Given $P=1+2i$ and $Q=3+4i$, find the equation of the perpendicular bisector of the hyperbolic line segment $[P, Q].$
I used the approach given in Groups and Geometry by Lyndon where you get the set of points $d(P, X) = d(Q, X)$. Using $X = u + iv$ results in the set of points $\{(u,v)|u^2+v^2-10u-12v+45=0\}.$ How can I express this as an equation?
This is the way I do it: use a hyperbolic rigid transformation to map one of your points to $i$ and the other to $ri$, with $r>0$. Then the midpoint of the transformed segment is $r^{1/2}i$, and the perp. bisector is the circle centered at the origin of radius $r^{1/2}$. Now apply the inverse fractional linear transformation. Needless to say, this is hardly the most efficient way of doing what you want, but at least it’s clear that it’s getting you the right result.