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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.

1921 questions
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Finding Möbius transformation from fixed points

Given a non-parabolic transformation which is also an orientation preserving isometry in the hyperbolic upper half plane union the boundary, if I know the two fixed points and they are two different irreducible fractions on the boundary, how can I…
user7017
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Is it possible to construct a seamless, boundless, finite surface solely with heptagons?

Note, the motivation for this question essentially comes from game design. I was wondering if it's possible and/or if it even makes sense to have a playing field that is both hyperbolic (the area you can reach is more than $\pi r^2$ for a given…
El'endia Starman
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Studying the hyperboloid model, what is represented by the conic sections?

I am trying to get my head around the hyperboloid model of hyperbolic geometry. Hyperboloid_model The article is much too technical for me, please improve. And was thinking the hyperboloid $ x^2 +y^2 - z^2= -1: z >0 $ has as asymptotic cone $ x^2…
Willemien
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another pythagorean theorem in hyperbolic geometry

on https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry it says However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the…
Willemien
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Hyperbolic triangle and two points in Poincare disk

Given a hyperbolic triangle $T$ and two points $p$ and $q$ in Poincare disk. Note that $p$ and $q$ are outside the triangle. If $p$ has shorter distances to the three vertices of $T$ than $q,$ can we claim that $p$ has shorter distances to all…
Miao J
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How to construct a circle in a the Poincare Disk model

How can I construct an circle with centre C going trough point P in a Poincare disk?. I found an script of how to do it in the "Poincaré Disk Model of Hyperbolic Geometry"toolkit from the geometers sketchpad,…
Willemien
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Parallel postulate in hyperbolic geometry question

Let $L =\{y=0\}\cap\mathcal{H}^{2}$, and let $P=(3,2,2)$. Show that the parallel postulate fails in $\mathcal{H}^{2}$ by giving two lines $L',L'' \in \mathcal{H}^{2}$ with $P\in L',L''$ and $L\cap L'=L\cap L'' = \varnothing$. I think a good idea…
AlexBowring
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Is the usual topology on the upper half plane same as that induced by Riemannian metric?

The upper half plane is a Riemannian manifold, with the Riemannian metric given by $(ds)^2 = (dx^2+dy^2)/y^2$ and thus has a metric topology induced by this metric. Is this topology same as the topology inherited as a subspace of $\mathbb{R}^2$ (or…
Kiran
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Preserved geodesic in $\mathbb{H}^n$ under hyperbolic isometry

reading my course I stumbled upon this fact: Let $f\in\text{Isom}(\mathbb{H}^n)$ be an hyperbolic function (exactly 2 fixed points on $\partial\mathbb{H}^n$ and no fixed point in the interior). Suppose these fixed points are $p$ and $q$ and let…
Leo
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Geodesics on the once-punctured torus.

Denote the once punctured torus by $\mathbb{T}$. The fundamental group $\pi_1(\mathbb{T})$ of the once-punctured torus is the free group with two generators. Now consider a Fuchsian representation $\rho: \pi_1(\mathbb{T}) \to \operatorname{PSL}(2,…
user7090
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Fundamental polygon of the pair of pants

I am reading a set of notes on Representations of Surface Groups made for a workshop at AIM. It states that a genus g surface with $k$ holes has a decomposition (I am assuming this is referring to that surface's fundamental domain) as a $4g+k$-gon…
user7090
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Möbius transformation that preserve distance for two pair of given points in $\mathbb{H}$.

I need to prove that for a given two pair of points $(z_1,z_2)$ and $(w_1,w_2)$ in $\mathbb{H}$ (Poincaré's upper half plane), where $d_{\mathbb{H}}(z_1,z_2)=d_{\mathbb{H}}(w_1,w_2)$, there is an Möbius transformation $m \in \text{Möb}(\mathbb{H})$…
Henrik B.
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Can constant negative $K$ coral shapes be crocheted?

Crocheted Surface shown here is entirely hyperbolic, made of negative Gauss Curvature patches. The Manifold smoothly grows unbounded exponentially into $\mathbb R^3$ ... as any one watching it gradually evolve by Crochet needles making new rings of…
Narasimham
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Pappus chain in an ideal triangle of Poincare disk of hyperbolic geometry

Let IaIbIc be an ideal triangle in Poincare disk. Denote: a, b, c = the hyperbolic lines whose ends are Ib and Ic, Ic and Ia, Ia and Ib respectively. r = the incircle of a, b and c.. A, B, C = the contact points of r and a, b, c respectively. O =…
seiichikiri
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Lower bound on the decay of the injectivity radius in hyperbolic manifold

Let $M$ be a complete orientable hyperbolique manifold with finite volume (3 dimensional in my case, typically a knot complement), that is essentially a quotient of $\mathbb{H}^3$. Let $r(x)=d_M(x,x_0)$ for some $x_0\in M$, and…
Isao
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