I'm referencing some material from Hyperbolic coordinates.
Consider the Half Plane (HP) $$HP=\big\lbrace(p,q):p\in \Bbb R, q>0\big \rbrace$$
and the Quadrant (Q)
$$Q= \big\lbrace(x,y):x<0, y<0\big \rbrace$$
I've checked my understanding that the coordinates are related through $p=\ln \sqrt{\frac{x}{y}}$ and $q=\sqrt{xy}$ with the inverse mapping given by $x=-qe^{p}$ and $y=-qe^{-p}$.
Consider the Unit Square (US)
$$US=\big \lbrace (r,u): r,u \in (0,1) \big \rbrace.$$
I've deduced that $r=e^x$ and $u=e^y$ with the inverse mapping given by $x=\log r$ and $y= \log u.$
I want to relate HP to US because I eventually want a model of Hyperbolic geometry on US. If my thinking is correct, US should be some image of the hyperbolic plane. If one wants distance preserving map then one's looking for an isometry. Anyway, I did $r=e^{-{qe^p}}$ and $u=e^{-q{e^{-p}}}.$ Then I arrived at $\ln \sqrt{\frac{r}{u}}= -q\cosh(p).$
I'm a little stuck at this point.