Parametrization of a rotating surface

Question

What is the parametrization of a surface obtained by rotating the circle $(y − 3)^2 + z^2 = 1, x = 0$ about the z-axis.

I came up with the parametrization $S(r,θ) = (r , 3+cosθ , sinθ)$, is it correct?

Answer 1

Choose a half-plane that represents the rotation of the graph by $\phi$ from the $(+y)z$-halfplane. The horizontal direction on this half-plane is given by the unit vector $(\sin \phi, \cos \phi, 0)$. Call this the "s" axis. The intersection of this half-plane with the surface looks exactly like the same in all of the half-planes. In particular, like it does when $\phi = 0$ and $s = y$. That is, it has equation $(s-3)^2 + z^2 = 1$, which can be parametrized as $(s, z) = (3 + \cos \theta, sin \theta)$.

Can you figure it out from there?