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Spherical basis change through euler angles

  1. Jun 27, 2012 #1
    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​
    Spherical coordinates: (rsss)
    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| ?
     
  2. jcsd
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