let $C =\{ (1+\frac12 \cos \phi,0,\frac12 \sin \phi) \; | \; \phi \in \mathbb{R}\}$
in some old lecture notes of geometry it is stated that :
$f(\mathbb{R^2}) = \bigcup_{\theta \in \mathbb{R}} R_{\theta}(C)$
where $f : \mathbb{\mathbb{R^2 \to R^3}}$ is defined by $f(\theta,\phi) = (\cos \theta(1+\frac12\cos \phi),\sin \theta(1+\frac12\cos \phi),\frac12 \sin \phi)$
and $R_{\theta}$ is the rotation of angle $\theta$ around the $z-$axis.
I'm a bit lost here, as I don't know how to explicitly write down a formula for $R_{\theta}$.
can someone clarify this ?