# 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.

1,786 questions
Filter by
Sorted by
Tagged with
32 views

1 vote
13 views

1 vote
61 views

### How does the hyperboloid model relate to "A Universal Model for Hyperbolic, Euclidean and Spherical Geometries"?

I just found A Universal Model for Hyperbolic, Euclidean and Spherical Geometries, after reading the HyperRogue game dev notes where it said the hyperboloid model (aka the Minkowski model) was the ...
• 3,472
45 views

### What is special about the "pentagrid" and "heptagrid" in Margenstern's work on Cellular Automata in Hyperbolic Spaces?

In his work he mainly focuses on the pentagrid {5,4} and heptagrid {7,3}: In what ways are these tilings special? How do they compare to hyperbolic tilings in general? I am wondering what insights ...
• 3,472
40 views

### Do we have a standard coordinate system for hyperbolic tessellations?

I am specifically interested in implementing animations for Cellular Automata in the Hyperbolic Plane. I have seen Coordinate systems for the hyperbolic plane on Wikipedia, but a lot of what Professor ...
• 3,472
17 views

### A brief explanation of Isometries and Mobius Transformations used in animating Hyperbolic Cellular automata?

I've spent the past while digging into the code I asked about in Where to begin with animating over a 2D hyperbolic tessellation?, my answer there is in regards to digging into the MagicTile project's ...
• 3,472
54 views

### Examples of hyperbolic and non-hyperbolic space for quasi-isometric spaces

Let $X$ and $Y$ are quasi-isometric spaces. I try to find an example for which one of these spaces will be hyperbolic, other is not hyperbolic. I know that for geodesic metric space if one of the ...
• 391
58 views

### In a Hypergeometric Distribution CDF with everything else held constant, should K be a linear function of N?

Using the Hypergeometric Distributon notation from Wikipedia, if I treat $k$, $n$, and $\Pr(X\ >\ k)$ as constants and solve for $K$ as a function of $N$ in Mathematica, the relation appears to be ...
• 1,554
1 vote
35 views

### How does curvature relate to angle measurement in hyperbolic geometry?

This question is about the relationship between curvature and angle measurement in hyperbolic geometry... Specifically, I am trying to understand the following excerpt from pp. 489-490 of Greenberg's ...
• 179
68 views

### Surface Group Representations

I am interested in Hyperbolic Geometry. I studied hyperbolic surfaces, the space of all marked hyperbolic structures on a surface (also known as the Teichmuller space of the surface), and the ...
• 107
1 vote
63 views

### Metric Derived From Differential on Hyperbolic Plane

I'm reading Katok's Fuchsian Groups, and I'm confused on how the metric on the unit disk model is derived from the differential $$ds = \frac{2|dz|}{1-|z|^2}.$$ To be more specific, we first have the ...
• 425
21 views

### Volume of hyperbolic submanifold of surface with a boundary component

Let $\Sigma$ be a compact surface of genus $k \geq 2$ having a single boundary component. Let $U \subset \text{Int}(\Sigma)$ be an open subset of the interior of $\Sigma$ with a Riemannian metric $g$ ...
• 185
41 views

### Volumes of hyperbolic submanifolds of closed surfaces

Let $\Sigma$ be a closed orientable surface of genus $k \geq 2$. Suppose $U \subseteq \Sigma$ is an open subset with a Riemannian metric $g$ on $U$ such that (1) the Gaussian curvature $K$ of $g$ is ...
• 185
43 views

### Is every loop on a hyperbolic surface freely homotopic to a geodesic?

Let $(S,g)$ be an orientable Riemannian 2-manifold having constant Gaussian curvature $K=-1$ and $\gamma$ a loop on $S$. Is $\gamma$ freely homotopic to a geodesic? Note the lack of completeness ...
• 185
39 views

### Geodesic curvature on hyperbolic manifold with boundary

Let $\Sigma$ be a compact oriented surface of genus $1$ having a single boundary component (i.e. $T^2$ minus an open disk) and let $g$ be a Riemannian metric on $\Sigma$ with constant Gaussian ...
• 185
1 vote
49 views

### Within hyperbolic space, are all sides of an ideal triangle parallel?; and is it possible for them all to be hyperparallel?

Question: I unfortunately have an extremely limited foundation in mathematics but I am trying to wrap my head around hyperbolic geometry in simple terms and I have spent all day trying to search for ...
37 views

### Algorithms for drawing hyperbolic tilings

I was looking at hyperbolic tilings on the Poincare disc model like these the other day, and I wondered how I might make my own. I have a basic understanding of what hyperbolic space is and how it ...
• 355
### Is $\text{PSL}(2,\mathbb{R})$ a semisimple Lie group? 