How to build a hexagon according to Poincaré model?

Question

Given a side, I know how to build a hexagon in the euclidean geometry. How can i build it in the hyperbolic geometry according to the Poincaré model? By translating every step using hyperbolic circle it doesn't work. I use geogebra or cabrì.

Sorry for my English. Thanks for help.

Answer 1

Just the construction:

let r be the euclidean radius of the Poincare disk Let C be the centre of the Poincare disk

let P be one of the vertices of the hexagon

• draw circle p around C going through P

• calculate $o = \frac{2 r^2}{\sqrt{3} |PC|} $

• draw a circle q around C with distance o

• rotate the ray CP $30^{\circ} $ giving ray $q_1$

• let $Q_1$ be the intersection of $q_1$ and circle q

• draw the circle $s_1$ with centre $Q_1$ going trough $P$

Then 5 times:

• Let $P_n$ be the intersection of $s_{n-1} $ and $p$

• rotate the ray $CQ_{n-1} 60^{\circ} $ giving ray $q_n$

• let $Q_n$ be the intersection of $q_n$ and circle $q$

• draw the circle $s_n$ with centre $Q_n$ going trough $P_{n-1}$

now you have the 6 vertices of your hexagon, and the sides are those parts of circles $s_n$ between them

Good luck