How to find the infinitesimal coordinate transform along a hyperbola?

[Sciencemaster]

TL;DR Summary

I've been told that the infinitesimal change in coordinates x and y takes the form δx=wy and δy=wx, and I was hoping someone could help me figure out why.

I've been told that the infinitesimal change in coordinates x and y as you rotate along a hyperbola that fits the equation b(dy)^2-a(dx)^2=r takes the form δx=bwy and δy=awx, where w is a function of the angle of rotation (I'm pretty sure it's something like sinh(theta) but it wasn't clarified for me so I'm not 100% sure). However, I'm not sure why this is the case, and I was hoping someone could show me how you get these infantesimal transformations and how they preserve the invariant r.

 

[fresh_42]

The last time I saw this point of view was when I wrote my insight article When Lie Groups Became Physics based on a book from 1911! I have tried to use rotation as an example (the equations in brackets [...]) so maybe it helps.