I am aware that one can have different parameterizations on $SU(2)$, e.g. in terms of hyperspherical polar coordinates. These are also geodesic coordinates.
One can also find parameterizations in terms of Euler angles.
However, I have difficulties to find clearly defined mappings in between the two coordinate systems in the literature.
So the question would be: How do I transfer from one to the other coordinate system?
Thanks.