1
vote
0answers
45 views

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 $0<r<1$ ...
0
votes
1answer
46 views

Can I solve for the fractional volume of a hyperboloid?

This looks like a homework problem because it is. I'm stuck at the portion where I solve for fractional volumes. Suppose you are a part of a team designing a water tank in the shape of a hyperboloid. ...
0
votes
0answers
67 views

Prove the Lobachevsky-Bolyai formula for the Klein model

I want to prove that e^(-d) = tan(Π(d)/d) in the Beltrami-Klein Model for the angle of parallelism in correspondence to the distance d, where d is the klein distance d(AB) = (1/2)|ln((AB,PQ)). A hint ...
2
votes
1answer
44 views

Need help with finding length of sides and angles of a triangle in upper half plane model

The given points are $i, 3i, 1 + 2i$ I know that the distance for points on a vertical line can be found by using the formula $$\ln\left|\frac{y_2}{y_1}\right|$$ So the distance between points $ i$ ...
1
vote
0answers
210 views

Calculating hyperbolic distance between two points

I was looking for a formula to calculate the hyperbolic distance between two planes and came across this formula $$\ln\left(\csc b-\cot b\over \csc a - \cot a\right)$$ where $a$ and $b$ are the ...
0
votes
1answer
36 views

conformal mapping

Prove that application ζ^-1:B^2---> H^2 is conform. ζ^-1(x1,x2,x3)=(x1/1+x3 , x2/1+x3) B^2 is unit disc and H^2 is hyperbolic 2-space and conformal mapping preserve angles.
1
vote
1answer
49 views

Questions on two elements in a Fuchsian group which have at least one common fixed point

This is a homework question I am unable to solve. Let $A,B \in PSL(2,R)=Aut(H)$. Assume none of them are elliptic and they have : Case 1) one common fixed point at the boundary of $H$ (i.e. they ...
3
votes
1answer
185 views

Set of points equidistant from two points in hyperbolic space.

Given two points $p,q$ in the hyperbolic plane, show that the set of points equidistant from $p$ and $q$ is a hyperbolic line. I am unsure how to proceed with this question. Would it be easier to use ...
3
votes
2answers
194 views

Reflection in a hyperbolic line formula

Let $H$ denote the upper half-plane model of hyperbolic space. If $L$ is the hyperbolic line in $H$ given by a Euclidean semicircle with centre $a\in \mathbb{R}$ and radius $r >0$, show that ...
1
vote
1answer
33 views

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 ...