# 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.

1,372 questions
Filter by
Sorted by
Tagged with
6 views

### how to find coordinates of a heptagons using the Poincare disk model? [closed]

I want to know how I can find cartesian coordinates of heptagons generated on a Poincare disk.
28 views

42 views

### Solving $R\space \sinh\frac{D}{R}=k$ for $R$

Does a solution exist for $R$ in this equation? I can't seem to solve it either analytically or numerically. $$R\space \sinh\frac{D}{R}=k$$
38 views

### How to derive relations between the sides and angles of equilateral hyperbolic triangles

I hope everyone in this community is staying safe, well and isolated. In this unprecedented situation I am starting to learn about some non-Euclidean geometry and explore down a fractal. In the ...
58 views

### When might Fenchel-Nielsen twist coordinates exceed 1/4?

When a compact Riemann surface of genus $g$ is cut up along $3g-3$ disjoint geodesic loops into $2g-2$ pairs of pants, the result is often described by giving Fenchel-Nielsen coordinates: one length ...
12 views

### How to find the dual point of a line geometrically?

I'm learning hyperbolic geometry by following Prof NJ Wildberger videos on YouTube. So far I know how to find the dual of a point (which is a line) geometrically (using pen, paper, and ruler). Here's ...
16 views

### Hyperbolic manifolds that are quotient of hyperbolic space

If $M=\mathbb{H}^n/\Gamma$ is a hyperbolic manifold ($\Gamma$ being a group of isometries of $\mathbb{H}^n$) can I conclude that it is complete? I know that if the covering map was a local isometry ...
115 views

### First uses of the Ping-Pong Lemma

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from ...
20 views

### Surfaces of non-constant negative curvature

Are there any nice models or books/papers with elementary discussions of surfaces with non-constant negative curvature, but negative everywhere, analogous to the Poincare disk for constant negative ...
45 views

### Geometric proof that the area of a hyperbolic triangle is proportional to its angle defect

The area of a spherical triangle with angles $\alpha$, $\beta$ and $\gamma$ on the 2-dimensional unit sphere is $\alpha + \beta + \gamma - \pi$. There is a nice geometrical proof of this fact that ...
26 views

### Model for $2$-dimensional manifold with constant negative sectional curvature

Let $M$ be a simply connected, complete manifold of dimension $2$ and constant sectional curvature $-k$ for $k>0$. I know I can model this manifold taking the circle of radius $1/\sqrt{k}$ and ...
12 views

### Particular configuration of planes and lines in three dimensional hyperbolic geometry

This is a question about three dimensional hyperbolic geometry. I don't know if there is any notion for this but let us call three lines H-parallel whenever they are pairwise disjoint, contained in ...
39 views

### area in hyperbolic geometry

suppose p is smaller or equal to q, and r is strictly greater than 0 and smaller or equal to s, where p,q,r,s are constants. how do I find the hyperbolic area of the region [p,q]x[r,s] in the ...
29 views

27 views

### For what reason straight lines must be on planes that go through the origin and how were the centers (origin) of the different geometries defined?

I've asked this to many mathematicians but I don't get a conclusive answer. Regarding origins (centers): - I understand that the origin in spherical geometry is the equidistant to all the points on ...
42 views

### Which automorphisms of the plane preserve the hyperbola?

Is there a reasonable characterization of all the "power-law" diffeomorphisms of finite order $f:\mathbb R^{>0} \times \mathbb R^{>0} \to \mathbb R^{>0} \times \mathbb R^{>0}$, of the form ...
22 views

### Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm?

Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm? Specifically, consider the poincare half-plane model of the 2d hyperbolic ...
16 views