# Tagged Questions

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

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### What is the geometric center and what is the other point?

In Euclidean geometry it is simple: In a triangle $\triangle ABC$ there is a single point $H_a$ on $BC$ such that the triangles $\triangle ABH_a$ and $\triangle ACH_a$ have the same area. the ...
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### Finding the equation of a hyperbola given the vertices and foci.

A hyperbola has the vertices $(0,0)$ and $(0,-16)$ and the foci $(0,2)$ and $(0,-18)$. Find the equation with the given information.
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### Triangle Inequality for Hyperbolic Metric (Logarithm of Cross-Ratio)

I need clarification in one step of this answer to my previous question. I was re-reading it, and it isn't clear to my why we can make the assumption of $\Im(p)< \Im(q)< \Im(r)$. (Immaginary ...
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### Identifying the conic given some conditions.

So I have to identify the conic which represents the centre of the circle which touches another circle externally, and also touches the x axis. Here's a link to the exact question with the equation ...
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### Lines in upper half-space

I'm teaching a tour-of-classical-geometry class this semester, and we are soon to introduce hyperbolic geometry. I am very inexpert in this subject, and I have a question about a compatibility of a ...
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### saccheri quadrilateral - how does base=summit violate hyperbolic parallel axiom?

I drew diagonals across the quadrilateral and was able to prove that the summit angles are right angles by SSS and CPCTC. Therefore the two congruent triangles creats a quadrilateral with an angle sum ...
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### Given two hermitian matrices of signature (2,1) there exists a Cayley transform between them?

Given a matrix $A\in M_{k\times l}(\mathbb{C})$ we define the hermitian transpose of $A$ as the matrix $A^*=\overline{A}^t\in M_{l\times k}(\mathbb{C})$. We say a matrix $H\in M_k(\mathbb{C})$ is ...
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### Distance in Poincaré disk from origin to a point given

Let $C$ circle $x^2+y^2=1$ find the distance (Poincaré disk) from $O=(0,0)$ to $(x,y)$ The distance in Poincaré is $d=ln(AB,PQ)$ where AB are a segment of the curve and P and Q are points in the ...
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### Formula for the midpoint in the hyperbolic geometry

I have two questions. First, is there a relatively simple formula for the midpoint of two points $a_1$ and $a_2$ in the disk with respect to the hyperbolic geometry? That is, the point on the ...
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### How to visualize the region $\mathbb{H}/\Gamma_0(4)$ and its cusps?

In number theory we learn that $\theta(z) = \sum q^{n^2}$ is a modular form with respect to $\Gamma = \Gamma_0(4)$. This boils down to two properties: $\theta(z)= \theta(z+1)$ this shift symmetry ...
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### Proof that three parallel lines don't be cutted by a transversal in Klein model

How do you prove that three parallel lines don't be cutted by a transversal? By definition parallel are Chords that meet on the boundary circle are limiting parallel lines. Then I built three ...
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### Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition
I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...