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

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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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0answers
7 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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1answer
233 views

Aristotle's Axiom in Hyperbolic Geometry

I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs ...
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1answer
26 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
6
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3answers
771 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
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1answer
31 views

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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1answer
38 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
40 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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2answers
49 views

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 ...
5
votes
2answers
373 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
8
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1answer
48 views

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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2answers
93 views

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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2answers
315 views

Models of hyperbolic geometry

Wikipedia states the following: [The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model ...
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0answers
46 views

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
57 views

ruling out non Pseudo-Anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
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0answers
17 views

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) \; ...
3
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1answer
43 views

“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
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 $\lambda \in \mathbb{R}^+$. ...
2
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2answers
50 views

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 ...
5
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0answers
58 views

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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0answers
56 views

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
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 ...
2
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1answer
32 views

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
62 views

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
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
107 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 ...
6
votes
1answer
153 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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0answers
34 views

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
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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0answers
50 views

Parallel postulate in hyperbolic geometry question

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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2answers
33 views

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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0answers
22 views

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
18 views

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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1answer
23 views

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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1answer
27 views

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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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
29 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
55 views

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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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
54 views

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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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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0answers
21 views

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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1answer
44 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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0answers
164 views

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides?

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides? I have been trying to visually see if that is possible or not.
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2answers
188 views

convex polygons in hyperbolic geometry

Does $\exists$ on the hyperbolic plane, a convex quadrilateral $Q$ and a convex pentagon $P$ with the same angle sum? I found this question to be rather interesting.
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1answer
128 views

Approximating Distances in the Hyperbolic Space

I am currently trying to understand the paper by Krioukov et. al. on hyperbolic networks, but since I do not have a background in hyperbolic geometry (or, in that sense, in geometry at all) I struggle ...
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1answer
280 views

Convert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels $x$ units in Euclidean space, how ...
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0answers
67 views

Does distance in hyperbolic space satisfy such properties which Euclidean distance have?

In Euclidean space $E^n$, the distance between two points $x, y$ is just $|x-y|$, and for each fixed $x_0$, the image $y\to\nabla_x|x_0-y|$ is $S^{n-1}$, so it satisfies ...
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0answers
19 views

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 ...