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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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Circle-Circle intersection area in the hyperbolic space

Is there any closed formula for the area of the intersection of two circles in the hyperbolic plane $\mathbb{H}^2$? The two circles have radii $R, R'$ and a distance of $d$ between centers. If possible, the curvature should be a parameter (though I…
HdM
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What does the logarithm of a hyperbolic line look like?

At a fixed point $p$ of hyperbolic $n$-space $H$, there is the exponential map from flat $n$-space to $H$ taking straight lines through the origin of the flat space to hyperbolic lines through $p$ in hyperbolic space. If $L$ is a hyperbolic line…
Hyperbolic Asker
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Verifying Möbius Transformations using Hyperbolic Geometry

Verify that every transformation from $$H = \left\{Tz = e^{i\theta} \frac{z-z_0}{1-z_0 z} \right\}$$ can be written as $Tz = \frac{az-b}{\bar{b}z+\bar{a}}$ with $|a|^2 - |b|^2 = 1$. The book gives the hint that we need to use algebraic manipulation…
user114657
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What is the radius of a regular right-angled dodecahedron in $\mathbb{H}^3$?

What is the distance from the center to a vertex of a regular right-angled dodecahedron in $\mathbb{H}^3$? A right-angled dodecahedron is a regular dodecahedron with all dihedral angles equal to $\frac{\pi}{2}$; e.g., see Example 5 in Chapter 13 of…
Geoffrey Sangston
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How to find an hyperbolic line that is perpendicular through two other hyperbolic lines?

Suppose $M$ and $N$ are two hyperbolic lines in $(\mathbb{D},\mathcal{H})$ that don't intersect (hyperbolic geometry). How do I find the hyperbolic line $L$ that is perpendicular to both $M$ and $N$? I know that we can find the perpendicular line…
Yeknic
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Topology of compactification of upper half plane

"A primer on mapping class groups" authors defined the topology on $\overline{\mathbb{H}^2}$. I am unable to visualize the open sets there. I would appreciate it if someone helps me to understand the topology on $\overline{\mathbb{H}^2}$. Thank you…
Lokenath Kundu
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Distance-preserving map from the upper half-plane to itself

Let $\mathbb H$ be the upper half-plane with the hyperbolic metric and let $f \colon \mathbb H \to \mathbb H$ be a distance-preserving function. I know that any orientation-preserving isometry (i.e. bijective distance-preserving map) is given by a…
Earthliŋ
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Polar coordinates and parametric equation of circle in hyperbolic plane

We know that the parametric equation of circles of radius r and center at origin is given by $\gamma (t) = (r\cos t, r \sin t)$ and any arbitrary closed curve can be written in the form $\gamma (t) = (r(t)\cos t, r(t) \sin t)$. Can someone help me…
ranadip ganguly
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Relation between Hilbert theorem and pseudosphere (also called hyperbolic plane or Bolyai–Lobachevsky plane)

The Hilbert theorem states that there exists no complete regular surface S of constant negative gaussian curvature $K$ immersed in $R^3$. Ok.. so I'm guessing that the surface of revolution of the Tractrix, which has constant negative curvature, is…
Diego
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What is the boundary at infinity in the hyperboloid model of the hyperbolic plane?

The boundary of the upper half-plane $\mathbb{H}$ model of the hyperbolic plane is the extended real line $\overline{\mathbb{R}}= \mathbb{R} \cup \{ \infty \}$. What is the boundary of the hyperboloid model of the hyperbolic plane? Is it the…
user7090
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Simple understanding of convex co-compactness

I was looking for the definition of the term "convex co-compact" in simple cases. But most references I find are looking into a little bit sophisticated notions such as mapping class group, Schottky subgroup and higher dimensional hyperbolic spaces.…
Frustrated Guy
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Proving an equality for an equilateral triangle in the Poincare model

I've been working a good while trying to establish an equality, but have made little success. Suppose you're working in the Poincare disk model inside an ambient Euclidean plane. If an equilateral triangle in the Poincare model has sides equal to…
yunone
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Deriving the distance between two distinct points on the Upper Half Plane $\mathbb{H}$

I am trying to derive the distance between two arbitrary points in hyperbolic space; the model I'm using is the upper half plane model. So the distance is just $\int_f \rho(z) dz$, where $\rho(z) = \frac{|z|^2}{\text{Im}(z)}$. Now I construct a…
user38268
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Regular triangulation of compact oriented hyperbolic space

Is there a good way of explicitly constructing a regular triangulation of a compact orientable hyperbolic 2-manifold, ideally with any desired vertex degree $\ge 7$? I only need the topology, not any geometric information.
Geoffrey Irving
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Computing the length of a hyperbolic circle

I have a problem with the proof of the following exercise. Define the circle of radius $r \gt 0$ around a point c $ \in $ $H^2$ as $C_r(c)= \; \{x \in H^2 \mid d(c,x)=r\}$ where d($\cdot$,$\cdot$) denotes the hyperbolic distance. Find the length of…
Polymorph
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