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
1
vote
1 answer

How to draw fundamental domain for a group generated by 2 elements acting on upper hlf plane

It is easy to draw a fundamental domain for a cyclic group as in Chapter 1 of the book: "Geometry and Spectra of Compact Riemann Surfaces" by Peter Buser. However, I am very confused when trying to find a fundamental domain for a group generated by…
user416933
1
vote
1 answer

Show that the Ford domain may be seen as the Dirichlet domain around $z_0 \to \infty$ (Series 2010)

I am working through Hyperbolic geometry by C. Series, ch 5.2. I came across the following which I do not understand. Consider the upper half plane model of the hyprbolic plane, $\mathbb{H}$. We consider a discrete group $G \subset…
Slugger
  • 5,556
1
vote
0 answers

Proof of Theorem 5.2.8 in Foundations of Hyperbolic Manifolds, John G. Ratcliffe

I am a university student in Japan ,and study mathematics. I can't understand below "As each Euclidean line $L_\alpha$ of $B^n$ is mapped by $\phi$ onto a hyperbolic line $\phi (L_\alpha)$ of $B^n$ whose endpoints are a distance at most $r$ from…
Shuto Mitsuhashi
  • 11
1
vote
2 answers

Show that in hyperbolic geometry (k=1), an inscribed circle of a triangle must have a diameter less than ln(3).

I have been given a hint that based off of that $(\angle AIB')^r+(\angle BIC')^r+(\angle CIA')^r=\pi$ and that each of these three angles is less than $\Pi(r)$. Using this information I must then apply the Bolyai-Lobachevsky formula to find x such…
dbmckie
  • 11
1
vote
2 answers

Isometry of $\mathbb H^2$ determined by its action on $\partial H^2$.

On pg. 282 of Farb and Margalit's A Primer on Mapping Class Groups, the following is mentioned: An isometry of $\mathbb H^2$ is determined by its action on $\partial \mathbb H^2$. I think here $\partial \mathbb H^2$ refers to the Gromov boundary…
caffeinemachine
  • 18,510
1
vote
0 answers

hyperbolic distance: definition or calculation

the hyperbolic distance d(z1,z2)d(z1,z2) is the absolute value of the log of absolute value of the cross ratio between z1,z2z1,z2 and the two points of the h-line that goes through z1,z2z1,z2 Why is this so? Why is this the definition of hyperbolic…
jan roelens
  • 11
  • 1
1
vote
0 answers

Special hyperbolic triangle

Can we visualize from any book/report reference a spiky curvilinear triangle with each vertex angle zero and area (by Gauss-Bonnet theorem) $\pi a^2$ on any surface of constant Gaussian curvature $K=-1/a^2$ in $\mathbb R^3 ?$ Can portions of it…
Narasimham
  • 40,495
1
vote
0 answers

Help me understand these equations for Schwartz triangles in Spherical/Hyperbolic geometries

I am trying to understand the following formula (from this shader). For integers (p q r), it creates $\vec{a},\vec{b},\vec{c}$, normal vectors to planes that define a Schwartz triangle with the correct interior angles when intersected with a) the…
Roy
  • 121
1
vote
0 answers

Four Colour Theorem in Hyperbolic Space

I'm currently working on a specialized game board, which will be a hyperbolic plane tiled with octagons, as a personal that I'm getting someone to crochet for me. Right now, the plan is to crochet a bunch of octagons and stitch them together. My…
1337w0n
  • 43
1
vote
1 answer

How to construct the angle bisector at an ideal point?

In the Poincare disk model I have two asymptotic lines (lines going towards the same ideal point) and I want to construct their angle bisector (or the line equidistant to both, or the line in who's reflection they change place ) How can I construct…
Willemien
  • 6,582
1
vote
1 answer

Shape of pseudosphere beyond cusp including frills

So far we have shape of pseudosphere defined below/upto its cuspidal boundary/edge. However, FullPseudosphereModelBeltrami has detail around (what appears to me) material added outside cusp radius that appears with small amplitude frills. Daina…
Narasimham
  • 40,495
1
vote
0 answers

Ratcliffe's Text- Theorem 12.3.4 Conical limits points and Euclidian boundary

QUESTION: So I tried to see what ratcliffe meant here, but I don't get it. Namely, how does he get a contradiction at the end. I don't see how $P$ being locally finite gives us a contradiction. I assumed that Ratcliffe wanted a contradiction by…
Enigma
  • 3,909
1
vote
1 answer

perpendicular bisector of a hyperbolic line

Given $P=1+2i$ and $Q=3+4i$, find the equation of the perpendicular bisector of the hyperbolic line segment $[P, Q].$ I used the approach given in Groups and Geometry by Lyndon where you get the set of points $d(P, X) = d(Q, X)$. Using $X = u +…
hyunst
  • 69
1
vote
1 answer

What's the difference between hyperbolic length and hyperbolic distance?

From what I understand, the hyperbolic distance between two points in hyperbolic space is the length of the line(semicircle) that connects those two points. The hyperbolic length then would be the sum of all the little tangents on any curve…
user338365
  • 327
1
vote
1 answer

Angle of rotation and center of a complex map

Let $\alpha (z)= \frac{z-1}{z+1} $ be a map on the upper half plane $\mathbb{H}$. What is its fixed point? What is the center and angle of this rotation? I already computed its fixed point and I got $i$. It has only one fix point which lies in…
hyunst
  • 69
1 2 3
…
18 19