11 July 2019 AD; St Benedict, St Pius I

Using the transformation of a complex plane into a half-plane we get a whole new set of angles and other parameters. The angles (I called them "Final Angles"), when added to the angles from before the transformation, give a spherical triangle (180+ degrees) and a spherical quadrangle (360+ degrees). Also when using pairs, triplets, and quadruplets from each set of the angles we get CP Violation Phase angles for Quarks and for Neutrinos.

It will be obvious in the main articles when I use concrete examples. Fortran calculates the complex angles in a mathematical way, i.e. angles in the first and second quadrant are positive, and angles in the third and fourth quadrant are negative. Those are the main angles which give all the mixing angles for subatomic particles. However, if another transformation is applied - where all the angles are getting positive values, the effects is that the main angles are being transformed into the positive real complex plane

The transformation is as follows:

first quadrant: theta final = theta

second quadrant: theta final = 180 degrees - absolute value |theta|

third quadrant: theta final = 180 degrees + absolute value |theta|

fourth quadrant: theta final = 360 degrees - absolute value |theta|

That's all. It will be very simple in the examples.

Three main articles follow:

1 - spherical geometry test (proving quantum numbers values)

2 - CP Violation Phase for Quarks (i.e angle 68.755 degrees)

3 - CP Violation Phase for Neutrinos (i.e. angle approx. 234 degrees)

The Link to very difficult article: CP violation