How to motivate the definition of distance in the upper half plane model of hyperbolic geometry? The book I am using just throws out complicated looking formulas involving ln, with no motivation. I did find a source that provided a detailed derivation for https://mphitchman.com/geometry/section5-3.html for the Poincare disk, which in principle I guess I could translate to the upper half plane, but wondering if there is another explanation?
The metric is $ds^2=\frac{dx^2 + dy^2}{y^2}$ for the Poincare upper half plane model of hyperbolic geometry.
For example for Euclidean geometry on $\mathbb{R}^2$, we can motivate the definition $ds^2=dx^2+dy^2$ via the Pythagorean theorem.
Here is my motivation so far I'm following in the text "Geometry:A metric approach with models" by Millman and Parker. And so we've so far defined the "lines" in the upper half plane to be the (semi) vertical lines and the semicircles centered on the x-axis. And we've defined "ruler" for a line l as distance preserving bijections $f:l \to \mathbb{R}$. So if we want a bijection from the part of the vertical line say x=2 in the upper-half plane with $\mathbb{R}$, then by considering the inverse function $\mathbb{R} \to \mathbb{R}_{\geq 0}$ we need a bijective strictly function $\mathbb{R} \to \mathbb{R}_{\geq 0}$, and there are many such, but the exponentials form a reasonable.
There is a bijective map $\mathbb{R}^2 \to H$ via $(x, t) \mapsto (x, e^t)$, but the hyperbolic metric pulls back to $$e^{-2t} dx^2 + dy^2$$
But I don't know what to make of that.