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
2
votes
2 answers

Construction of equilateral triangle in Poincare disc model

Points A and B are given in Poincare disc model. Construct equilateral triangle ABC. Any kind of help is welcome.
user55529
  • 141
2
votes
1 answer

Is the hyperbolic paraboloid a model for hyperbolic plane?

I have been doing some outreach work on conveying notions of hyperbolic space to people with limited math backgrounds. One idea I like to use is to talk about the hyperbolic plane as being like the opposite of the sphere. For instance, if I start…
j0equ1nn
  • 3,429
2
votes
1 answer

On ideal quadrilaterals in hyperbolic geometry

Consider the ideal quadrilateral $Q_x$ comprising geodesics $S_1$ between $0$ and $\infty$, $S_2$ between $x$ and $\infty$, $T_1$ between $1$ and $x$, and $T_2$ between $1$ and $0$. We have hyperbolic isometries $A$ and $B$ that send $S_1$ to $T_1$…
WhySee
  • 724
2
votes
1 answer

Hyperbolic reflections and inversions in different models

I have a problem with understanding the following exercise. Consider a hyperbolic line l in the hyperboloid model given by l={x $\in P(R^{2,1})$|$\langle x,n\rangle =0, \langle x,x\rangle =-1 $ and $x_3 \gt 3$} where n$\in P(R^{2,1})$ with…
Polymorph
  • 1,225
2
votes
1 answer

Is this proof that there are circles in hyperbolic plane that cannot be inscribed in a triangle correct?

I want to show that there are circles in the hyperbolic plane, which cannot be inscribed in a triangle. Is it enough for me to say that, since the largest possible triangle is an ideal triangle and the largest radius for a circle inscribed in an…
yagod
  • 397
2
votes
1 answer

Translation in Poincare disc model

Suppose I have a set of points $x_i$, among them $x_0 = (1/2, 1/2)$ on a unit disk. What operation $F$ do I apply to those points, so that they would behave as points in Poincare disc model after such translation that $x_0$ end up in the middle of…
Jan Rzymkowski
  • 195
2
votes
1 answer

Right angles in hyperbolic pool

(This uses a bit of physics) So I learned today the following fact from physics: Imagine you have two pool balls of the same mass. You hit the first one, and it collides into the second. Then their velocities after the collision will be…
Nehsb
  • 358
2
votes
2 answers

How to draw Hyperbolic Circle in the Klein-Beltrami Disk

I'd like to know how to draw an hyperbolic circle in the Klein-Beltrami model just knowing its center and its radius, or a center and a point on it. Apparently, this hyperbolic circles are represented by ellipses as shown on this Wikipedia…
Kii
  • 267
2
votes
1 answer

Convex Ideal quadrilaterals are not all the same

In hyperbolic geometry, ideal triangles are all congruent to each other. Convex Ideal quadrilaterals ( a quadrilateral where all 4 points are ideal points) all have the same area $(2\pi)$ But I think for the rest they are not all congruent for…
Willemien
  • 6,582
2
votes
1 answer

Hyperbolic Rotation

I'm studying the transformations of the upper half plane model of Hyperbolic Geometry. I got stuck studying the elliptic isometries (which are the hyperbolic rotations). Can anyone help me to find the explicit formula of an elliptic isometrie?
0212user
  • 379
2
votes
2 answers

Aristotle's Axiom in Hyperbolic Geometry

I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs of an angle and make some construction based on…
roboguy12
  • 581
  • 2
  • 7
  • 18
2
votes
1 answer

Freeness of the group generated by two hyperbolic isometries

If $f$ and $g$ are two hyperbolic isometries of the hyperbolic space $\mathcal H^n$, we know that $f$ has 2 fixed points on $\partial \mathcal H^n$ and similiarly for $g$. Is it true that $f$ and $g$ generate a non-abelian free group if…
paf
  • 4,403
2
votes
1 answer

Poincaré cylinder

The Poincaré disk model of the hyperbolic plane is the open disk ${\rm int}(D^2)$ with a certain metric $d_H(x,y)$. What happens if I take the open tube ${\rm int}(D^2)\times\Bbb R$ with the…
Akiva Weinberger
  • 23,062
2
votes
1 answer

radii of horoballs

Consider the horoball $$B_h=\left\{(z,t)\in\mathbb{H}^3\mid t>h>0\right\}.$$ If $T\in\mathbb{P}SL(2,\mathbb{C})$ is an isometry of the hyperbolic space which does not fix the point at infinity, then $T$ maps $B_h$ to a horoball which is tangent to…
fatoddsun
  • 2,169
2
votes
2 answers

Formula for length of diagonal in a Lambert quadrilateral

Given a Lambert quadrilateral $AOBF$ where the angles $ \angle FAO , \angle AOB , \angle OBF $ are right, and $F$ is opposite $O , \angle AFB$ is the acute angle , and the Gaussian curvature = -1 (so all is standard) Then a whole lot of equations…
Willemien
  • 6,582
1 2 3
…
18 19