# Tagged Questions

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.

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### Good data structure for hyperbolic tiling

Say you're doing something computational where each data point is a tile in a (not necessarily Euclidean) 2-dimensional tiling, for instance, a Life-like cellular automata. You might want a data ...
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### simplification of the area of a hyperbolic circle (BONOLA, S 53)

I'm trying to understand the S-53 of "Non-Euclidean Geometry" (BONOLA, R.) in which the formula for the area of a circle of radius r: $$2\pi k^2(\cosh\frac rk -1)$$ is somehow reduced by only applying ...
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### Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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### What's the connection between “hyperbolic” inner product spaces and the hyperbolic plane?

In Jacobson's Basic Algebra I, in Kaplansky's Linear algebra and geometry and in Artin's Geometric algebra, a hyperbolic plane is defined to be a two-dimensional, nondegenerate inner product space ...
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### Why are Euclidean and hyperbolic lengths proportional to first order?

In his book “Three-Dimensional Geometry and Topology”, Thurston constructs a Riemannian metric for the Poincare disk model and begins as follows. He says that, given any (hyperbolic) line segment $s$ ...
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### What is the hyperbolic plane equivalent to translation in euclidean space

in euclidean plane one can move polygons like rectangles, triangles etc. around by isometries, e.g. translations. For instance if we consider a rectangle with midpoint $0\in\mathbb{R}²$ then the image ...
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### Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
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### Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
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### Simple proof of existence of hyperbolic triangles

I've studied the hyperbolic plane by analytically building up the hyperboloid model, the Klein—Beltrami disc, the Poincaré disc, and the half-plane model from scratch. Now I'd like to prove that, ...
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### Geometries (Euclidean and Projective)

We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it's true in Coordinate Geometry. It makes ...
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### hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

On http://en.wikipedia.org/wiki/Hypercycle_%28geometry%29 I found the statement. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their ...
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### How to construct a circle in a the Poincare Disk model

How can I construct an circle with centre C going trough point P in a Poincare disk?. I found an script of how to do it in the "Poincaré Disk Model of Hyperbolic Geometry"toolkit from the geometers ...
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### Distance-preserving coordinate transformations for the poincaré disc

Following this question, I'm looking for a coordinate transformation which leaves distances unchanged. Does such a transformation exist? The isometries for the poincaré disk looked promising, but only ...
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### Representation of cell location in hyperbolic plane

I want to represent an order-5 square tiling (image from Wikipedia; more text below image): Obviously for a simple grid I can uniquely refer to a given square by its ...
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### What are Straight lines in the Gans Disk model of the Euclidean plane?

The answer of Blue ( http://math.stackexchange.com/a/1464/88985 ) to Hyperbolic critters studying Euclidean geometry made me interested in the Gans Disk model of the euclidean plane. Blue writes: ...
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### Hyperbolic Triangles and Uniform thinness

My textbook states that all triangles in hyperbolic space are uniformly thin in the following way: If $ABC$ is a triangle and $x$ is a point on one side, then there exists a point $y$ on one of the ...
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### Show how to map the semicircle $x^2 +y^2 = 1$, $y > 0$, onto $(x−1)^2+y^2 = 4$, $y > 0$, by a combination of $z \to z+l$ and $z \to kz$.

I need some help with this one! One can begin to understand the geometric significance of linear fractional transformations of the half plane by studying the simplest ones, $z \to z+l$ and $z \to kz$ ...
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### hyperbolic geometry proof with parallel lines

We are assuming hyperbolic geometry in this proof. Prove that for every line $l$ and external point P (im assuming point $P$ is not on line $l$), there are an infinite number of distinct lines ...
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### Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
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### Showing reflections are hyperbolic isometries in $\mathbb{D}$.

I am interested in showing that isometries in $\mathbb{D}$ are either conformal self-maps in $\mathbb{D}$ or they are compositions of conformal self-maps with $z\mapsto \bar{z}$. It is given that ...
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### Invariance of measure on upper half plane

The upper half plane has the measure $|y|^{-2}dxdy$. Show that it is invariant under the action of $SL(2, \mathbb{R})$. I don't understand what any of this means. First, I don't understand what they ...
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### Hyperbolic inversions are transitive on unit vectors at $x \in D$

Consider the Poincaré model in which the hyperbolic plane is the interior of a disk $D$, and a point $x$ in it with two vectors $v$ and $w$ of the same length attached. The reflection with respect to ...