# General Hippopede Parametrization

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.