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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.

The prototypical example of hyperbolic geometry in two dimensions of Gauss-Lobachevsky-Bolyai in which the parallel postulate of Euclidean geometry is replaced by a new postulate of at least 2 parallel lines through an external point not on the given line with sum of interior angles of a geodesic triangle smaller than $\pi$ radians.

1921 questions
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Solving for the Moebius transform that sends $(x_0, y_0)$ to $(x_1, y_1)$ with two known fixed points

Suppose $M$ is a hyperbolic Moebius transformation with fixed points at $(0, 0), (1, 0)$ which, when applied to the complex $(x_0, y_0)$, yields the result $(x_1, y_1)$. How do I solve for $M$ given $x_0, y_0, x_1$, and $y_1$?
Gabriel Silk
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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 include that, from his presentation, it seems that…
Red Banana
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Find the hyperbolic distance in the upper hyperbolic plane

Let $A=(0,112), B=(0,126), C=(98,112)$ be points in the hyperbolic upper half plane H. Find the hyperbolic distances $d_h(A,B), d_h(A,C), d_h(B,C)$. Every answer should be in the form of a logarithm. So I worked out just $BC$ since I am sure $AB$…
mika
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Compute hyperbolic length of the arc of the circle

Compute the hyperbolic length of the arc of the circle $ x^2 + y^2 = 25$ that lies between (3, 4) and (4, 3). From my notes I know the formula is $$ \ln \frac{{\csc \beta - \cot \beta }}{{\csc \alpha - \cot \alpha }} $$ however I don't know how…
user214010
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Problem in understanding models of hyperbolic geometry

I recently started reading The Princeton Companion to Mathematics. I am currently stuck in the introduction to hyperbolic geometry and have some doubts about its models. Isn't the hyperbolic space produced by rotating a hyperbola? That is, isn't…
Akshit
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Homothetic transformation in the Poincaré upper half plane

I am interested in finding homothetic transformations in the Poincaré upper half plane. I heard that unlike $\mathbb{R}^n$ we don't have an homothetic transform for every $\lambda \in \mathbb{R}^+$. So I have now two questions. Do homothetic…
Chevallier
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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 hyperbolic lengths of all relevant distances (distance…
Willemien
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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 $\pi$, but what about the total perimeter? that is, the…
lurscher
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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 that gave me a gradient of $-\frac{y}{x}$. How do I go…
hohner
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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 won't work anymore, Lobatschewsky assumes that light…
Willemien
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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 distances go towards infinity. But I don't…
Willemien
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calculate the curvature of a surface with a Lambert quadrilateral

I was wondering how can I calculate the curvature of a surface? For example: Given a Lambert quadrilateral ABCD (see http://en.wikipedia.org/wiki/Lambert_quadrilateral ) with: $ DA \bot AB $, $ AB \bot BC $, $ BC \bot CD $ and we know the length of…
Willemien
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Distance from point to line segment in Poincaré disk model

I'm trying to build a geometric datastructure in hyperbolic space. For that purpose, I'm using the Poincaré disk model. The distance between two points can be calculated with the hyperbolic law of cosines, as mentioned here. As mentioned in the…
user155598
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Definition of complex hyperbolic geometry

I am trying to read about complex hyperbolic geometry.But I couldnot find a basic definition for it. Is it just the special case of hyperbolic geometry where we work with complex numbers in the model. Or is there something more to it? Can someone…
user148638
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Triangular tiling of hyperbolic plane

I'm currently trying to read an interesting paper having to do with embedding graphs in hyperbolic spaces. Namely, "Geographic Routing Using Hyperbolic Space" by Robert Kleinberg. Link:…
gogurt
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