Here are my questions, hope you can help in some way:

1. Let f:S^2->S^2 S^2={x^2+y^2+z^2=R^2|R is real} be a spherical isometry, prove that there exists an orthogonal transformation A in O(3) such that: f=A|S^2, i.e the restriction of A to the sphere is f.

2. for which real number, a, the transforamtion:

f(x,y)=((x+y)/(sqrt(2))+1,(x-y)/(sqrt(2))+a) for each (x,y) in R^2 would be a reflection.

for two ofcourse without (1,a), i.e g(x,y)=f(x,y)-(1,a) is a reflection, I think that for a=0 it's also a reflection, but don't know how to find the line with respect to which the reflection is invariant.

for one, I think intuitively a translation and glide reflection would be denied, cause they wouldn't conserve the length of two distinct points on the sphere (not sure how to show this), and from this that we can't translate by a constant I think it will be proven quite easily the statement.

Any hints?

Thanks in advance, by the way my previous username was DangerMan, but I don't seem to recall which email address I've given for the registration.