Parametrizations of the 3-sphere

[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.

 

[mfb]

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

What do you mean with "equivalent"?

 

[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

 

[mfb]

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]