Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
1 answer

Projection of point onto closest point on geodesic in hyperbolic geometry (hyperboloid model)

Lets say we have a point $p\in\mathbb{H}^n$ in hyperbolic space (with curvature 1). And a geodesic, starting at the origin $(1,0,\ldots,0)$ in direction $\mathbf{v}$. What I'd like to do, is to find the point $v^*$ on the geodesic described by…
ndrizza
  • 1,348
3
votes
1 answer

Given a hyperbolic triangle's sides (or angles), is there an easy way to determine whether it is inscribed in a circle, horocycle, or hypercycle?

If I have a hyperbolic triangle, specified by edge lengths or angles, is there an easy way to determine whether it is inscribed into a circle, a horocycle, or a hypercycle?
Marek14
  • 883
3
votes
0 answers

Hyperbolic Quadrilateral

Suppose we are given four angles $\alpha$, $\beta$, $\gamma$ and $\delta$. For a quadrilateral to exist with interior angles $\alpha$, $\beta$, $\gamma$ and $\delta$ in the hyperbolic plane we must have $\alpha +\beta+\gamma+\delta< 2\pi$. But what…
P.S
  • 550
3
votes
1 answer

Projection on geodesic lines in $\mathbb{H}^n$

Good morning everyone, I was wondering wether or not is the projection on a geodesic line in $\mathbb{H}^n$ $1$-lipschitz for the hyperbolic distance. I asked myself this question because i ran into the following result : if $a,b\in \mathbb{H}^n$…
Selim Ghazouani
  • 2,234
  • 15
  • 24
3
votes
2 answers

Hyperbolic Crown

A $\textit{hyperbolic crown}$ is a hyperbolic annulus bounded by a closed geodesic $C$ on one side, and a chain of bi-infinite geodesics on the other. Each adjacent pair of bi-infinite geodesics bounds a “boundary cusp”. Show that a hyperbolic…
WhySee
  • 724
3
votes
1 answer

Approximating Distances in the Hyperbolic Space

I am currently trying to understand the paper by Krioukov et. al. on hyperbolic networks, but since I do not have a background in hyperbolic geometry (or, in that sense, in geometry at all) I struggle to understand some points they are making. For…
HdM
  • 215
3
votes
2 answers

Area of an hyperbolic triangle made by two geodesic and an horocycle

Let D be the disk Poincare' model for the hyperbolic plane. Let A be a point on the boundary $\partial D$ of the disk, consider two geodesics emanating from A and ending in other two points B and C of the boundary, in addition consider an horocycle…
giulio bullsaver
  • 197
3
votes
1 answer

Gauss Bonnet theorem validation with hyperbolic circles

How to verify the theorem in case of a hyperbolic circle radius $r$ for constant negative Gauss curvature $ K=-1/a^2 $ and constant geodesic curvature $k_g $ ... e.g., like here... $$k_g=1/r \tag1 $$ $$ \int k_g ds + \int\int K dA = 2 \pi \tag2…
Narasimham
  • 40,495
3
votes
1 answer

Length and area in hyperbolic geometry

I am reading a book about modern geometries by Michael Henle. He gives formulas for length of a curve and area of a region (in upper half plane model: $l(\gamma)=\int _a^b \frac{|z'(t)|}{y(t)}dt, A(R)=\iint_{R} \frac{dxdy}{y^{2}})$ as definitions.…
Idan
  • 782
3
votes
1 answer

Hyperbolic circles are euclidean circles in the Poincaré Half Plane Model

Consider the metric space $(\mathbb{H}²,d_{\mathbb{H}^2})$, where $d_{\mathbb{H}^2}$ is the hyperbolic Cayley Klein metric, i.e., $ d_{\mathbb{H}^2}(A,B) = |log ((AA_{\infty}. BB_{\infty}) / (BA_{\infty}. AB_{\infty}))| $ Here $A_{\infty}$ and $…
user286485
3
votes
1 answer

Triangle inequality for hyperbolic metric of logarithm of cross ratio.

Consider the Poincaré Half-Plane model of the Hyperbolic Space $ \mathbb{H}^2 $. I need to proof that the following d function is a metric.$ d:\mathbb{H}^2\rightarrow\mathbb{R}, d(A,B) = \big{|} log…
user286485
3
votes
2 answers

What about the Fuchsian groups make them stand out?

Why do we stop at Fuchsian groups (I.e. discrete subgroups of automorphisms of the hyperbolic plane) when we study things like quotients and what not? Is there a maximalist or universality property behind that distinction?
user98246
3
votes
1 answer

How to solve an hyperbolic Angle Side Angle triangle?

If from an hyperbolic triangle $ \triangle ABC$ the angels $\angle A$, $\angle B$ and side $c$ are given. (ASA triangle, a side and both adjacent angles are given) How can I calculate the remaining angles and sides? (or at least one of them) In…
Willemien
  • 6,582
2
votes
1 answer

Exponential map in the Poincaré upper half plane

I have a question regarding the Poincaré upper half plane. Is there a simple way to express the exponential map? I have been looking unsuccessfully on internet for an expression... Thanks for any help on how to get one.
Chevallier
  • 1,062
2
votes
2 answers

Showing geodesics in $\mathbb{H}^2$ have unique common perpendicular or common endpoint

If we let $\mathbb{H}^2$ be the hyperbolic plane and we let $\gamma_1,\gamma_2$ be geodesics which do not intersect. I have a question which asks me to show that either $\gamma_1$ and $\gamma_2$ have a unique common perpendicular or that they have a…
hmmmm
  • 5,616
1 2 3
…
18 19