$\mathbb{H}^2$" />I've been trying to self-study differential geometry lately, but I'm struggling with some of the ideas of hyperbolic space. I came across a problem here, to where the points P and Q are $(1,1)$ and $(2, \sqrt{2})$ respectively, and I'm to find the hyperbolic distance between them by the following steps:
i.) I should find the coordinates of the endpoints of the purple geodesic $\gamma$ on which the points P and Q lie
ii.) It's given that when we invert the green semi-circle C, whose center is at the right-hand endpoint of $\gamma$ and whose radius is twice that of $\gamma$, we get the vertical blue line with which the points P' and Q' reside. I should then calculate the coordinates of the images P' of P and Q' of Q on the line using similar triangles, and lastly
iii.) Calculate the exact hyperbolic distance between P' and Q', and deduce the hyperbolic distance between P and Q justly.
I'm familiar with the workings of Mobius transformations and conformal mappings in $\mathbb{C}$, but here I'm not sure how to envision things, and am not seeing a clear way to begin.
Where do you get stuck?
Step i) is finding the semicircle whose endpoints are on the $x$-axis that contains the points $P$ and $Q$. This is basically an exercise in basic algebra. It suffices to find the center of the full circle, which is the point on the $x$-axis whose distances to $P$ and to $Q$ are equal and therefore equal to the radius.
To do Step ii), recall or figure how the distance from a point inside the large semicircle to the semicircle is related to the distance from the inverted point to the semicircle. You could also try to find the formula for the Mobius transformation corresponding to the inversion. But if you take this approach, I suggest first sliding everything horizontally so that the center of the larger semicircle is at the origin.
Step iii) uses the fact that the vertical line is a geodesic and the formula for the hyperbolic metric.