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.

1921 questions

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1 answer

Circles inscribed in regular polygons in hyperbolic geometry

Does the radius of a circle matter when determining the number sides of a regular polygon in hyperbolic geometry? The sides must be tangent to the circle. Can't I just use an equilateral triangle every time?

hyperbolic-geometry

asked May 07 '15 at 19:24

Dani Car

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2 answers

Why do we use cosh to define the angle between two vectors in hyperbolic geometry?

I can kind of see why this works when we use the regular dot product, but I don't understand why this is still true when we use the dot product adapted for hyperbolic geometry?

hyperbolic-geometry

asked Mar 30 '15 at 21:55

boblo

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1 answer

Trisection of a hyperbolic line/segment

I'm wondering how to trisect a line/segment in $\mathbb{H}^2$ (using the Poincaré Disk model). Bisection of a hyperbolic line seems rather straightforward (e.g. as described in the paper Compass and Straightedge in the Poincaré Disk by…

hyperbolic-geometry

asked Mar 10 '15 at 23:24

Ailurus

1,192

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1 answer

constructing an segment with a known length in hyperbolic geometry

I am studying Arlan Ramsay's and Robert Richtmeyer's " Introduction to hyperbolic geometry" On page 255/6 it gives how to construct an segment with an absolute length of $ \ln (\sqrt{2} +1) $ (via the construction of a Sweikart triangle, a right…

hyperbolic-geometry

asked Mar 06 '15 at 11:26

Willemien

6,582

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1 answer

calculating Hyperbolic distance

To calculate the hyperbolic distance we use the formula $$\left|\frac{w-z}{1-\bar wz}\right|$$ I want to apply this to the following pair of points: \begin{align*} w&=\frac{-1}{\sqrt{3}}\space +\space \frac{1}{\sqrt{3}}i…

hyperbolic-geometry

asked Feb 24 '15 at 13:27

Nessa

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1 answer

A casual definition of Hyperbolic Space

I'm writing an article about a lecture that mentioned hyperbolic space. I wondered if anyone had a friendly way of describing it to the general public. (I will rewrite any definitions in my own words, or credit you. No plagiarism here!)

hyperbolic-geometry

asked Feb 15 '15 at 00:17

ReidScience

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1 answer

Geometry of a hyperbolic triangle

How can we find the upper bound of the hyperbolic distance from any point on the side $AB$ to either $AC$ and $BC$ for the hyperbolic triangle $\triangle ABC$? Help will be appreciated.

hyperbolic-geometry

asked Feb 27 '12 at 23:36

gareth

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2 answers

Advantage of using Hyperbolic Trigonometric functions?

$\DeclareMathOperator{\sech}{sech}\DeclareMathOperator{\csch}{csch}$Is there any added advantage of using Hyperbolic Trigonometric functions? Since you can always use normal trigonometric functions in all…

hyperbolic-geometry

asked Jan 29 '15 at 12:47

RE60K

17,716

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2 answers

Ideal quadrilateral in $\mathbb{H^2}$ can be mapped to triangle with vertices $-1,0,\infty, x$ where $x \in \mathbb{R}$

Why can we always map vertices of an ideal quadrilateral in $\mathbb{H^2}$ to $-1,0,\infty, x$ where $x \in \mathbb{R}$? I'm not realising why this can always be done? I.e why $x$ is always real.

hyperbolic-geometry

asked Dec 02 '14 at 02:05

Joesph Sutton

-1

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1 answer

Calculating side in quadrilateral in Poincare disc

If I have a quadrilateral ABCD inside the Poincaré disc such that $\angle A=\angle B=\frac{2\pi}{3}$, $AD=BC$ and we know the hyperbolic lengths of sides $AB$ and $CD$, how can I calculate the hyperbolic length of $AD$ in terms of $AB$ and $CD$?

hyperbolic-geometry

asked Dec 07 '18 at 18:35