I study Poincare model as hyperbolic model. But honestly, I don't understand with this topic. I try to find some examples but nothing. Here is my problem.
- Find the equation of the hyperbolic line (euclid circle) through $P(1/3,0)$ and $Q(0,1/3)$.
- Distance between $P$ and $Q$.
In my textbook, distance in Poincare is $\left\lvert\ln\dfrac{(PQ')(QP')}{(PP')(QQ')}\right\rvert$. But how to find $P'$ and $Q'$?
First, I use geogebra to solve it.
Draw a circle $C$ with $x^2+y^2=1$. Suppose $R$ and $S$ are inverse points $P,Q$ of circle $C$. By using inversion formula $|OP||PR|=1$ so $R=(3,0)$ and $S=(0,3)$. Now, suppose circle $D$ through $P,Q,R$. If $\ell$ is perpendicular bisector of $P,Q$ then $\ell$ pass the centre of circle $D$ in $(1.67,1.67)$. So the equation of the circle $D$ is $(x-1.67)^2+(y-1.67)^2=4.56$.
Intersection of two circles are $P'=(0.94,-0.34)$ and $Q'=(-0.34,0.94)$. Calculate $|PQ'|, |QP'|, |PP'|, |QQ'|$ by using distance formula in euclidean. Then
$PQ=\left\lvert\ln\dfrac{(PQ')(QP')}{(PP')(QQ')}\right\rvert=\left\lvert\ln\dfrac{(1.16)(1.16)}{(0.7)(0.7)}\right\rvert\approx 1.01019$.