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    $\begingroup$ You might find it helpful to simply use $\phi'=\arctan(p'_y/p'_x)$, which accounts for both the positive and negative case! Or, in mathematica, ArcTan[px, py], which accounts also for the $p'_x=0$ case. $\endgroup$ Commented Mar 27, 2021 at 0:53
  • $\begingroup$ @John Taylor You made a typo in formula for $\bf {v}_N$ in Latex. $\endgroup$ Commented Mar 27, 2021 at 12:59
  • $\begingroup$ @AlexTrounev : thanks! $\endgroup$ Commented Mar 27, 2021 at 14:19