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:


Converting
$$
x^2-y^2=1
$$
to polar coordinates with $x=r\cos(\theta)$ and $y=r\sin(\theta)$, and substituting $\theta=\phi-\frac\pi4$ :
$$
\begin{align}
1
&=r^2\left(\cos^2(\theta)-\sin^2(\theta)\right)\\
&=r^2\cos(2\theta)\\
&=r^2\cos(2\phi-\tfrac\pi2)\\
&=r^2\sin(2\phi)\\
&=2r^2\sin(\phi)\cos(\phi)\\
&=2uv
\end{align}
$$
where $u=r\cos(\phi)$ and $v=r\sin(\phi)$. Therefore, rotating $x^2-y^2=1$ by $45^\circ$ gives $uv=\frac12$.
