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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
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Connected components of a subset of E

Let $E$ be a real vector space of dimension n+1 with a symmetric bilinear form B of signature (n,1). Let $H=\{x \in E : B(x,x) <0\}$. Somewhere I saw that it has two connected components. Can anybody please tell me what are its connected components?
budi
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hyperbolic confusion: Is an apeirogon even a (closed) polygon?

Via Tesselation of the upper half plane via Ford Circles I was introduced to Ford circles ( https://en.wikipedia.org/wiki/Ford_circle note the wikipedia article has been updated since that question) Ford circles made it possible for me to…
Willemien
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Product of two elliptic isometries with distincts centers

I'd like to know why is the product of two elliptic isometries of the hyperbolic upper plan (or of the unitary disk) with distincts fixed points is parabolic or hyperbolic? PS: I only need it for dimension $2$ $ie$ in $PSL(2,\mathbb{R})$ Thank you!
WrabbitW
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Parallelism preservation of hyperbolic rigid motions on the half plane model

I need to proof (Under Hilbert axiomatization) that hyperbolic rigid motions, with respect to the metric $ d:\mathbb{H}^2 \times\mathbb{H}^2\rightarrow\mathbb{R}: d(A,B) =\left| \log \left( \frac{|AA_{\infty}| \ |BB_{\infty}|}{|AB_{\infty}| \ …
user286485
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Prove $A \cdot B\leq -1$, where $A$ and $B$ are in $\mathbb{H}^2$

Let $A$ and $B$ be in $\mathbb{H}^2$. I need to prove that the lorentzian dot product between $A$ and $B$ is less than or equal to $-1$. I have no idea where to start.
John.P
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Hyperbolic coordinates

You can uniquely specify any point in 2D Euclidian space using 2 numbers: the distance from the infinitely long X-axis, and the distance from the infinitely long Y-axis. How do you uniquely specify a point in 2D hyperbolic space? Can you do it with…
MathematicalOrchid
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When a ray of an horocircle passing through the origin intersects the y axis.

In the following figure, $h(A,B)$ is an horocycle centered in A passing over B. $\Theta(h)$ is the angle of parallelism of the segment $h$ and $S$ is the well known intersection of a chord of an horocycle centered in $\Omega$ passing over the origin…
user286485
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Tiling on Poincaré disc

Is there anyone to help me tile on a Poincaré disc? In fact, I'm going to tile triangle tiles on a surface in hyperbolic geometry ; is there any algorithmic method to do so?
Seeen Jim
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When $\cosh yx/2=\pm 1 $?

When $\cosh \frac{xy}{2}=\pm 1 $? is it correct to say $xy/2=cosh^{-1}(\pm1)$ Then $xy=2 \cosh^{-1}(\pm1)$ I think there is better solution for this problem? any idea?
Tony
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Poincaré disk model of hyperbolic plane

Can someone please explain trigonometry in Equations (2) to (8) of: PoincareDisk ?
Narasimham
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What is the largest possible sum of all the angle measures of a $\Delta$ in hyperbolic space?

$\Delta ABC$ exists in hyperbolic geometry. What is the maximum value for $m\angle A+m\angle B+m\angle C$?
user253055
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Hyperbolic segment from $(0,0)$ to $(0,0)$

Can there be a segment on a hyperbolic plane that goes from point $(0,0)$ to $(0,0)$ in the hyperbolic plane. There are some rules, though for this to work: 1) The segment must apply to the rules of segments in hyperbolic space. 2) The segment a…
user253055
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Relationship between Hyperboloid model of hyperbolic space and disc model / confused by a picture.

I am confused by this picture: https://en.wikipedia.org/wiki/File:HyperboloidProjection.png What is wrong with projecting from the origin, and using the disc at $t = 1$? After doing some computation to verify my gut, it doesn't seem like the…
Elle Najt
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circumscribe a regular polygon around a circle in hyperbolic geometry

In the hyperbolic plane, let a circle of radius r be given. If we want to circumscribe a regular polygon with n sides around this circle (i.e., if we want the sides of the polygon to be tangents of the circle), how large must n be?
Elizabeth Hill
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Hyperbolic quadrilaterals : Opposite sides of the quadrilateral cannot intersect

Suppose that a hyperbolic quadrilateral $ABCD$ satisfies $h(A, B) = h(C, D), h(B, C) = h(A, D)$. Mark each of the following claims about the quadrilateral as true or false: Opposite angles of the quadrilateral are supplementary. (FALSE) Adjacent…
Mark
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