# General “Conics” of higher degrees?

A general conic has the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. I understand that there are certain properties of this equation that make it special and allow us to classify the different types of curves it can represent (ellipses, parabolas, and hyperbolas, distinguished by their eccentricity), all of which can be derived by slicing cones. But why does it stop there? Are there not interesting properties of an equation like

$$Ax^3 + Bx^2y^2 + Cy^3 + Dx^2y + Exy^2 + Fx^2 + Gxy + Hy^2 + Ix + Jy + K = 0$$

Or is there not a generalized equation for these curves such as $$C = \sum_{i = 1}^n(A_ix^i + B_ix^{i-1}y^{i-1} + C_iy^i + D_ix^{i-1}y^{i-2} + E_ix^{i-2}y^{i-1})$$

Basically my question is what is the name for curves of this general form? Are they important, do they have interesting properties, have they been studied thoroughly? I've never seen equations like these discussed before anywhere, so I was wondering if they're considered mathematically "important" like general conics are.

• Be reassured, the cubics (and also quartics and higher degree) have been well studied. en.wikipedia.org/wiki/Cubic_plane_curve But these topics are too much advanced and complicated for normal curricula. – Yves Daoust Aug 13 '14 at 12:24