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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.

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Hyperbolic Regular Polygons: Construction and Visualization

I am working on visualising a polygon in a hyperbolic space using a HyperbolicRegularPolygon class in SageMath, which constructs regular polygons in the hyperbolic plane. The construction algorithm involves several steps: Algorithm: URL Compute the…
Rowing0914
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Understanding Assumed Angles in Umemoto's Hexagon Subdivision

In Umemoto's thesis 1 on Dirichlet fundamental domains for Fuchsian groups, Theorem 24 involves assumed angles for subdivided hexagons in the proof (Fig.18 on page 35). However, the rationale behind these angles is unclear. For instance, I thought,…
Rowing0914
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Visualisation of isotopy

How can I visualise the meaning of isotopy that appears while defining Teichmuller space? Can you suggest a picture where two maps are not isotopic? I want more clarification about isotopyic maps. Isotopy is sometimes referred to as homotopy via…
Subash Chandra Behera
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How is the Poincare disc model or the upper half plane model a representation of hyperbolic space?

I am currently working on these two models and I don't understand the connection between them and hyperbolic space. In case of spherical geometry one can imagine everything well as a 2 dimensional sphere in 3 dimensional space and all the major…
struggling_student
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Question on a proof of convexity of Poincare metric

Let two mutually disjoint geodesics $L_1$ and $L_2$ in $\Bbb H$ be given, where $\Bbb H$ is an upper half plane in $\Bbb C$ with hyperbolic metric $\rho$. Then $\rho(z,w)$ is strictly convex on $\{(z,w)\in\Bbb C^2\mid z\in L_1,w\in L_2\}$.…
one potato two potato
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Radius of inversion circle in the Poincare Disk model

Consider two points $P$ and $Q$ inside the Poincare disk, at distances $r_1$ and $r_2$ respectively from the origin $O$. Furthermore, $P$ and $Q$ form an angle of $\theta$ at the origin. Is there any closed form result for the radius of the…
Saurav
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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 interpretation of Teichmuller space as a representation…
user1138886
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Uniqueness of translations in hyperbolic plane

I am self-studying hyperbolic geometry and am stuck on the following. Let $\mathbb{D}$ be the Poincare disc model of the hyperbolic plane with $[3^7]$ tiling on it. Let $T$ be a hyperbolic translation which is also a symmetry of the tiling such that…
KAK
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Angle sums in the upper half plane model

I am a little confused. I see many sources state that the sum of the angles in a hyperbolic triangle is always less than $\pi.$ Yet in the upper half plane model, hyperbolic triangles are represented by triangles with arcs of circles for sides, and…
user960774
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Isometry from Half-Plane metric to Poincare Disk metric

If I take the metric of the upper half-plane model $$ (ds)^2=\frac{(dx)^2+(dy)^2}{y^2} $$ there is obviously an isometry which should provide the metric of the Poincare disk model $$ (ds)^2=…
apg
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barycentre in hyperbolic geometry :

In euclidean geometry we know the formulas for midpoint and barycentre of a finite set of points, so can we find similar formulas in hyperbolic geometry ? In the Klein disk, Ungar cited in his book "A gyrovector space approach to Hyperbolic…
M-S
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In poincare disc model we are given two points A and B. Find point C such as ABC is right angled triangle

Here is the answer: Let l be the h-symmetry of OA (O is center of absolute). In respect to l, A goes to O and B goes to some point B'. Now we construct a circle c(0,B') Then we apply l again and c(O,OB') goes to c(A,AB) We now find h-symmetry of OB…
JurgenKlop
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A concurrency of Euler lines

I have obtained the concurrency of Euler lines in Poincare disk with Geogebra. I want to know its synthetic proof.
seiichikiri
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General Method to find area of super triangle

Is there a general way to find a area of a super triangle? I know the definition of a super triangle is the following: In the disk model of hyperbolic plane, the area bounded by three points M, P, and Q is called a super triangle if at least one of…
mathwizz123
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Structure of Euclidean boundary of Dirichlet region having infinite area

Let $\Gamma$ be a Fuchsian group. Choosing any point $p\in \Bbb H^2$ not fixed by any non-identity element of $\Gamma$, we can construct a hyperbolically convex connected fundamental region, called Dirichlet region, denoted by $D_p$. Also, $\partial…
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