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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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Showing that hyperbolic circles are Euclidean circles and vice versa in the Poincaré upper half plane

This is exercise 1.11 in the book Fuchsian groups by Svetlana Katok. Show that every hyperbolic circle in the upper half plane $\mathbb{H}$ is a Euclidean circle (with a different center, of course), and vice versa. I am aware of the fact that we…
TheGeekGreek
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Why are hyperbolic circles in the upper half plane also Euclidean circles?

In a few places I've seen talking about the upper half plane as a model of hyperbolic geometry, it's mentioned offhand that circles (in the sense of a set of points equidistant from a given point under a given metric) in the Euclidean and hyperbolic…
pseudocydonia
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Going from Metric to Distance Function in the Poincaré Half Plane

Let the Poincaré Half Plane be the set $\{(x, y) \in \mathbb{R}^2 : y > 0\}$. It is a known result that the the metric $ds^2 = \frac{dx^2 + dy^2}{y^2}$ yields a distance function $f$ such that its output is the length of the geodesic between two…
Fomalhaut
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Distance formula for points in the Poincare half plane model on a "vertical geodesic".

In comment at https://math.stackexchange.com/a/1381829/88985 at Distance of two hyperbolic lines is says (as i interpreted it) that the distance between two points $(a,r)$ and $(a, R)$ in the Poincaré half-plane model (…
Willemien
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Measuring distance on the Poincare disk

I've seen several different ways to measure distance on the Poincare disk i.e Riemann metric/manifold (which I don't understand). However the method we're taught is using $\tanh^{-1}$ and complex numbers.…
George1811
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Distance from a point to a line in the hyperbolic plane

I have two questions: What is the distance from a point to a line in the hyperbolic plane? Fix a line $L$ in the hyperbolic plane. What does the set of points of distance $d$ from $L$ look like?
Hyperbolic Asker
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The absolute ruler unit of hyperbolic geometry

In this article, the following is said: On the other hand, there is no absolute standard of length in Euclidean geometry. If one wants to construct a segment of any particular length, one requires a ruler of some type. If you constructed a…
tryst with freedom
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Difference between a hyperbolic line and a geodesic

The setting for hyperbolic space in this question will be the upper half plane. Now I know that to measure the distance between two points $p$ and $q$ in the upper half plane, we take $ \inf \int_\gamma \frac{1}{\Im(z)} |dz| $, where $\gamma$ is…
user38268
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Measuring angles in the Beltrami Klein model of Hyperbolic geometry

I am learning bits of hyperbolic geometry and the wikipedia page gives two such standard models for it ; the Beltrami Klein (BK) model and the Poincare (P) disk model. Now as I understand it hyperbolic geometry has exact analogues for every concept…
smilingbuddha
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Isometries of hyperbolic 3-space!

The Klein model of the hyperbolic 3-space is defined as follows $$\mathbb{H}^3=\{x\in\mathbb{R}^{3,1}:\langle x,x\rangle_{3,1}<0\}/\sim$$ where $x\sim x'$ if and only there exists $\lambda$ such that $x =\lambda x' $, and $\mathbb{R}^{3,1}$ is the…
Faster Dam
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Do there exist space-filling curves that fill the whole hyperbolic 2d plane? If so, can they be visualized?

Just as mentioned in a similar question, in Euclidean plane, there exist space-filling curves that fills the whole plane. So for hyperbolic plane, do there exist some space-filling curves?
Mountain
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Hyperbolic geometry: 2/3 ideal triangle in Poincare Disk

I would like to know the following. (1)How to construct the incircle of 2/3 ideal triangle. (2)How to calculate its radius. If there were any literature concerning above, I would like to know.
seiichikiri
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Spheres in hyperbolic spaces

I've been playing too much HyperRogue and it's given me ideas. I'm currently working on a text-based RPG, and there's a part of it where the PC is immersed in a hyperbolic space (notwithstanding whether they could even survive in such an…
Romain Echaniz
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List of hyperbolic geometry models

When I studied (as an amator) non-euclidean geometry I read that big number of models of hyperbolic geometry was created in 19th and 20th century. Where can I find any list and short description of such models? Usually three models are described:…
Arek
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Are points on the boundary of the Poincare Disc Model considered part of the hyperbolic plane?

Are the points that lie on the outside edge of a Poincare Disc considered to be in the hyperbolic plane? In other words, are points C, D, J, Y, and Z inside the plane?
Dana Hill
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