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Rotation in spherical coordinates

  1. Aug 27, 2013 #1
    Hi guys,

    This isn't really a homework problem but I just need a bit of help grasping rotations in spherical coordinates.

    My main question is,

    Is it possible to rotate a vector r about the y axis by an angle β if r is expressed in spherical coordinates and you don't want to convert r cartesian coordinates to do the rotation (just using 1 rotation matrix?

    I've been searching the web for a while in hope for a simple explanation and a straight forward rotation matrix so I can play with it / test it and the only things I have found are based in cartesian coordinates.

    Of course the y axis rotation is just an example for me to learn from but in this case.

    I know that in cartesian coordinates the rotation matrix will be:

    cos β.....0.....-sin β
    0..........1.......0
    sin β.....0......cos β

    But you can only apply this to vectors which are represented in cartesian coordinates right? How can I write this rotation matrix in spherical coordinates so I can applying to a vector writing in sphereical coordinates?

    Would anyone be able to give me a hand?

    Edit:
    The correct thing to do would be this right?


    [Broken]

    What I'm trying to ask is if there is an easier way?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Aug 28, 2013 #2

    haruspex

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    Maybe I'm being dumb, but I don't understand what the vector on the right hand end of that equation represents. Is it supposed to be a position vector expressed in spherical coordinates? You can't write conversion to Cartesian in matrix form since it is not a linear transformation.
    To get Cartesian coordinates from applying the rightmost matrix, the vector it's applied to would consist of (r, r, r)T.
     
    Last edited by a moderator: May 6, 2017
  4. Aug 28, 2013 #3

    I like Serena

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    Hi linda! :smile:

    The vector on the right hand would represent a vector in a local cartesian system. A system that is defined for a specific point in spherical coordinates. It changes when that point changes.

    That transformation looks correct.
    It would transform a localized vector to another localized vector after a rotation by an angle beta with respect to the y axis through the point where the vector is localized.
    I do not see a way to significantly simplify that.

    In advanced computer graphics, a typical way to simplify it (or rather reduce the amount of memory and number of operations to evaluate it), is by doing the same thing using octonions.
    But I suspect that is not what you're looking for.
     
  5. Aug 28, 2013 #4

    haruspex

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    More thoughts...
    You could do it very easily if you had a way to transpose the Cartesian coordinates while working only in polar form. Suppose you want to swap x and z. The new polar angles are θ', ϕ'.
    sin θ cos ϕ = cos θ'
    sin θ sin ϕ = sin θ' sin ϕ'
    (whence tan ϕ = tan θ' sin ϕ')
    cos θ = sin θ' cos ϕ'
    So it's easy to find θ and ϕ from θ' and ϕ'.
     
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