# Tagged Questions

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

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### Hyperbolic metric geodesically complete

Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means ...
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### Curvature of a hyperbolic plane

Consider a projective plane and a real quadric. According to the Klein-Beltrami-model the inside of the quadric is a hyperbolic plane. Klein proved that this plane has a constant negative curvature. ...
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I was introduced the Poincaré Disc model of hyperbolic geometry. The concept idea was clear but I had some questions about it that I could not figure out myself. I understand that the geometric ...
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### Proof verification: Cross Ratio

Prove: If $[z_1,z_2,z_3,z_4] \in \mathbb{R} \cup \{\infty \}$, then $z_1,z_2,z_3,z_4$ are either concyclic or collinear. My proof below uses the geometric interpretation of cross ratio. I am not ...
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### Geodesic sphere in $\mathbb H^2$

I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that ...
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### Hyperbolic plane shrinking

A very small area of the hyperbolic plane looks more Euclidean as the curvature approachs 0. Any more evidence? Or reference would help? Thanks
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### Does the Ray Casting Algorithm works in Poincare's Disk to detect if point is inside Polygone?

As the Ray Casting Algorithm looks to me like a geometric construction on geodesics, and geodesics are redefined in Poincare's Disk, I feel this method would also work in hyperbolic geometry. Is this ...
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### Classification of Möbius Transformations

We know how to classify the points on a surface,by looking the Gaussian curvature at a point in order to guess the shape of the surface near that point.On the other hand we classify the Möbius ...
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### Seeking proof to a Hyperbolic polygon conjecture

In the course of writing a(n Honours) thesis, I'm searching for a proof to a conjecture that appears very likely to be true. Many results will rely upon it. My own attempts to prove it have been ...
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### Simple proof of the existence of lines in the hyperbolic space

Let $\mathbb{H}^n$ be the hyperbolic space defined as warped product: $$g_{\mathbb{H}^n} = dr^2 + \sinh(r)^2 g_{\mathbb{S}^{n-1}}.$$ What is the easiest way to show that there exist at least one ...
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### Geodesic curvature of a curve in the hyperbolic plane

Consider the curve $\gamma$ given by $y=b$ in the upper half-plane equipped with the hyperbolic metric $$\dfrac{dx^2+dy^2}{y^2}$$ Calculate the geodesic curvature of $\gamma$. The problem I'm ...
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### Metric relations in lambert quadrilateral

I already found the relations in a rectangle triangle (6 formulas for the sides) and for a general ordinary triangle (sine and cosine hyperbolic laws). But now I'm trying to find them for a triangle ...