Deforming the Nephroid into the Cayley Sextic and Beyond

In $\mathbb{R}^2$, let $\Gamma$ be a fixed circle with centre $C$ and radius $r$,and $\ell$ a straight line at distance $d$ of $C$. We study the curve which is the envelope of the circles whose centre lies on $\Gamma$ and which are tangent to $\ell$. When $d=0$ this curve is a nephroid, when $d=3r/2$ it is a Cayley sextic.