The rational parametrization looks like the Cayley transform, which IIRC does not cover the whole circle (you can't reach (-1, 0) with a finite mu). Does this happen here as well?
This is a stereographic projection representation of an angle. The number should be seen as a point on the real projective line, which includes a point at infinity.
If you’re using floating point, you can use the number ∞ for this. You’ll need some special cases in the code for it.
