  We study two classes of quantum spheres and hyperboloids which are
$*$-quantum spaces for the quantum orthogonal group $\mathcal{O}(SO_q(3))$. We
construct line bundles over the quantum homogeneous space of invariant elements
for the quantum subgroup $SO(2)$ of $SO_q(3)$. These are associated to the
quantum principal bundle via corepresentations of $SO(2)$ and are given by
finitely-generated projective modules $\mathcal{E}_n$ of rank $1$ and even
degree $-2n$. The corresponding idempotents, representing classes in K-theory,
are explicitly worked out. For $q$ real, we diagonalise the Casimir operator of
the Hopf algebra ${\mathcal{U}_{q^{1/2}}(sl_2)}$ dual to
$\mathcal{O}(SO_q(3))$.
