1

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?

hbghlyj
  • 2,115
hyunst
  • 69
  • Why don't you make a Google query with "perpendicular bisector of a hyperbolic line segment " ? This is the way I have obtained at once: (http://mat.uab.es/~juditab/mediatriuA.htm) with maybe a precision if you work on the half plane or the disk model... – Jean Marie Apr 06 '17 at 15:01
  • 1
    Yes, I am working on the upper half plane model. Based on the link you showed me, the midpoint of this hyperbolic line would be the same as taking [P,Q] to be a euclidean line? – hyunst Apr 06 '17 at 15:18
  • Please incorporate that clarification into your Question (edit). If you work out the equation of the "hyperbolic line" through $P,Q$ and take the intersection with the equation for points equidistant from $P$ and $Q$, you should have your answer. – hardmath Apr 07 '17 at 17:53

1 Answers1

2

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.

Lubin
  • 62,818
  • Is not hyperbolic distance between $i$ and $ri$, $ln(\sqrt{r})$? – Tedebbur Jan 04 '20 at 18:01
  • I guess that may be right, @Serpenche, but it’s been quite a while since I’ve thought about these things: $\frac12\ln r$. – Lubin Jan 04 '20 at 20:40
  • Sorry, I should have said differently, if $r>1$ the distance is $ln(r)$, right? – Tedebbur Jan 05 '20 at 01:42
  • I guess in all cases you take the absolute value of the log, @Serpenche. – Lubin Jan 05 '20 at 05:00
  • Do you know how to construct Fundamental Domain for $SL(2, \mathbb{Z})$? I know this is irrelevant here, but I posted my question and could not receive any answer and I have an exam in a few days. – Tedebbur Jan 05 '20 at 07:59
  • I hesitate, ’cause I’m a little rusty on this. Isn’t it the region above the unit circle and between the lines $x=\pm\frac12$? – Lubin Jan 05 '20 at 14:54