(x,y)|~y>0, x\in \mathbb{R}\}$. I'm not involved with differential ...
- 780
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Connections between Modular forms and Hyperbolic geometry
I am taking a course each on Modular forms and Hyperbolic geometry currently and I have begun to like the nice connections that exists between them. I am still a beginner in both these subjects and ...
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Help with definition of hyperbolic angle
I am currently reading about the Hyperbolic Space in the book Metric Spaces of Non-Positive Curvature of Martin R. Bridson and André Haefliger and they define the space $\mathbb{E}^{n,1}$ as $\mathbb{...
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Reference Request on Sullivan's paper
I am reading Sullivan's 1985 Non-wandering paper(for the paper, see https://www.math.stonybrook.edu/~bishop/classes/math627.S13/Sullivan-1985-Nonwandering.pdf). Section 3 in the paper says that a ...
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On the characterization of the hyperbolic metric on a circle domain
Recall a domain in the Riemann sphere is called a circle domain if every connected component of its boundary is either a circle or a point. Circle domain is interesting as Koebe asks whether any ...
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A question on Lorentzian matrices from “Foundations of Hyperbolic Manifolds” by John G. Ratcliffe
I’m stuck understanding the implication from questions 3 and 4 in Exercise 3.1 of the book Foundations of Hyperbolic Manifolds, hoping someone can help! I have provided some details at the bottom ...
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Do Schläfli symbols unambiguously represent gemetric shapes?
According to Wikipedia, the tesseract is the four-dimensional analogue of the cube and has the Schläfli symbol {4,3,3}, and Wikipedia features the following visualization:
However, when looking at it,...
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How can we find sizes in a uniform tiling of the hyperbolic plane?
Given a certain uniform tiling of the hyperbolic plane (for example, one given by its vertex configuration, if that specifies a tiling unambiguously, or a tiling specified by some other means, ...
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How does the Liouville Current of a constant curvature hyperbolic surface depend on the metric?
I am currently studying hyperbolic metrics on surfaces right now, and want to understand the construction of the Liouville Current for a metric $\varphi$ on my (orientable, compact) surface $S$. In ...
- 1
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Definition of convergence for the Gromov boundary
BirdsongHaeflinger "Metric spaces of non-positive curvature" defines the Topology of the compactification $\bar X=X\cup \partial X$ via generalized rays. A sequence $c_n$ of such rays ...
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Motivation for Hyperbolic Groups - Soft Question
I took a Geometric Group Theory course this semester. A very big part was hyperbolic groups. What I felt was a little bit lacking in this course was - why do I need hyperbolic groups? What is the ...
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Point-slope form for geodesics using the Poincaré disk model
The equation for graphing hyperbolic geodesics using the Poincaré disk model embedded in standard 2D space is $x^2+y^2+\frac{u_2(v_1^2+v_2^2+1)-v_2(u_1^2+u_2^2+1)}{u_1v_2-u_2v_1}x+\frac{v_1(u_1^2+u_2^...
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Is this a valid way to measure distance in Hyperbolic Geometry?
I've been looking for a way to measure distance in hyperbolic geometry, but I've come to a dead end-- I have to integrate this beast:$$\sqrt{\frac{\gamma^2}{(e_1e_4-e_2e_3)^2}\left((e_4+e_2\tau^{-2})^...
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Gradient on the hyperbolic plane
Consider the hyperbolic plane
\begin{align}
\mathbb{H}^2=\{(x,y)\in \mathbb{R}_+\times \mathbb{R}\}
\end{align}
with some metric $g$.
Is there something like the Voss-Weyl formula for the gradient? I ...
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Is it a true hyperbolic icosahedron?
Take a regular icosahedron. For each face, take the image of the circumcenter of the icosahedron by the reflection about the plane containing the face. Now subtract the sphere centered at this point ...
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Cartesian Hyperbolic Plane Geodesics
Let the ordered pair $(x,y)\in\mathbb{R}^2$ be the unique point in the hyperbolic plane arrived at by starting at [an arbitrary point called the origin] and going [east, an arbitrarily chosen ...
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Divergence operator on the hyperbolic plane $\mathbb{H}^2$
I want to figure out the difference between the divergence operator on $\mathbb{R}^2$ which I denote by $\text{div}_{\mathbb{R}^2}$ and the divergence operator $\text{div}_{\mathbb{H}^2}$ on the ...
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Geodesics in hyperbolic space $ \mathbb{H}^n$
I know that there are a lot of questions in this topic, but none of them has been useful to me. So that's mine;
I need to describe the geodesics of $\mathbb{H}^n$ in the two models of it:
$ \mathbb{H}...
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How does the hyperboloid model relate to "A Universal Model for Hyperbolic, Euclidean and Spherical Geometries"?
I just found A Universal Model for Hyperbolic, Euclidean and Spherical Geometries, after reading the HyperRogue game dev notes where it said the hyperboloid model (aka the Minkowski model) was the ...
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What is special about the "pentagrid" and "heptagrid" in Margenstern's work on Cellular Automata in Hyperbolic Spaces?
In his work he mainly focuses on the pentagrid {5,4} and heptagrid {7,3}:
In what ways are these tilings special? How do they compare to hyperbolic tilings in general? I am wondering what insights ...
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Do we have a standard coordinate system for hyperbolic tessellations?
I am specifically interested in implementing animations for Cellular Automata in the Hyperbolic Plane. I have seen Coordinate systems for the hyperbolic plane on Wikipedia, but a lot of what Professor ...
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A brief explanation of Isometries and Mobius Transformations used in animating Hyperbolic Cellular automata?
I've spent the past while digging into the code I asked about in Where to begin with animating over a 2D hyperbolic tessellation?, my answer there is in regards to digging into the MagicTile project's ...
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Examples of hyperbolic and non-hyperbolic space for quasi-isometric spaces
Let $X$ and $Y$ are quasi-isometric spaces. I try to find an example for which one of these spaces will be hyperbolic, other is not hyperbolic.
I know that for geodesic metric space if one of the ...
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In a Hypergeometric Distribution CDF with everything else held constant, should K be a linear function of N?
Using the Hypergeometric Distributon notation from Wikipedia, if I treat $k$, $n$, and $\Pr(X\ >\ k)$ as constants and solve for $K$ as a function of $N$ in Mathematica, the relation appears to be ...
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How does curvature relate to angle measurement in hyperbolic geometry?
This question is about the relationship between curvature and angle measurement in hyperbolic geometry... Specifically, I am trying to understand the following excerpt from pp. 489-490 of Greenberg's ...
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Surface Group Representations
I am interested in Hyperbolic Geometry. I studied hyperbolic surfaces, the space of all marked hyperbolic structures on a surface (also known as the Teichmuller space of the surface), and the ...
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Metric Derived From Differential on Hyperbolic Plane
I'm reading Katok's Fuchsian Groups, and I'm confused on how the metric on the unit disk model is derived from the differential $$ds = \frac{2|dz|}{1-|z|^2}.$$
To be more specific, we first have the ...
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Volume of hyperbolic submanifold of surface with a boundary component
Let $\Sigma$ be a compact surface of genus $k \geq 2$ having a single boundary component. Let $U \subset \text{Int}(\Sigma)$ be an open subset of the interior of $\Sigma$ with a Riemannian metric $g$ ...
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Volumes of hyperbolic submanifolds of closed surfaces
Let $\Sigma$ be a closed orientable surface of genus $k \geq 2$. Suppose $U \subseteq \Sigma $ is an open subset with a Riemannian metric $g$ on $U$ such that (1) the Gaussian curvature $K$ of $g$ is ...
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Is every loop on a hyperbolic surface freely homotopic to a geodesic?
Let $(S,g)$ be an orientable Riemannian 2-manifold having constant Gaussian curvature $K=-1$ and $\gamma$ a loop on $S$. Is $\gamma$ freely homotopic to a geodesic? Note the lack of completeness ...
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Geodesic curvature on hyperbolic manifold with boundary
Let $\Sigma$ be a compact oriented surface of genus $1$ having a single boundary component (i.e. $T^2$ minus an open disk) and let $g$ be a Riemannian metric on $\Sigma$ with constant Gaussian ...
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Within hyperbolic space, are all sides of an ideal triangle parallel?; and is it possible for them all to be hyperparallel?
Question:
I unfortunately have an extremely limited foundation in mathematics but I am trying to wrap my head around hyperbolic geometry in simple terms and I have spent all day trying to search for ...
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Algorithms for drawing hyperbolic tilings
I was looking at hyperbolic tilings on the Poincare disc model like these the other day, and I wondered how I might make my own. I have a basic understanding of what hyperbolic space is and how it ...
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Is $\text{PSL}(2,\mathbb{R})$ a semisimple Lie group?
I am learning Hyperbolic Geometry. Few days ago I attended a seminar on hyperbolic geometry.
The professor introduced us with Teichmüller space and later he presented Teichmüller space is as a ...
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