1
vote
0answers
59 views

The distance between two distinct points in the upper half plane

I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization $\sigma(t): x=r\cos(t), y=r\sin(t),\; ...
4
votes
0answers
200 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
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 ...
4
votes
1answer
491 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
2
votes
1answer
56 views

hyperbolic trigonometric relation

Let $F$ be a hyperbolic once-punctured torus, and $G=\pi_1(F)$. Fix a discrete, faithful representation $\rho\colon G\to\mathbb{P}SL(2,\mathbb{R})$ and an element $g\in G$ corresponding to a ...
5
votes
1answer
501 views

Generalized Laws of Cosines and Sines

I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources) (i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
4
votes
6answers
225 views

Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?

I can faintly visualize some way of deducing this formula with exponential functions but forgot it. How do you remember it? Suppose you just forget whether it is plus-or-minus there, how do you find ...