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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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Show that the distance ρ, between two points with the same ordinate on the lines x = 0 and x = 1, goes to 0 when y approaches infinity.

I want to show that in the half-plane model the distance ρ, between two points with the same ordinate on the lines $x = 0$ and $x = 1$, goes to $0$ when $y$ approaches infinity. I need to show that lim f(n) = 0, but I do not know hot to get…
Hanna T.
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Hyperbolic circles in the hyperbolic model

Let H be the hyperboloid model for the Poincare' disk. Geodesics of H are given by intersections with H of planes { w | $$ = 0}, v being a space like vector Is it possible to represent hyperbolic circles as intersection of planes with the…
giulio bullsaver
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Hyperbolic plane - Asymptotic lines Proof

"Prove: If two lines in the hyperbolic plane are asymptotic, then they do not admit a common perpendicular" I'm trying to use something related to the angle of parallelism, but I guess it's not the right thing to do. Can someone suggest something? I…
Dalton
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Points on the sides of hyperbolic triangle

Let D be a triangle in a hyperbolic plane. Prove that there exist points A,B,C on three different sides of D such that diameter of the set {A,B,C} is bounded by a constant which is independent from the choice of D
Patricia
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modify the measure distance of poincare disk

I'm a new guy studying the distance metric in Poincare Disk model. The measure distance between two random points u and v on the Poincare disk is described as follows: $$ d\left( \mathbf u, \mathbf v \right) = arcosh \left( 1+2\frac{||\mathbf u-…
qiang zhang
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A way to go from the Hilbert Plane and its axioms + the hyperbolic axiom to a surface of constant negative curvature

It is clear to me that a surface of constant negative Gaussian curvature satisfies the hyperbolic axiom (more than one 'straight' line not meeting a given 'straight' line). Hartshorne (Geometry: Euclid and Beyond p. 374) defines the hyperbolic…
luysii
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Regular polygons in non-euclidean geometries

How is a regular hexagon described in elliptic and hyperbolic geometries in $\mathbb R ^3$?
Narasimham
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Why doesn't Mob(H) act transitively on the set P of pairs of distinct points of H

Where H is the hyperbolic plane. I am stuck because we know mob(H) acts transitively on the set of hyperbolic lines and we we know two points make up a hyperbolic line so Why doesn't Mob(H) act transitively on the set P of pairs of distinct points…
MathIsHard
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Hyperbolic distance between two coordinates

Find the hyperbolic distances between the following pairs of points ( 4/3, 0,5/3) , (3/4, 0,5/4) ∈ H^+ where H^+ = {x,y,z ∈ R^3 | z^2-y^2-x^2=1 and z>0} Own work: The formula I have been given is d(w,z) = ln({1 + |(w-z)/(1-($\bar…
user407151
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Show that any side of a hyperbolic triangle in the hyperbolic plane lies in the closed $δ$-neighborhood of the union of the two other sides.

Prove that there exists a constant $δ ∈ [0,∞)$ such that the following holds. Let T be an arbitrary non-degenerate hyperbolic triangle in $\mathbb D$ with vertices $a, b, c ∈ \mathbb D$, and let $p_a ∈ [b, c]$,$ p_b ∈ [a, c]$ and $p_c ∈ [a, b]$ be…
user2973447
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hyperbolic distance of non-elliptic möbius transformation via trace

Reading through a book of Farb & Magalit (p. 271/in pdf 288 shortly before Prop 10.3) I stumbled upon (after some cleaning) the following equation for hyperbolic metric: $dist(z, Az)=cosh^{-1}(trace(A)/2)$ for some non-elliptic $A \in…
ctst
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Calculating the area bounded by 2 hyperbolas

I have some trouble calculating the area bounded by two hyperbolas(in the first quadrant) in a analytical way. The functions are: $y=\sqrt{a^2+cx^2}$ $x=\sqrt{a^2+cy^2}$ I've tried hyperbolic substitutions, but it did not lead to an elegant…
R.Y
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Does 3 points on Poincaré Disk geodesic lie on the same Poincaré Half Plane geodesic?

This may be a trivial question, IMO the answer should be yes. Given a geodesic $\delta$ on the Poincaré Disk's model with $A, B, C \in \delta$ And given that $f(x)$ is an isometry from the Poincaré Disk to the Poincaré Half-Plane. Do $f(A)$, $f(B)$…
Kii
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Showing that the inversion of a circle can be written in a certain way

Let $\phi = \dfrac{1}{r} ∘ T_{-O_C}$, $O_C$ the center of $C$, $T_{-O_C}$ is a translation. I want to show that the inversion of a circle $C \in \mathbb{C}$ can be written as: $$\iota_C = \phi ∘ \iota ∘ \phi^{-1}$$ I have to show 3 things:…
aribaldi
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Triangle on Beltrami pseudosphere with angle sum $180^\circ$

What characteristic lines on the pseudosphere can form a triangle whose internal angle sum is $180^\circ$?
Narasimham
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