Comment by srean

That's right. However, one disadvantage of using the tan value (the y/x ratio) over that of (cos,sin) tuple is that the former loses information on whether the y coordinates or the x coordinate was negative.

So, if you use the tan representation you have to carry that information separately. Furthermore, the code needs to correctly handle zero and infinity.

Tan of the half angle takes care of the first problem and is related to the stereographic transform. This works modulo one full rotation.