# Tagged Questions

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

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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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### 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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### 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 (...
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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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### “Square,” line-preserving models of the hyperbolic plane

The Klein model of the hyperbolic plane is a line-preserving map from $H^2$ to the disk. Is there a model of the hyperbolic plane which is a line-preserving map from $H^2$ to $[0,1]^2$? By line-...
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### What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
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### Lazy mathematician: what are the real lengths in an Ideal Lambert quadrilateral?

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral An Ideal Lambert quadrilateral (my term, ...
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### Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry

I was trying to compare the metric tensor at the wikipedia pages of the Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model and the metric tensor of the Poincare disk model at https:...
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### Show that $\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4$

I have been reading about "mean value theorems in number theory" such as $$\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4$$ How to prove such a result? One source says it is ...
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### What is the name of this (circumscribed) triangle?

I am meeting the following triangle more and more in my investigations of ideal triangles in the Beltrami Klein model of hyperbolic geometry. That made me wonder: is there a name for it? (And does it ...
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### An interesting point of a triangle. (Help needed to prove a statement.)

Consider a triangle whose sides are segments of $\color{red}{\text{line}}$, $\color{blue}{\text{line}}$, $\color{green}{\text{line}}$ falling in the circum-circle $c$. Let $\color{red}{\text{P}}$,\$\...
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### Finding angles of hyperbolic triangles

I am trying to learn about how to find the angles of hyperbolic triangles. Now below is a problem: It has all the steps but I am not understanding the concept (the ones that are underlined in green ...
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### Hyperbolic Ideal Triangle

I have everything pretty much figured out everything but I need help proving the unique point formed by the three perpendiculars in the picture