I'm looking for ideas on how to modify the Lemniscate of Bernoulli to include some asymmetry across one axis.
The parametric equations of the lemniscate are
$$x(t)=\frac {a\sqrt{2}\cos t }{\sin^2t+1} \qquad y(t)=\frac {a\sqrt{2}\cos t\sin t }{\sin^2 t+1}$$
The type of curve I'm trying to fit looks like this. The boundary of the figures defines the curve. It's symmetric across the $x$-axis, but across the $y$-axis that I need the asymmetry.
Perhaps, by defining some angle parameter, the amount of rotation of the contact points around the focus of the lobe where the tangent line touches the curve on either side could be determined and also another parameter to determine whether the lobes are bent to the left or right and by how much. I guess it's more like a bent or warped elliptic curve, the two lobes of the curve intersecting at a Riemann zero.