I know that general second degree curve, i.e. $ax^2 + by^2 + 2hxy + 2gx + 2fy + c=0$ gives us the equation of different cross sections of a cone.
Similarly, what does a third degree* curve actually represent? Different cross sections of another solid structure, if so then which one? And what does a forth degree? ..fifth?
$^* =ax^3+ by^3 + 2dx^2y + 2ey^2x + 2jx^2 + 2ky^2 + 2hxy + 2gx + 2fy+ c=0$