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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.

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hyperbolic equilateral triangle : $\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = \frac{1}{2}$

I met this problem in Ratcliffe's Foundations of Hyperbolic Manifolds. Please help me prove this. In an equilateral triangle with side length $a$ and angle $\alpha$, $$\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) =…
Peachy Chiu
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boundary at infinity of $\mathbb{H}^2$

In hyperbolic geometry what does it mean when they say the boundary at infinity of $\mathbb{H}^2$? The only idea I came up with was a horizontal line to represent the horizon and to lines meeting at a point on that line, but they are not actually…
user121895
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working in hyperbolic geometry

I wonder if anyone can provide me with a simple step-by-step proof in hyperbolic geometry of a fact that does not hold in Euclidean geometry. I imagine an answer to be a series of statements, such that later statements follow from earlier ones. It…
Adam
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Problem about alternate angle on poincare disc model.

If two alternate angles are same, two poincare lines are parallel. (i.e. If two poincare lines cut by a transversal have a pair of congruent alternate interior angles, then the two poincare lines are parallel.) I want to show this statement by using…
jakeoung
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prove that the sum of the angles in any triangle is less than 180 in hyperbolic geometry (or poincare model).

We could use poincare disc model as a hyperbolic geometry model. I have difficulty understanding poincare disc model. So is there someone to help?
jakeoung
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Every Hyperbolic manifold is a quotient of a Hyperbolic space by a certain discrete group

I started reading about Hyperbolic manifolds here: https://en.m.wikipedia.org/wiki/Hyperbolic_manifold and I didn't understand the following paragraph in the first section of Rigourous definition: Every complete, connected, simply-connected…
user32415
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It is possible to isometrically embed $H^2$ space in $E^6$ space. Are there practical equations to do so?

I was shown this paper which proves that hyperbolic $H^n$ space can be isometrically embedded in Euclidean $E^{6n-6}$ space. While I am able to understand pieces of the proof, I am not an advanced mathematician and am having trouble pulling out much…
Tim Morris
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Difficulty in understanding a proof from Katok's Fuchsian Groups.

I am currently reading Fuchsian Groups by Svetlana Katok. The following proposition appears on page 30: There is a misprint. It is supposed to be $z_0$ not $w_0$. The next paragraph shows that $\text{E}$ is closed. I am not able understand why…
Ajin Shaji Jose
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Easy explanation of non-abelianness of hyperbolic curves

I'm looking for easy proofs (or just an easy proof) of the following statement: Let X be a hyperbolic Riemann surface, i.e., $X$ is a Riemann surface and the universal covering of $X$ is the complex upper half plane. Then the fundamental group of X…
Uiterloo
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Does every triangle in the hyperbolic plane have an incircle?

It is known that NOT all triangles on the hyperbolic plane have a circle that contains the triangle and passes thru all its 3 vertices. IOW, the circumcircle is not a universal property of triangles. What about the incircle? If every triangle does…
Nandakumar R
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Express a point $v$ in terms of $v_1$ and $v_2$ and the hyperbolic distance.

In the hyperbolic plane, we have this following result : On the geodesic boundary $\left(v_{1}, v_{2}\right)$, for example, we have, for $v \in\left(v_{1}, v_{2}\right)$, $$ v=\frac{\sinh \left(c-d_{\mathbb{H}^{2}}\left(v_{2},…
M-S
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How to find (u,v) coordinates of center of inscribed circle on the Poincare plane?

I have three points with their coordinates (u1,v1) , (u2,v2) , (u3,v3) on the unit circle in Poincare model. Is there a formula giving coordinates of centre of inscribed circle in triangle builded on those three points?
Looorean
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Find the reflection of $1 + i$ through the line $l$ with endpoints $3$ and $9$.

Find the reflection of $1 + i$ through the line $l$ with endpoints $3$ and $9$. My approach: Let AB be a line l with endpoints $(3,0)$ and $(9,0)$. Then, the line l lies along the x-axis so the image of the point $(x, y)$ becomes $(x, -y)$. So we…
mathwizz123
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A compact way to define a geodesic circle of a two sheeted hyperboloid.

Upon reading up on the hyperboloid model of the hyperbolic space I have found a pretty way of obtaining geodesic line segments on the hyperboloid of two sheets connecting two given points. I became interested with finding geodesic circles of a given…
b sulyok
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Hypercycle are not lines

I was trying to show in $\mathbb{H}$ -Poincaré half-plane model- that hypercycles defined for a given line $l$ and a given $a$ in $\mathbb{R}$ as $$H(l)=\{z \in \mathbb{H} : d_{\mathbb{H}}(z,l)=a \} $$ are not lines. To prove something I tried to…
Augusto Matteini
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