How to determinate a toric section?

Question

I just found an a class of curves named toric section.
Essentially like conic sections but instead of a cone intersected by a plane, it is a torus.

They have this equation (except for rototranslations) $$\left(x^{2}+y^{2}\right)^{2}+ax^{2}+by^{2}+cx+dy+e=0$$ Some toric sections are:

1. Bernoulli lemniscate
2. Booth lemniscate
3. Cardioid
4. Anellus
5. Cassini ovals
6. Limacon
7. Ippoid

Is there a list of all toric section and how to determinate in by the equation?
Like for conic section, associating a matrix of coefficients.
Maybe it can be usefull considering this matrix $$\begin{pmatrix}1&0&0&0\\0&a&0&c\\0&0&b&d\\0&c&d&e\end{pmatrix}$$