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:


Write it in polar coordinates:
$$x^2 - y^2 = 1 \ \to \ r^2(\cos^2 \theta - \sin^2 \theta) r^2 = r^2\cos(2\theta) = 1$$
$$xy=1 \ \to \ r^2\sin\theta\cos\theta=\frac{1}{2}r^2\sin(2\theta) =1$$
Or:
$$\frac{1}{2}r^2\sin(2\theta)=\frac{1}{2}r^2\cos(2(\theta + 45^\circ))=\left(\frac{r}{\sqrt{2}}\right)^2\cos(2(\theta + 45^\circ))$$
So you see that $y=1/x$ is indeed a rotated copy, up to a multiplicative factor . 
