Parametrizations of the 3-sphere

  • Thread starter jk22
  • Start date
  • #1
704
23
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.
 

Answers and Replies

  • #2
35,137
11,380
Just check x^2+y^2+z^2+w^2?

What do you mean with "equivalent"?
 
  • #3
704
23
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
 
  • #4
35,137
11,380
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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