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

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Relativity and Projective Geometry [migrated]

How do you identify the cross ratio equation of projective geometry from the hyperbolic geometry of relativity? Specifically, what relativistic variables would correspond to A,B,C,D in the standard ...
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Relativistic Projective Geometry

If we assume that space-time has an extra two dimensions so that there is more symmetry between space (with 3) and time (now with 3). What would the corresponding cross ratio equation look like if we ...
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1answer
317 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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Isometries of hyperbolic space

The metric tensor for the Poincaré ball model of hyperbolic geometry is $$ g_{ij} = \frac{\delta_{ij}}{(1 - \lvert \mathbf{r} \rvert^2)^2} $$ where $\mathbf{r}$ is the position in the ambient ...
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What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
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1answer
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Poincaré disk model of hyperbolic plane

Can someone please explain trigonometry in Equations (2) to (8) of: PoincareDisk ?
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Calculate the inradius of a cell in hyperbolic {p,q,r} tiling?

As written in the title, I need to calculate the inradius of a cell in hyperbolic tiling with Schlafli symbol {p,q,r}. You can link to a document which have the formula or write the formula here. That ...
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1answer
30 views

What are the most important issues to consider in upper-half plane model?

I hope you can help me, I need to do a research project about this model of hyperbolic geometry. Honestly, I've never studied the subject, and I'm not sure that subjects should give more importance. ...
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Is it possible to distinguish rest and movement in hyperbolic universe?

Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed. In Euclidean space, all points would move along ...
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What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
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3answers
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Motivations for Hyperbolic Geometry

Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You ...
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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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Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
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6answers
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Books for Hyperbolic Geometry.

I want to read hyperbolic geometry. Can any one suggest some good books on the topic.
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11answers
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What are the interesting applications of hyperbolic geometry?

I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...
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Why can't the pseudosphere be completed in $R^3$?

Without appealing to Hilbert's theorem on the non-embeddability of complete hyperbolic surfaces in $R^3$, is there a way to "see" that one can't extend the pseudosphere / surface of revolution of a ...
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Linear Isoperimetric Inequality is invariant under quasi isometry

Suppose $X$ and $Y$ are quasi-isometric. Show that $X$ satisfies a linear isoperimetric inequality iff $Y$ satisfies a linear isoperimetric inequality. My idea: Suppose $X$ satisfies a linear ...
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Algebraic conditions for the directions coming from a hyperbolic configuration of point

Consider hyperbolic $3$-space $H^3$, thought of as the open unit ball in $\mathbb{R}^3$, where geodesics are represented by arcs of circles etc. (the well known Poincare model of $H^3$). Let $B$ ...
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N-filling implies 3N/2 - geodesic filling

Suppose X is a geodesic space and c is a rectifiable loop.Show that if c admits an N-filling then c admits a 3N/2- geodesic filling. I suspect that there is nothing to prove indeed but still I ...
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Progression of a point along geodesics under the action of hyperbolic Möbius transformations

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
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1answer
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Why to include the $C$ in the formula for the distance in hyperbolic geometry?

I'm reading Penrose's: Road To Reality. First he gives the Lambert formula and later, he says that if you want, you can include the $C$ of the Lambert area formula. But It's not clear why I would ...
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3answers
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Expression of the Hyperbolic Distance in the Upper Half Plane

While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia. ...
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1answer
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Poincaré hyperbolic geodesics in half-plane and disc models

EDIT1: The derivation of geodesics of the two models follows in a straightforward manner from the metric. For the half-plane we have in Cartesian coordinates $$ ds^2 = (dx^2 + dy^2)/y^2 \tag{1} ...
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1answer
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How (not) to plot a Hyperbola? [closed]

I am trying to plot a Hyperbola in Wolfram Alpha. Its giving me a strange graph. How to correct that?
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1answer
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Geodesic on hyperboloid and Poincare's Disk Model

I have two questions that 1. Why geodesic on hyperboloid corespond the arc in the Poincare's Disk Model? The hyperboloid : $x^2 + y^2 - z^2 = -1, \hspace{.15cm} z>0$ When any plane through ...
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2answers
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Error when computing geodesics in hyperbolic half plane

It is known that the geodesic equations for the upper half plane equipped with the hyperbolic metric are $$x''=\frac{2x'y'}{y},$$ $$y''=\frac{(y')^2 -(x')^2}{y}.$$ It is also well known that the ...
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Equation of line in hyperbolic space

After a slightly peculiar dream the other night, I find myself suddenly inspired to do numerical simulations in three-dimensional hyperbolic space. For this to work, I need an equation of line in ...
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Geodesics in upper half-plane model of $\mathbb{H}$

On this page in Schlag's book on complex analysis, he is discussing the upper half-plane model of $\mathbb{H}^2$. He says for all $z_0\in \mathbb{H}$ $$\{T'(z_0) \mid T \in PSL(2, \mathbb{R}) ...
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3answers
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Distance formula for points in the Poincare half plane model on a “vertical geodesic”.

In comment at http://math.stackexchange.com/a/1381829/88985 at Distance of two hyperbolic lines is says (as i interpreted it) that the distance between two points $(a,r)$ and $(a, R)$ in the ...
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Cross Ratio of two rays through origin

There are two fixed and two variable concurrent rays of unit length in 3 space through the origin. How should the spherical coordinates of the two variable points be related to result in a constant ...
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2answers
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Is the cross ratio the unique invariant under projective transformations up to multiples?

I have been studying the actions of $PSL_2(\mathbb{R})$ on the hyperbolic plane recently, and the hyperbolic distance $d(z_1, z_2)$ is the absolute value of the log of absolute value of the cross ...
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The shape of the hyperbolic curves coordinates

Any one has an idea about hyperbolic coordinates ? and how to imagine it ? Indeed I am trying to find the shape of the coordinate curves far away from origin ! and what is the shape of them at $u=0 ...
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1answer
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Equidistant points to a hyperbolic line

consider the Poincare upper half-plane model of hyperbolic plane $\mathbb{H}^2$ and a hyperbolic line $\ell\subset \mathbb{H}^2$ (or geodesic if you want). I would like to visualize the set of points ...
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On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?

Wikipedia has the answer in the case of the Poincaré disk model. When the point $\mathbf{v}$ is translated to the origin, then the point $\mathbf{x}$ is translated to $$\frac{(1 + 2\mathbf{v} \cdot ...
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1answer
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Mapping from Poincare's disk model to UHP

I have a question that : How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function $$f(z) = \frac{z + i}{iz+1}$$ But I want to know the geometric ...
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2answers
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Distance of two hyperbolic lines

Consider the upper-half plane model of the hyperbolic plane $\mathbb {H}^2.$ Now consider two lines in it given as $\ell_1:=\lbrace { (x, y)\in \mathbb {H}^2 \vert x^2 +y^2=r^2\rbrace}, ...
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1answer
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Proof of the ultraparallel theorem in the Beltrami Klein model

I was reading (and editing) the proof mentioned at https://en.wikipedia.org/wiki/Ultraparallel_theorem#Proof_in_the_Beltrami-Klein_model and noticed it is not correct. (the ultra parallel theorem is ...
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1answer
50 views

Triangular Area on hyperbolic surface

I have read numerous paper over area calculation in hyperbolic geometry but just can't seem to understand how to calculate a triangle's area in hyperbolic geometry. It would be nice to have a proof ...
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2answers
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N-polygons in hyperbolic geometry

Let $N$ be an integer and we have two $N$-polygon $A_{1}A_{2}\ldots A_{N}$ and $A'_{1}A'_{2}\ldots A'_{N}$ such that the length of geodesic $A_{i}A_{i+1}$ is equal to the length of geodesic ...
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Curvature of hyperbolic surface

So from what I understand the curvature of a surface by calculated by inversing the radius of the osculating circle. But if a hyperbolic surface have a negative curvature, wouldn't that imply the ...
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1answer
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rotations and SU(1,1)

I'm interested in the isometries of the hyperbolic plane, i.e. the mappings which leave the geometric properties of objects invariant. I'm working with the Poincare disc model. My lecture notes ...
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Search for coverings in hyperbolic tessellations

Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane? For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), ...
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What's the right way to calculate hyperbolic distance on the hyperboloid model?

I see in the Wikipedia article on the hyperboloid model and also in this other Math.SE question about the hyperboloid model that this is how you calculate distance on the hyperboloid model: Let $u = ...
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2answers
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Isometric group of hyperbolic 3-dim manifolds

In the book : Foundation of Hyperbolic Manifolds There is a theorem that any finite subgroup of $ Isom(\mathbb{E}^n) $ fixes a point. And I hope to solve the following question : Any ...
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2answers
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What is the largest possible sum of all the angle measures of a $\Delta$ in hyperbolic space?

$\Delta ABC$ exists in hyperbolic geometry. What is the maximum value for $m\angle A+m\angle B+m\angle C$?
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Corollary of Wolpert lemma

Recall Wolpert Lemma. Let $S$ be a surface with genus greater than 2, let $[X,f]$ and $[Y,g]$ two points of $T(S)$ (Teichmüller space) and let $\phi \colon X \to Y$ a $K$ quasi conformal homeo. Then ...
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1answer
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Heat kernel formula on hyperbolic plane well defined

Consider the heat kernel for the hyperbolic plane $\mathbb{H}^2$ and the corresponding heat kernel: $$k(x,y,t)=\frac{C}{t^{\frac{3}{2}}}\cdot ...
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1answer
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Hyperbolic segment from $(0,0)$ to $(0,0)$

Can there be a segment on a hyperbolic plane that goes from point $(0,0)$ to $(0,0)$ in the hyperbolic plane. There are some rules, though for this to work: 1) The segment must apply to the rules of ...
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1answer
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Are bounded geodesics in the modular surface closed?

Let $M=\mathbb{H}/SL(2,\mathbb{Z})$ be the modular surface (which is noncompact but finite volume with the volume induced by the constant negative curvature metric inherited from $\mathbb{H}$). Any ...
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1answer
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Finite hyperbolic geometry with ideal points

I was browsing "Thinking Geometricly: A Survey in Geometries" by Thomas Q. Sibley, 2015 and on page 388 it mentions a finite hyperbolic geometry of order 3 (3 points per line) consisting of 13 ...