It's an interesting fact to me that if you consider any curve in the standard $x-y$ plane obeying:
$$ Ax^2 + Bxy+ Cy^2 + Dx + Ey + F = 0$$
For a fixed choice real numbers $A,B,C,D,E,F$ that such a curve is geometrically similar to a curve yielded by the intersection of the plane in some particular orientation in $\mathbb{R}^3$, with a double-cone of some particular angle and orientation.
Can a similar statement be made say for the class of cubic curves:
$$ Ax^3 + By^3 + Cx^2y + Dxy^2 + Ex^2 + Fy^2 + Gxy + Hx+ Iy + J = 0$$
I.E. is there some parametrized family of surfaces so that an intersection of one such surface with a plane always results in a curve that is geometrically similar to one the cubic curves aforementioned.
And if so, what would be an example of such a family of surface?