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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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What is the relationship between hyperbolic geometry and Einstein's special relativity?

I am a third year math student writing a term paper on hyperbolic geometry and I would like to understand its relationship with special relativity. I have read that the hyperboloid model of hyperbolic geometry, also known as the Minkowski model,…
Sid
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Distance in the Poincare Disk model of hyperbolic geometry

I am trying to understand the Poincare Disk model of a hyperbolic geometry and how to measure distances. I found the equation for the distance between two points on the disk as: $d^2 = (dx^2 + dy^2) / (1-x^2-y^2)^2$ Given two points on the disk, I…
Chris Johnson
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Geodesic Uniqueness in the Hyperbolic Plane

I am studying Hyperbolic Geometry. At this part, I have proved that semicircles and straight lines orthogonals to the real axis are geodesics in the hyperbolic plane. But how I proof that this geodesics are uniques? That it does not exist others…
Pierre
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How does one apply the formula for $(ds)^2$ on the hyperbolic half-plane?

We commonly say that the metric on the upper half-plane model of the hyperbolic plane is given by $$(ds)^2=\frac{(dx)^2+(dy)^2}{y^2}.$$ (There are a lot of articles that start this way and I've been guilty of writing it myself.) The $ds$ means the…
j0equ1nn
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Hyperbolic metric on the torus?

Here is a silly mistake I am making: where exactly is the mistake? I know that torus cannot hold a metric of constant curvature -1 ( hyperbolic metric ). But what if I do this: The upper half-plane and $\mathbb{C}$ are diffeomorphic by a…
Learning Math
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how to construct a line on a poincare disk?

Given the Euclidean coordinates of two points (p1, p2) and (q1, q2) in the unit circle, how do I construct the Euclidean circle x^2 + y^2 + fx + gy+1 representing the hyperbolic d-line on the poincare disk containing these points?
nilo de roock
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Where does the hyperbolic metric come from?

In hyperbolic geometry, the metric is often defined as $$ds=\frac{\sqrt{dx^2+dy^2}}{y}$$ Where did this metric come from? I have thought long and hard about this question, but have no satisfactory answer.
algebraically_speaking
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Construction of Hyperbolic Circles With a Given Radius

In the Poincare Disk Model, Hyperbolic circles (i.e. the locus of all points with a given distance from a center point) are also circles in the euclidean sense, but with the euclidean center different from the hyperbolic center. My question is,…
Nathaniel Bubis
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Reflections generating isometry group

I was reading an article and it states that every isometry of the upper half plane model of the hyperbolic plane is a composition of reflections in hyperbolic lines, but does not seem to explain why this is true. Could anyone offer any insight?…
Fredrick Henry
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Mapping the Poincare disk model to the Poincare half plane model

I am puzzeling with the following: Given a point $ A = ( a_x, a_y) : a_x^2+ a_y^2 \le 1 $ in the Poincare Disk model ( https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model ) to which point does this point map on the Poincare half plane model.…
Willemien
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The law of sines in hyperbolic geometry

What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the constant only in Euclidean and spherical…
Apotema
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Interpretation of Hyperbolic Metric and Möbius Transforms

I was wondering if someone could explain the interpretation of the following results. In hyperbolic geometry, we say that lengths are invariant under the action of Mob($\mathbb{H}$) if given any piecewise-differentiable curve $f:[a,b]\rightarrow…
user38268
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Poincaré disk with hyperbolic metric is a metric space

I am trying to prove that the Poincaré disk $D=\left\{ z \in \mathbb{C} : |z|<1 \right\}$ equipped with the hyperbolic metric given by $d_{D}(z_1,z_2)=\inf \{ L_{D}(\gamma) \mid \gamma \text { is a continuously differentiable path with endpoints } …
wwinters57
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Is two-dimensional hyperbolic geometry unique up to isomorphism?

This may have been answered here somewhere, but I'm unable to find it. Is 2D hyperbolic geometry is unique up to isomorphism (or up to whatever's appropriate)? I know there are at least four models -- Poincare disk, half-plane, etc., but I assume…
Robert Gethner
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Mirrors in a hyperbolic space

Consider a space with hyperbolic geometry (instead of Euclidean). We know that two objects are similar iff they are congruent. It is said that you can't capture a personal $3\times 4$ picture of yourself in such spaces:). I am interested to know if…
Qurultay
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