I need to parametrize the intersection of the sphere
$$ x^2+y^2+z^2=4R^2 $$
and the cylinder
$$ (x-2R+a)^2+y^2=a^2. $$
If $a=R$, then I learned that
$$ \alpha(t)=R(1+\cos2t)\,\partial_x+R\sin2t\,\partial_y+2R\sin t\,\partial_z\qquad t\in[0,2\pi) $$
parametrizes the hippopede. However, I need a hint on what substitution I could make for which variable in this more general scenario.
Thanks in advance!