Background:
Many math libraries provide hypot(y,x) and atan2(y,x) as numerically robust routines for converting Cartesian coordinates to polar coordinates, with hypot providing the magnitude and atan2 providing the angle. The two parameter version of the inverse tangent, having complete knowledge of the quadrant from which the point originated, can recover any angle between -pi and pi, whereas atan(y/x) loses that information and can only represent half of that range.
Question:
Is there any analog to atan2 for computing the hyperbolic angle from Cartesian coordinates, or do hyperbolas sidestep the issues that needed to be addressed for circles? More generally, what is the canonical polar decomposition of a split-complex number - is information about the quadrant where the point originated maintained, or is the conversion to polar form intrinsically lossy?
If we introduce real coordinates $\rho,\,phi$ with $x=\rho\cosh\phi,\,y=\rho\sinh\phi$ so that $y=x\tanh\phi$, the sign of $x$ determines that of $\rho$, so the situation isn't analogous to ordinary polar coordinates needing $\operatorname{atan2}$. There's a further difference: you can only describe points with $|x|>|y|$.