1. Apr 25, 2013 #1

   jk22

   

   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.

    

   jk22, Apr 25, 2013
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3. Apr 25, 2013 #2

   mfb

   

   Insights Author

   2016 Award

   Staff: Mentor

   Just check x^2+y^2+z^2+w^2?
   
   What do you mean with "equivalent"?

    

   mfb, Apr 25, 2013
4. Apr 25, 2013 #3

   jk22

   

   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

    

   jk22, Apr 25, 2013
5. Apr 25, 2013 #4

   mfb

   

   Insights Author

   2016 Award

   Staff: Mentor

   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]

    

   mfb, Apr 25, 2013

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