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Parametrizations of the 3-sphere

  1. Apr 25, 2013 #1
    i got the 2 following parametrizations :
    x=sin a sin b sin c
    y=sin a sin b cos c
    z=sin a cos b
    w=cos a

    And
    x=sin a sin b
    y=sin a cos b
    z=cos a sin c
    w=cos a cos c

    Are those really 2 parametrizations of the 3-sphere and are they equivalent ?
    Thanks.
     
  2. jcsd
  3. Apr 25, 2013 #2

    mfb

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    Just check x^2+y^2+z^2+w^2?

    What do you mean with "equivalent"?
     
  4. Apr 25, 2013 #3
    the check is ok. I mean there exist a diffeomorphism between them. But I couldn't find such a transformation between the two sets of parameters
     
  5. Apr 25, 2013 #4

    mfb

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    Well, if I rename the second parameters to d,e,f:

    x=sin a sin b sin c = sin d sin e
    y=sin a sin b cos c = sin d cos e
    z=sin a cos b = cos d sin f
    w=cos a = cos d cos f

    =>
    a=acos(cos d cos f)
    b=acos(cos d sin f / sin a) (where you can insert a here)
    c = [even longer expression where you can insert b and a]
     
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