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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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What's the right way to calculate hyperbolic distance on the hyperboloid model?

I see in the Wikipedia article on the hyperboloid model and also in this other Math.SE question about the hyperboloid model that this is how you calculate distance on the hyperboloid model: Let $u = (x_0,x_1,x_2)$ and $v = (y_0,y_1,y_2)$ be two…
El'endia Starman
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Hyperbolic Geometry and Circles

How does the angle of parallelism relate to the arc of a circle and a point outside? In Hyperbolic Geometry, I'm trying to figure out what happens to the "visibility" of a circle when a point outside increases in distance away from the circle. So…
Dani Car
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Scale and the models of the hyperbolic plane

I was reading somewhere (sorry I always forget where) that the scale of the Poincare Half plane is y (the vertical) So at the boundary line the scale is $ 0 $ or $ ( 1 : \infty ) $. at the horocycle $ y = 1 $ the scale is $1:1$ and at $ y =…
Willemien
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How to solve an Hyperbolic triangle when all is given except angle C and side c)

Another Hyperbolic triangle problem (all given except angle $\angle C$, and side $c$) I thought that after asking How to solve an hyperbolic Angle Side Angle triangle? I could solve all hyperbolic triangles, but I am still stumped with SSA and AAS…
Willemien
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Commutator of hyperbolic isometries

Let $f,g \in PSL(2,R)$ be isometries of $\mathbb{H}^2$ of hyperbolic type. Let $h=[f,g]$ be their commutator. Is there an explicit geometric criterion to determine if $h$ is hyperbolic, parabolic or elliptic? With "explicit geometric criterion" I…
Lor
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Constructing a quadrilateral in hyperbolic plane

Suppose we have a set of angles, $\{\alpha_1, \alpha_2, \alpha_3, \alpha_4 \}$, and we want to construct some quadrilateral at Poincare Disc Model with angle $\alpha_i$ at vertex $i$. The question is, how to do it? I know how solve this problem for…
Month
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Determine an hyperbolic midpoint

I am asked to determine the hyperbolic midpoint of the points $0,\frac{1}{2} \in \mathcal{P}$ Q: how do I determine the hyperbolic midpoint and what is actually meant by the midpoint?
Mainviel
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How does one formally verify this picture from hyperbolic geometry?

Suppose we consider some hyperbolic circle with center $iz$ using the upper-half plane model of hyperbolic geometry, and in the interior we have a point $x+iy$. How does one prove that $iy$ is also part of the interior, as the picture seems to…
ctcbole
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Spherical Geometry is to the Sphere just as Hyperbolic Geometry is to the....?

I need to write up a quickie description of Hyperbolic Geometry for non-mathematicians. I am trying to say "Hyperbolic Geoemtry is the Geometry of the surface of a ____" I remember that there is, in fact, a term for the surface I am thinking of. It…
Ilan Weinschelbaum
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Constructing shapes in hyperbolic space

I'm trying to get started writing a game that uses the order-4 dodecahedral honeycomb in hyperbolic space. I'm representing points as 4-vectors of the form $\left(\begin{smallmatrix}h\\x\\y\\z\end{smallmatrix}\right)$ where…
Jeremy List
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Distance-preserving coordinate transformations for the poincaré disc

Following this question, I'm looking for a coordinate transformation which leaves distances unchanged. Does such a transformation exist? The isometries for the poincaré disk looked promising, but only conserve angles, not distances. Edit: I'm using…
user155598
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Hyperbolic geometry and orientation reversing isometries.

In Quasi-cluster algebras from non-orientable surfaces by Dupont and Palesi, one can read the following on page 11: I don't understand why the 'following relations' in the image included hold. Applying $D$ to the point $(u,0)$ gives $(\mu u, 0)$.…
Ricky Blake
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Non-Euclidean Translations and Rotations

If $f(z) = z + 1$ and $g(z) = -\frac{1}{z}$ show that $$ g f g^{-1}(z) = \frac{z}{1-z}. $$ I don't know how to solve this question please help.
John Doe
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Solve $d_h(A,B)$ on a Poincare Disc

Consider △ABC on a poincare disc. On △ABC, $\angle C = \theta(radian)$, $d_h(B,C)=d_h(A,C)=b$ In this situation, solve $d_h(A,B)$. To me, it is hard because I have no experience. Is there someone to help?
jakeoung
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Poincare disc model problem. find $d_h(A,B)$

Consider $\triangle ABC$ on a poincare disc. On $\triangle ABC$, $\angle C=90^\circ$, $d_h(B,C)=a$ and $d_h(A,C)=b$ In this situation, find $d_h(A,B)$. I'm taking a course but I cannot follow the class. Is there someone to explain in detail?
jakeoung
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