## spherical basis change through euler angles

More exactly, I want to translate polar coordinates to spherical ones, knowing the euler angles that define the polar plane in the spherical basis.

Polar coordinates: (rpp)
it's actually the position of a satellite on its orbit
its position relative to the planet so φ is the elevation angle(latitude), not the polar angle
Euler angles: (α,β,γ)
the orbit parameters: longitude of ascending node, inclination, argument of periapsis
What I end up with (looks correct as far as I can tell) :
rs=rp
θs=α+atan2(u,v)
φs=atan2(|u,v|, sin(θp+γ)sin(β))
with:
u=cos(θp+γ)
v=sin(θp+γ)cos(β)

But the calculation of |u,v| (length) is killing my timings with the square root.
So my question is : is there a way to simplify those equations ? And specifically to get rid of the |u,v| ?
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