Poincare disc model problem. find $d_h(A,B)$

Question

Consider $\triangle ABC$ on a poincare disc.

On $\triangle ABC$,

$\angle C=90^\circ$, $d_h(B,C)=a$ and $d_h(A,C)=b$

In this situation, find $d_h(A,B)$.

I'm taking a course but I cannot follow the class. Is there someone to explain in detail?

Answer 1

The hyperbolic version of the Pythagorean theorem goes like this:

$$\cosh c = \cosh a \cosh b$$

$a$ and $b$ are the legs, $c$ the hypothenuse.

This formula is a special case of the hyperbolic law of cosines.

If you square both sides, this looks almost like the Euclidean version, except that you have multiplication instead of addition, and of course those $\cosh$ functions. This might help remembering the form.