First of all, I take the surface patch for sphere
$$\sigma(u,v)=(\sin u\cos v, \sin u\sin v, \cos u)$$
And then I calculated its second fundametal form. And I got the following result
$$ -du^2-\sin^2 u \ dv^2$$
Second of all, I take different surface patch for the sphere.
$$\sigma(u,v)=\cos u\sin v, \sin u\sin v, \cos v)$$
And again I calculated the second fundamental form for it.
I obtained that $$ \sin^2 v du^2+ dv^2$$
The second fundamental forms are different since the surface patches are different. However, my instructor says that although the results are different, in fact, they are not completely different.
I think its reason. But I cannot find. Please explains why they are not completely different? Thanks a lot.