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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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L is a family of hyp-lines passing through a pt. Not sure how this implies rr'=(c−Re(p))c'

Lemma 1. Let $p \in \mathbb{H}$, and assume $l$ is a family of hyp-lines passing through $p$ such that $l$ is of the form $l = \{c +re^{i\theta} | 0 < θ < π\}$. For simplicity, assume the functions $c : \mathbb{R} → \mathbb{R}$ and $r : \mathbb{R} →…
nando
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Show that if a Mobius transformation, $m$, is parabolic then it's displacement, $disp(m)=0$

For any element m of $Mob^+(\mathbb H)$ other than the identity, define its displacement $disp(m)$ to be $disp(m) = inf\{d_\mathbb H(z, m(z)) | z ∈ \mathbb H\}$ Show that if $m$ is parabolic then $disp(m)=0$ (m is parabolic if it has one fixed…
user2973447
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Proving perpendicular lines in hyperbolic geometry

Suppose ABO is an asymptotic triangle and angle A is congruent to angle B. If M is the midpoint of the finite side AB, prove that line MO is perpendicular to AB. I constructed this such that O is the point lying on the fundamental circle and points…
user384316
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Relation between one-sheet and two-sheet hyperboloid, and hyperbolic space

I'm trying to get an intuitive grasp of the relations between the one- and two-sheet hyperboloids and the two-dimensional hyperbolic sphere. Anthony Zee uses the two-sheet hyperboloid $x^2+y^2-z^2=-1$ to derive the two-dimensional hyperbolic sphere…
Andy Miles
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Finding a Mobius transformation which takes any 2 points in upper half plane to points with same imaginary part

I have done most of the below question but I am struggling with the justification of the hint given.The problem is: Given distinct points $z$ and $w$ in $\mathbb H$, we define the perpendicular bisector of the hyperbolic line segment joining $z$ and…
user2973447
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Hyperbolic straight line segment of shortest length is perpendicular

Given a hyperbolic straight line $C$ and a point $p$ not on $C$, the hyperbolic straight line segment of shortest length connecting $ p $ and $C$ is perpendicular to $C$. I've tried considering the hyperbolic straight line segment connecting $-1$…
james b
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How to find distance from emitter using TDOA?

I have the three receivers set up: $A (0.5, 0)$ $ B (-0.5, 0)$ $ C (0, 1)$ I know that the signal arrives at $C$ first. It then arrives at $B$ $2.63464*10^{-4}$ later. It finally arrives at $A$ last $7.07023 * 10^{-4}$ seconds after it arrives at…
the fool
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Hyperbolic Distance Confusion

On the hyperboloid model for hyperbolic space, the hyperbolic distance between the two points is NOT the Euclidean length of the geodesic connecting those two points. So my question is, if hyperbolic distance doesn't measure that, then what DOES it…
Eric Terry
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In hyperbolic geometry, exactly how big is a dodecahedron composed entirely of right angles?

Specifically, I need to know the distance from the center to the vertices, and the distance to the faces.
DanielLC
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fixed point of a parabolic commutator

For a hyperbolic isometry $g$ of the hyperbolic plane, denote by $ax(g)$ the oriented geodesic running from the repelling fixed point $p_g$ and the attracting one $q_g$. Let $G$ be the fundamental group of a hyperbolic once-punctured torus and…
fatoddsun
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hyperbolic trigonometry when one angle =0

The hyperbolic trigonometry functions don't really help when you have one angle =0 (the remaining lenght of side $AB$ becomes ${\infty}-{\infty}$ ) Given a triangle $\triangle AB \Omega$ with $\angle \Omega =0 $ then by this fact alone we get $…
Willemien
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Rotate a point around another point in the Poincaré Hyperbolic Disk

Suppose I have a point $P = (x_1,y_1)$ in the Poincaré disk model. How do I rotate it about another point $Q = (x_2,y_2) \neq(0,0)$ by a Euclidean angle $\alpha$? If $Q = (0,0)$ this is simple, just apply a normal Euclidean rotation, but I can't…
Lewy Blue
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What is the radius of the inscribed circle of an ideal triangle

I wanted to calculate the radius of the inscribed circle of an ideal triangle. and when i dat calculate it i came to $\ln( \sqrt {3}) \approx 0.54 $ (being arcos(sec (30^o)) but then at…
Willemien
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What are the most important issues to consider in upper-half plane model?

I hope you can help me, I need to do a research project about this model of hyperbolic geometry. Honestly, I've never studied the subject, and I'm not sure that subjects should give more importance. Although I considered two issues: 1) Möbius…
brbrbrbr
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Geodesic on hyperboloid and Poincare's Disk Model

I have two questions that 1. Why geodesic on hyperboloid corespond the arc in the Poincare's Disk Model? The hyperboloid : $x^2 + y^2 - z^2 = -1, \hspace{.15cm} z>0$ When any plane through origin cut the hyperboloid, the intersecting curve is a…
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