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

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Advantage of using Hyperbolic Trigonometric functions?

Is there any added advantage of using Hyperbolic Trigonometric functions? Since you can always use normal trigonometric functions in all cases: $$\left.\begin{array}{ccc} ...
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1answer
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Hyperbolic Geometry: Question about the Transitivity of Möbius transformations

I was confronted with this exercise in the book Hyperbolic Geometry by Anderson which states: In each case, find $m \in Möb(\mathbb{H})$ such that the property holds, or prove that no such $m$ ...
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1answer
33 views

Is there a name for this special, “most parallel” ultraparallel line in hyperbolic geometry?

Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P. However, there's one line M which is ...
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1answer
33 views

Commutator of hyperbolic isometries

Let $f,g \in PSL(2,R)$ be isometries of $\mathbb{H}^2$ of hyperbolic type. Let $h=[f,g]$ be their commutator. Is there an explicit geometric criterion to determine if $h$ is hyperbolic, parabolic or ...
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1answer
13 views

Relation between length of arc of horocycle and length of chord?

In Hyperbolic geometry: What is the relation between the length of the arc of a horocycle between two points and the length of the chord (segment) between the two points? Also what is the relation ...
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3answers
48 views

Complex hyperbolic Trigonometry

When faced with the equation $\cos{z}=\sqrt{2}$ I want to solve for z so I break it up into a sum $z=x+iy$ and get: $\cos{z}=\cos{x}\cosh{y}-i \sin{x} \sinh{y}$ equating real and imaginary parts I ...
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1answer
29 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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2answers
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Poincare disk and Poincare half plane

My book claims that the Möbius transforms are isometries of the Poincare half plane model. Thus, the metric is preserved under these maps. But I know that the Poincare disk can be derived from ...
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1answer
39 views

Triangle Identity leads to another Euclidean parallel.

Referring to TriangleIdentity by 伍柒貳 a while ago, considering $\bigtriangleup$ ABC, it is proved that: $$\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C (1*) $$ I want to take angle $A = ...
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1answer
42 views

Constructing a quadrilateral in hyperbolic plane

Suppose we have a set of angles, $\{\alpha_1, \alpha_2, \alpha_3, \alpha_4 \}$, and we want to construct some quadrilateral at Poincare Disc Model with angle $\alpha_i$ at vertex $i$. The question ...
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1answer
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Bi-asymptotic geodesics in Visibility manifolds

I'm thinking about some properties of geodesics in visibility spaces. Here I give some definitions: A Riemannian manifold $(M,g)$ with riemannian distance $d$ is said to be a visibility manifold if ...
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constant-curvature Riemannian metric for Bring's surface

There is a well-known and very symmetric space that is called either "Bring's curve" or "Bring's surface", depending upon the context. (Bring was a Swedish mathematician in the 18th century.) Let's ...
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We can get the geodesic only by the first geodesic equation in hyperbolic plane?

The two geodesic equations are: $$\frac d{ds} (Eu' + Fv') = 1/2(E_u {u'}^2 + 2F_u u'v' + G_u {v'}^2) $$ $$\frac d{ds} (Fu' + Gv') = 1/2(Ev{u'}^2 + 2F_v u'v' + G_v{v'}^2) $$ And in the hyperbolic ...
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0answers
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Comparing two fundamental domains for $\Gamma(2)$

(0). My question concerns the relation between two different fundamental domains for the group $$ \Gamma(2)= \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix}\in SL_2(\mathbb Z) \; ...
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1answer
21 views

Homothetic transformation in the Poincaré upper half plane

i am interested in finding homothetic transformation in the Poincaré upper half plane. I heard that unlike $\mathbb{R}^n$ we don't have an homothetic transform for every $\lambda \in \mathbb{R}^+$. ...
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Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
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2answers
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hyperbolic tangent vs tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (in a right triangle). Is there a similar definition for the hyperbolic tangent? The reason ...
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Mapping a line in the hyperboloid model of $\mathbb{H}^2$ to a circle in the Poincaré model

In the hyperboloid model of $\mathbb{H}^2$ a point $P(x,y,z)$ is the intersection of the vector $(x,y,z)$ with the upper sheet of the hyperboloid $x^2 + y^2 - z^2 = -1$, and a line $L(a,b,c)$ is the ...
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1answer
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“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
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1answer
57 views

Models of the hyperbolic plane

Hilbert's theorem tells us that there is no immersion in $\mathbb{R}^3$ with negative Gauß curvature that is complete. Despite, there are some models of surfaces with negative Gauß-curvature like the ...
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1answer
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Proving triangle inequality for hyperbolic distance using contours

I need to prove the triangle inequality for hyperbolic distances. Could someone give me some pointers? I've tried something, but I'm not sure... Is this valid? Could someone look at $\color{red}{(1)}$ ...
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2answers
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Universal Cover of a Surface with Boundary. What does Cantor set on Boundary Correspond to?

I am trying to understand in more detail the answer to: Universal Cover of a Surface (with Boundary) It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a ...
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1answer
108 views

The notion of a right-angled hexagon in hyperbolic geometry

I was hoping someone would help me understand better what a "right-angled hexagon" is in hyperbolic geometry. I know these are glued together somehow to produce hyperbolic pairs-of-pants. The only ...
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Completion of hyperbolic 3-manifold (Thurston's lecture note)

I need help with a part of Thurston's lecture note. p.55~56 in the 4th lecture note states that when obtaining hyperbolic manifold by gluing polyhedra having some ideal vertices, the set of ...
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1answer
173 views

Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases

There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic): $$ \frac{l(a)}{\sin\alpha}= \frac{l(b)}{\sin\beta}= ...
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1answer
17 views

Determine an hyperbolic midpoint

I am asked to determine the hyperbolic midpoint of the points $0,\frac{1}{2} \in \mathcal{P}$ Q: how do I determine the hyperbolic midpoint and what is actually meant by the midpoint?
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1answer
154 views

Curvature of De Sitter's space: where does the sign comes?

Consider $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, where: $${\rm d}s^2 = {\rm d}x^2 + {\rm d}y^2 - {\rm d}z^2.$$ We have both the hyperbolic space: $$\Bbb H^2(-1) = \{(x,y,z) \in \Bbb L^3 \mid ...
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Pasch's axiom in the hyperbolic plane

Q: I am asked to show that pasch's axiom holds true in the hyperbolic plane. Pasch's axiom Idea: I was thinking of about a circle $\Gamma$ whose centre is the origin of the complex plane. ...
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0answers
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On: geometry ; incidence axioms and a given set of points and straight lines

I need help on the following problem set: Let $P = \{ A, B, C, D, E \}$ be a set with five elements and let $$ \mathfrak{g} := \left \{ ...
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Bilinear form and cross product in hyperbolic geometry

I'm reading Patrick J. Ryan's Euclidean and non-Euclidean geometry, page 152. There is a bilinear form defined by $b\left( {x,y} \right) = {x_1}{y_1} + {x_2}{y_2} - {x_3}{y_3}$ on ${\mathbb{R}^3}$ and ...
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1answer
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how to construct an hyperbolic (8,3) tiling

how can I construct an hyperbolic (8,3) tiling ( see https://en.wikipedia.org/wiki/Octagonal_tiling ) in the Poincare Disk model or Klein Disk model of hyperbolic geometry ? or: What are the ...
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2answers
28 views

Ideal quadrilateral in $\mathbb{H^2}$ can be mapped to triangle with vertices $-1,0,\infty, x$ where $x \in \mathbb{R}$

Why can we always map vertices of an ideal quadrilateral in $\mathbb{H^2}$ to $-1,0,\infty, x$ where $x \in \mathbb{R}$? I'm not realising why this can always be done? I.e why $x$ is always real.
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1answer
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Relation between $\Delta \subset PSL(2, \mathbb{R})$ and $\pi_1(S)$ where $S \cong \mathbb{H^2}/\Delta$.

Suppose $S \cong \mathbb{H^2}/\Delta$ where $\Delta$ is a discrete subgroup of $PSL(2, \mathbb{R})$ I am told that $\Delta \subset PSL(2, \mathbb{R})$ is canonically isomorphic to $\pi_1(S)$. I am ...
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1answer
31 views

Perimeter of $(p,q)$ tiling of the hyperbolic plane

Consider a $(p,q)$ regular tiling of the hyperbolic plane projected on the Poincare disc (that is, a tiling of q p-gons joining at each vertex). Obviously the area of all tilings converge to $\pi$, ...
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1answer
60 views

Strong contraction of hyperbolic space

I'm trying to study Hyperbolic geometry, but I can not understand the following statement. Let $X$ be a $δ$-hyperbolic space. Then, there exists $M > 0$ such that for any geodesic $γ$, and ...
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1answer
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Axioms vs Models in projective and hyperbolic geometry

I'm studying Projective Geometry. The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. ...
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1answer
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Is the group generated by two loxodromic isometries with a fixed point in common cocompact?

If you have two distinct loxodromic isometries of the hyperbolic plane $\gamma_1, \gamma_2$ such that they have a fixed point in common. For simplicity let's take the half plane model and let the ...
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0answers
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Side and angle relations in a hyperbolic quadrilateral.

Let $PQRS$ be hyperbolic quadrilateral, i.e. a quadrilateral in $\mathbb{H}$ whose sides are hyperbolic geodesic. Let length$(PQ)=l_1$, and length$(PS)=l_2.$ Also $\angle SPQ=\theta_1$, $\angle ...
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2answers
31 views

Is this a Trapezium?

I once read that in hyperbolic geometry, two hyperbolas can be parallel. In a trapezium, you have four sides and a pair of parallel lines, therefore is it possible to have a trapezium with two ...
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1answer
45 views

Discrete faithful representation in $PSL(2,\mathbb R)$ and horocycles in hyperbolic space

Let $S$ be a closed oriented surface of genus $g>1$. Is the following true ? Let $\alpha,\beta\in \pi_1(S)\backslash \{1\}$ and $\rho:\pi_1(S)\rightarrow PSL(2,\mathbb R)$ be a discrete ...
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3answers
60 views

Where does the hyperbolic metric come from?

In hyperbolic geometry, the metric is often defined as $$ds=\frac{\sqrt{dx^2+dy^2}}{y}$$ Where did this metric come from? I have thought long and hard about this question, but have no satisfactory ...
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Prove by showing that CW is asymptotic parallel to DZ in the direction of W and Z

If XABY is a biangle and WCDZ is a figure formed so that AB=CD. angle BAX = angle DCW, and angle ABY = angle CDZ then WCDZ is a biangle. If Two lines AX and BY are asymptotic parallel in the ...
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2answers
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How to show the Poincaré disk is hyperbolic for some $\delta$

I am trying to prove that the Poincaré disk, $\mathbb{D}$, is $\delta$-hyperbolic with respect to the slim triangle definition for hyperbolicity. I have been stuck for a while on where to begin, ...
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1answer
31 views

Semiregular tilings of the hyperbolic plane

Consider the irregular quadrilateral tiling of the Euclidean plane depicted by the log-log coordinate grid: I'm wondering if in the Hyperbolic plane exist some analog of this kind of tiling where ...
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0answers
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Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
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2answers
17 views

Prove that the gradient of the tangent to $xy=d^2$ is $-\frac{c^2}{4d^2}$

I have this question: Given that $y=mx+c$ is a tangent to $xy=d^2$ prove that $m=-\frac{c^2}{4d^2}$. I'm not sure what direction to take - I tried differentiating the hyperbola equation, but ...
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1answer
36 views

measuring curvature

Suppose you are transported to an 2 dimensional hyperbolic world, ( a plane (2 dinensional) manifold with a constant negative curvature ) the only geometrical tools you have are a ruler, a pencil, ...
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66 views

(compact, non-empty boundary )Surface Geodesics on Hyperbolic Geometry

I have a basic knowledge of hyperbolic geometry . I am trying to understand the meaning of "a compact surface S with non-empty boundary(which is neither a disk nor an annulus )with a complete ...
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2answers
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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 ...