# Tagged Questions

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

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### Fundamental group of a closed hyperbolic surface is Gromov hyperbolic

Does anyone have a reference for the proof of the result in the title? Thanks!
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### Fundamental solution of Poisson equation in the Hyperbolic Plane

If we consider the Poisson's equation $$-\Delta u=f(x), \ \ \mbox{in} \ \ \mathbb{R}^n,$$ we can construct the fundamental solution $$u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)f(y)dy,$$ where $\Gamma$ is ...
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### Fundamental domains of infinite-index subgroups of $SL(2,\mathbb{Z})$

While discussing modular forms associated to different subgroups $\Gamma$ of $SL(2,\mathbb{Z})$, there appeared to be a heuristic relationship between the index $[SL(2,\mathbb{Z}) \colon \Gamma]$ and ...
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### measuring the curvature of a pringle

Assuming for the sake of argument that a pringle shaped potato chip has a constant negative curvature, (which according to http://math.stackexchange.com/a/617610/88985 it is not ) see also picture ...
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### Trying to understand Lobatschewsky's parallax formula.

Lobatschewsky gave a method to calculate the curvature of space (see Bonola “Non-Euclidean Geometry” § 45) But I don't understand his method. Can somebody explain? I understand that the method now ...
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Beltrami made (out of thin paper or stiff cloth) a model of a surface of constant negative Gauss curvature. The original might have resembled a large saddle shaped Pringles chip, and frills might have ...
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### Definition of hyperbolic trig functions

I was doing some homework for my complex analysis class and ran into a personal question. I haven't worked a lot with hyperbolic trig functions (e.g. $\sinh (x)$, $\cosh(x)$, etc.) so this question ...
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### Exponential map in the Poincaré upper half plane

I have a question regarding the Poincaré upper half plane. Is there a simple way to express the exponential map? I have been looking unsuccessfully on internet for an expression... Thanks for any ...
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### How does one formally verify this picture from hyperbolic geometry?

Suppose we consider some hyperbolic circle with center $iz$ using the upper-half plane model of hyperbolic geometry, and in the interior we have a point $x+iy$. How does one prove that $iy$ is also ...
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### Spherical Geometry is to the Sphere just as Hyperbolic Geometry is to the…?

I need to write up a quickie description of Hyperbolic Geometry for non-mathematicians. I am trying to say "Hyperbolic Geoemtry is the Geometry of the surface of a ____" I remember that there is, in ...
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### 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 ...
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### “projective maassbestimmung” in Automorphic Functions by Fricke + Klein

I was reading a copy of Fricke and Klein's Theory of Automorphic Forms, and I came across the phrase projective maassbestimmung in the first chapter. Google translate returns: maßbestimmung as ...
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### Area of pseudospherical segment

Surface area of segment of a sphere radius $a$ at the equator, between two parallels, is given by $2 \pi a (z_2-z_1)$,where $z_2, z_1$ are heights of spherical segment at radii of parallel circles ...
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### “Hyperboloid like surface” as hyperbolic plane / pseudosphere

A pseudosphere is an surface wth a constant negative curvature. In most publications, it is almost given that the tracioid (rotated tractrix) is the surface that has a constant negative curvature, ...
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### Tractricoid as a pseudosphere (surface with constant negative curvature)

How to motivate/calculate/prove see that the tractricoid, i.e. a tractrix rotated about its asymptote, has a constant negative curvature? What are the hyperbolic lines on a tractricoid and how to see ...
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### Good data structure for hyperbolic tiling

Say you're doing something computational where each data point is a tile in a (not necessarily Euclidean) 2-dimensional tiling, for instance, a Life-like cellular automata. You might want a data ...
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### simplification of the area of a hyperbolic circle (BONOLA, S 53)

I'm trying to understand the S-53 of "Non-Euclidean Geometry" (BONOLA, R.) in which the formula for the area of a circle of radius r: $$2\pi k^2(\cosh\frac rk -1)$$ is somehow reduced by only applying ...
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### Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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### What's the connection between “hyperbolic” inner product spaces and the hyperbolic plane?

In Jacobson's Basic Algebra I, in Kaplansky's Linear algebra and geometry and in Artin's Geometric algebra, a hyperbolic plane is defined to be a two-dimensional, nondegenerate inner product space ...
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### Why are Euclidean and hyperbolic lengths proportional to first order?

In his book “Three-Dimensional Geometry and Topology”, Thurston constructs a Riemannian metric for the Poincare disk model and begins as follows. He says that, given any (hyperbolic) line segment $s$ ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 $z \to kz$ ...