Is $x^2-y^2=1$ Merely $\frac 1x$ Rotated -$45^\circ$? Comparing their graphs and definitions of hyperbolic angles seems to suggest so aside from the $\sqrt{2}$ factor:

and:


Almost. $$x^2-y^2=1\iff (x+y)(x-y)=1$$
By rotating by $-45^\circ$ you move the point $(x,y)$ to $(\hat x,\hat y) = \frac1{\sqrt 2}(x-y,x+y)$, so what you really get is $\hat y = \frac1{\hat x\sqrt 2}$
