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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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Computing the hyperbolic distance between two points

In http://users.monash.edu/~jpurcell/papers/hyp-knot-theory.pdf, p. 22, it was left as an exercise that The hyperbolic distance between the endpoints of the perpendiculars on the line from 0 to $\infty$ is $|\ln|z||$ (exercise). However I was…
wilsonw
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Angles in hyperbolic space.

Ive been thinking about the following question. Let $p \in H^2,$ and $L$ be a complete geodesic in $H^2.$ Prove that there is a unique comlete geodesic $L'$ through $p$ and orthogonal to $L$ at some point $q \in L.$ Moreover, the line segment from…
Rafael Vergnaud
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Angle between lines in Hyperbolic Geometry

I have the following hyperbolic lines and I'm asked to find the angle between them. The lines are $ε_0 = \{ z \in \mathbb{C}: Im(z) >0, Re(z) = 0\}$ and $ε_{1,2} = \{z \in \mathbb{C}: |z-1| = 2\}$. With some algebra I found that they intersect in $w…
Alex Matt
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Mapping geodesic lines in hyperbolic space

Suppose we have two distinct $\gamma_1$ and $\gamma_2$, non-intersecting geodesic lines with disjoint endpoints on the hyperbolic plane. Assuming $\gamma_i$ cuts the hyperbolic plane into two half-planes $A_i$, $B_i$ satisfying $A_1 \subset A_2$ and…
math
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Finite index Fuchsian subgroup

Suppose I have a polygon $F$ and a group $G$ of isometries of the upper half plane, $\mathbb{H}^2$, such that $F$ tessellate $\mathbb{H}^2$ under the action of $G$. Can we always say that $G$ has a finite index discrete subgroup which acts freely on…
Arun
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Emulating distance on Poincaré disk for different curvatures

The distance $d_K^H(x,y)$ between two points on the hyperboloid $H_K$ with curvature $K<0$ can be emulated on the distance $d_{-1}(x,y)$ of the hyperboloid $H_{-1}$ of curvature ($K=-1$) as follows: $$ d_K^H(x,y)=R\cdot d_{-1}^H(x/R,y/R) $$ where…
ndrizza
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use the origin lemma to show that there exists a hyperbolic reflection that maps 2 perpendicular d-lines to 2 perpendicular diameters of $\mathscr D$

Use the Origin Lemma to show that, given two d-lines meeting at right angles, there is a hyperbolic reflection mapping them to two perpendicular diameters of $\mathscr D$. my work: case 1 : two d-lines are diameters, then the hyperbolic reflection…
Rico
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Orthogonal Projection of Point in Ambient Space onto Hyperboloid

Let the following denote the set of points on the $n$-dimensional hyperboloid manifold $$ \mathbb{H}^n_K=\left\{x\in\mathbb{R}^{n+1}\,\middle|\,\langle x,x\rangle_*=-r^2=\frac{1}{K} \,\land\,x_1>0\right\} $$ where $K<0$ is the sectional curvature of…
ndrizza
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Hyperbolic isometry and line segments

I was trying to apply Poincare's Polygon theorem, for that I had to give a pairing of sides, i.e., to have an isometry of the hyperbolic plane that will take a side of a polygon to another side (of the same length). So my question is that - Given…
Agneedh Basu
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Hyperbolic circles, horocycles and hypercycles

How do you write a formula for Hyperbolic circles, horocycles and hypercycles in Normal form give a point of rotation, or translation? Specifically, a hyperbolic rotation (hyperbolic circle) about the point 1/2 A parallel displacement (horocycle)…
Melissa Johnson
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Poincaré disk construction

I am trying to understand how the Poincaré disk is constructed using the stereographic projection for the hyperboloid $x^2+y^2-z^2=-1$. So I want to project a line from the fixed point $(0,0,-1)$ to a point on the hyperboloid. The parametric…
wwinters57
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Find the equation of the hyperbolic line through $A=(3,4)$, perpendicular to hyperbolic line $x^2+y^2=25$, $ y>0$

I don't know how to solve this exercise in Hyperbolic Geometry. Find the linear equation of hyperbolic line which passes through point $A=(3,4)$ and it is perpendicular to hyperbolic line with linear equation $x^2+y^2=25$ , $ y>0$.
Sotiris Z.
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Perpendicular bisectors of hyperbolic lines

I want to prove the following basic property of hyperbolic lines in $IR^{2,1}$. If x $\in$ $H^2$ and l is a line in $H^2$ then there is a unique line l' through x orthogonal to l. I want to prove this in the hyperboloid model. Let…
Polymorph
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Stabilizer of the orthogonal group of the Minkowski form acting on the upper sheet of hyperboloid space

I was studying this paper and on page 8 the author defines the orthogonal group associated to the Minkowski linear form in the usual way ( $A \in O(1,n) \Longleftrightarrow \langle Ax | Ay \rangle = \langle x|y \rangle \ \forall x, y \in…
Werner Germán Busch
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Are circles touching at the real line considered parallel in the hyperbolic plane?

In James Anderson's Hyperbolic Geometry, two lines in the upper half-plane model of the hyperbolic plane are said to be parallel if they are disjoint. Suppose $l_1$ is the half-circle centered at $(0, 0)$ with radius $1$, and $l_2$ is the…
nodim
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